Triangulated categories of mixed motives

Denis-Char les Cisinski, Frédér ic Déglise T r iangulated categor ies of mix ed motiv es June 28, 2019 Spr ing er N ature vi Abstract This book discusses the constr uction of triangulated categor ies of mixed motiv es ov er a noether ian scheme of ﬁnite dimension, e xtending V oe v odsky’ s def- inition of motives o v er a ﬁeld. In par ticular , it is sho wn that motives with rational coeﬃcients satisfy the f ormalism of the six operations of Grothendieck. This is achie v ed b y s tudying descent properties of motives, as well as by comparing diﬀer - ent presentations of these categories, follo wing and e xtending insights and cons truc- tions of Deligne, Beilinson, Bloch, Thomason, Gabber , Levine, Morel, V oev odsky , A y oub, Spitzwec k, Röndigs, Østv ær and others. In par ticular , the relation of mo- tiv es with K -theory is addre ssed in full, and we prov e the absolute pur ity theorem with rational coeﬃcients, using Quillen’ s localization theorem in algebraic K -theory together with a variation on the Grothendieck -Riemann-R och theorem. Using res- olution of singular ities via alterations of de Jong-Gabber , this leads to a version of Grothendieck - V erdier duality for constr uctible motivic sheav es with rational co- eﬃcients o v er rather general base schemes. W e also study v ersions with integral coeﬃcients, constructed via shea v es with transf ers, f or which we obtain par tial re- sults. Finally , w e associate to any mix ed W eil cohomolo gy a sy stem of categories of coeﬃcients and well behav ed realization functors, establishing a correspondence betw een mixed W eil cohomologies and suitable sys tems of coeﬃcients. The results of this book hav e already served as ground reference in many subsequent works on motivic sheav es and their realizations, and pointers to the most recent de velopments of the theor y are giv en in the introduction. Contents Introduction A Historical background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii A.1 The conjectural theor y described by Beilinson . . . . . . . . . . . . xiii A.2 V oev odsky’ s motivic comple x es . . . . . . . . . . . . . . . . . . . . . . . . xiv A.3 Morel and V oev odsky’ s homotop y theor y . . . . . . . . . . . . . . . . xv A.4 V oev odsky’ s cross functors and A youb’ s thesis . . . . . . . . . . . . xvi A.5 Grothendiec k six functors f or malism . . . . . . . . . . . . . . . . . . . . xvii B V oe v odsky’ s motivic comple xes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xx C Beilinson motiv es . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxii C.1 Deﬁnition and fundamental proper ties . . . . . . . . . . . . . . . . . . . xxii C.2 Constr uctible Beilinson motiv es . . . . . . . . . . . . . . . . . . . . . . . . xxiv C.3 Comparison theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxv C.4 Realizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxviii D Detailed org anization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxx D.1 Grothendieck six functors f or malism (Part 1) . . . . . . . . . . . . . xxx D.2 The constructiv e par t (Part 2) . . . . . . . . . . . . . . . . . . . . . . . . . . xxxiv D.3 Motivic comple xes (Part 3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxxv D.4 Beilinson motives (Part 4) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxxviii E De velopments since the ﬁrst arXiv version . . . . . . . . . . . . . . . . . . . . . . xxxix E.1 Nisnevic h motiv es with integral coeﬃcients . . . . . . . . . . . . . . xxxix E.2 Étale motiv es with integral coeﬃcients and ` -adic realization xl E.3 Motivic stable homotop y theor y with rational coeﬃcients . . xli E.4 Duality , weights and traces . . . . . . . . . . . . . . . . . . . . . . . . . . . . xlii E.5 Enriched realizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xlii Thanks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xliii Notations and conv entions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xliii Part I Fibred categories and the six functors formalism 1 General deﬁnitions and axiomatic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 vii viii Contents 1.1 P -ﬁbred categor ies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.1.a Deﬁnitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.1.b Monoidal structures . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.1.c Geometric sections . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 1.1.d T wists . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 1.2 Mor phisms of P -ﬁbred categor ies . . . . . . . . . . . . . . . . . . . . . 17 1.2.a General case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 1.2.b Monoidal case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 1.3 Structures on P -ﬁbred categories . . . . . . . . . . . . . . . . . . . . . . 20 1.3.a Abs tract deﬁnition . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 1.3.b The abelian case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 1.3.c The triangulated case . . . . . . . . . . . . . . . . . . . . . . . . . 23 1.3.d The model categor y case . . . . . . . . . . . . . . . . . . . . . . 25 1.4 Premotivic categor ies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 2 T r iangulated P -ﬁbred categories in algebraic geometry . . . . . . . . . . 31 2.1 Elementar y proper ties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 2.2 Ex ceptional functors, follo wing Deligne . . . . . . . . . . . . . . . . . 35 2.2.a The suppor t axiom . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 2.2.b Ex ceptional direct image . . . . . . . . . . . . . . . . . . . . . . 37 2.2.c F ur ther properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 2.3 The localization proper ty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 2.3.a Deﬁnition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 2.3.b F irst consequences of localization . . . . . . . . . . . . . . 47 2.3.c Localization and ex chang e proper ties . . . . . . . . . . . 49 2.3.d Localization and monoidal structure . . . . . . . . . . . . 52 2.4 Pur ity and the theorem of V oev odsky -R öndigs-A y oub . . . . . . 56 2.4.a The stability proper ty . . . . . . . . . . . . . . . . . . . . . . . . . 56 2.4.b The pur ity proper ty . . . . . . . . . . . . . . . . . . . . . . . . . . 61 2.4.c Duality , pur ity and or ientation . . . . . . . . . . . . . . . . . 67 2.4.d Motivic categor ies . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 3 Descent in P -ﬁbred model categor ies . . . . . . . . . . . . . . . . . . . . . . . . . 81 3.1 Extension of P -ﬁbred categories to diag rams . . . . . . . . . . . . 81 3.1.a The general case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 3.1.b The model categor y case . . . . . . . . . . . . . . . . . . . . . . 84 3.2 Hyperco v ers, descent, and der iv ed global sections . . . . . . . . . 97 3.3 Descent ov er schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 3.3.a Localization and Nisnevic h descent . . . . . . . . . . . . . 108 3.3.b Proper base chang e isomor phism and descent by blo w-ups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 3.3.c Proper descent with rational coeﬃcients I: Galois e x cision . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 3.3.d Proper descent with rational coeﬃcients II: separation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 4 Constructible motiv es . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 4.1 Resolution of singularities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 Contents ix 4.2 Finiteness theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 4.3 Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 4.4 Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 Part II Construction of ﬁbred categories 5 Fibred der iv ed categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 5.1 From abelian premotives to triangulated premotives . . . . . . . 167 5.1.a A belian premotiv es: recall and ex amples . . . . . . . . . 167 5.1.b The t -descent model categor y structure . . . . . . . . . . 169 5.1.c Cons tructible premotivic comple xes . . . . . . . . . . . . 179 5.2 The A 1 -derived premotivic category . . . . . . . . . . . . . . . . . . . . 184 5.2.a Localization of triangulated premotivic categor ies . 184 5.2.b The homotopy relation . . . . . . . . . . . . . . . . . . . . . . . . 190 5.2.c Explicit A 1 -resolution . . . . . . . . . . . . . . . . . . . . . . . . 195 5.2.d Constructible A 1 -local premotiv es . . . . . . . . . . . . . . 199 5.3 The stable A 1 -derived premotivic category . . . . . . . . . . . . . . . 201 5.3.a Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201 5.3.b Symmetr ic sequences . . . . . . . . . . . . . . . . . . . . . . . . . 202 5.3.c Symmetr ic T ate spectra . . . . . . . . . . . . . . . . . . . . . . . 205 5.3.d Symmetr ic T ate Ω -spectra . . . . . . . . . . . . . . . . . . . . . 207 5.3.e Constructible premotivic spectra . . . . . . . . . . . . . . . 214 6 Localization and the univ ersal der iv ed e xample . . . . . . . . . . . . . . . . . . 216 6.1 Generalized der iv ed premotivic categories . . . . . . . . . . . . . . . 216 6.2 The fundamental e xample . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220 6.3 Near ly Nisne vich shea v es . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221 6.3.a Support proper ty (eﬀectiv e case) . . . . . . . . . . . . . . . 222 6.3.b Support proper ty (stable case) . . . . . . . . . . . . . . . . . 225 6.3.c Localization for smooth schemes . . . . . . . . . . . . . . . 226 7 Basic homotop y commutative algebra . . . . . . . . . . . . . . . . . . . . . . . . . . 227 7.1 Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227 7.2 Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233 Part III Motivic complex es and relativ e cy cles 8 Relativ e cycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243 8.1 Deﬁnitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243 8.1.a Category of cy cles . . . . . . . . . . . . . . . . . . . . . . . . . . . 243 8.1.b Hilbert cy cles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245 8.1.c Specialization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248 8.1.d Pullback . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252 8.2 Intersection theoretic proper ties . . . . . . . . . . . . . . . . . . . . . . . . 259 8.2.a Commutativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259 8.2.b Associativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260 8.2.c Projection f or mulas . . . . . . . . . . . . . . . . . . . . . . . . . . 262 x Contents 8.3 Geometric properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264 8.3.a Constructibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264 8.3.b Samuel multiplicities . . . . . . . . . . . . . . . . . . . . . . . . . 268 9 Finite cor respondences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274 9.1 Deﬁnition and composition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275 9.2 Monoidal structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280 9.3 Functoriality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281 9.3.a Base chang e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281 9.3.b R estriction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283 9.3.c A ﬁniteness property . . . . . . . . . . . . . . . . . . . . . . . . . 284 9.4 The ﬁbred categor y of cor respondences . . . . . . . . . . . . . . . . . 284 10 Shea v es with transfers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286 10.1 Presheav es with transfers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286 10.2 Sheav es with transf ers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288 10.3 Associated sheaf with transfers . . . . . . . . . . . . . . . . . . . . . . . . . 289 10.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298 10.5 Comparison results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 300 10.5.a Change of coeﬃcients . . . . . . . . . . . . . . . . . . . . . . . . 300 10.5.b R epresentable qfh -shea v es . . . . . . . . . . . . . . . . . . . . 300 10.5.c qfh -shea ves and transfers . . . . . . . . . . . . . . . . . . . . . . 301 11 Motivic comple x es . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303 11.1 Deﬁnition and basic proper ties . . . . . . . . . . . . . . . . . . . . . . . . . 304 11.1.a Premotivic categories . . . . . . . . . . . . . . . . . . . . . . . . . 304 11.1.b Constructible and geometric motiv es . . . . . . . . . . . . 306 11.1.c Enlarg ement, descent and continuity . . . . . . . . . . . . 307 11.2 Motivic cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 310 11.2.a Deﬁnition and functoriality . . . . . . . . . . . . . . . . . . . . 310 11.2.b Eﬀective motivic cohomology in weight 0 and 1 . . 313 11.2.c The motivic cohomology r ing spectrum . . . . . . . . . 317 11.3 Or ientation and pur ity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319 11.4 The six functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323 Part IV Beilinson motives and algebraic K -theory 12 Stable homotopy theory of schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . 331 12.1 Ring spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331 12.2 Orientation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331 12.3 Rational categor y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334 13 Alg ebraic K -theor y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334 13.1 The K -theor y spectr um . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334 13.2 Periodicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335 13.3 Modules o v er algebraic K -theor y . . . . . . . . . . . . . . . . . . . . . . . 335 13.4 K -theor y with suppor t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337 13.5 Fundamental class . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 338 13.6 Absolute purity f or K -theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 340 Contents xi 13.7 T race maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342 14 Beilinson motiv es . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345 14.1 The γ -ﬁltration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345 14.2 Deﬁnition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 348 14.3 Motivic proper descent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353 14.4 Motivic absolute pur ity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354 15 Constructible Beilinson motiv es . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 356 15.1 Deﬁnition and basic proper ties . . . . . . . . . . . . . . . . . . . . . . . . . 356 15.2 Grothendiec k 6 functors formalism and duality . . . . . . . . . . . 357 16 Compar ison theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359 16.1 Comparison with V oev odsky motives . . . . . . . . . . . . . . . . . . . 359 16.2 Compar ison with Morel motiv es . . . . . . . . . . . . . . . . . . . . . . . 362 17 Realizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371 17.1 T ilting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371 17.2 Mixed W eil cohomologies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 376 Ref erences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 387 Inde x . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 401 Inde x of proper ties of P -ﬁbred tr iangulated categor ies . . . . . . . . . . . . . . . . 402 Introduction A Historical backgr ound A.1 The conjectural theory described b y Beilinson In a landmark paper , [ Be ˘ ı87 ], A. Beilinson s tated a series of conjectures which oﬀers a complete renew al of the traditional theory of pure motiv es inv ented b y A. Grothendieck. Namel y , he proposes to extend the notion of pure motives to that of mixed motives with two models in mind: mix ed Hodg e str uctures deﬁned b y P . Deligne on the one hand [ Del71 , Del74 ], perverse sheav es on the other hand deﬁned in [ BBD82 ]. One of Beilinson ’ s main innov ations, motivated b y the second model, is to consider a triangulated v ersion of mixed motives in which one could hope to ﬁnd the more in v olv ed theor y of abelian mix ed motiv es through the concept of t-s tr uctures. This hoped f or theory was conjecturally descr ibed b y Beilinson in [ Be ˘ ı87 , 5.10] under the name of motivic complexes . It was modeled (see loc. cit. , paragraph A) on the theor y of étale ` -adic sheav es and their der iv ed category as introduced ﬁfty years ago b y Grothendieck and M. Ar tin. The major achiev ement of Grothendieck and his collaborators in [ A GV73 ] was to deﬁne a theory of coeﬃcients sys tems relative to any scheme with a collec- tion of operations, f ∗ , f ∗ , f ! , f ! , ⊗ , Hom , satisfying a set of formulas now called the Gro thendieck six functors formalism (see section A.5 in this introduction f or more details) 1 . This f or malism, f ormulated in the language of triangulated categor ies, ultimately encodes a very g eneral duality theory . Note ho we ver that the complete duality theor y f or ` -adic sheav es w as completed only recently by the w ork of Gabber [ ILO14 ]. The theor y w as also conjectured to be deeply linked with Quillen algebraic K - theory (see [ Be ˘ ı87 , 5.10, §B]). In fact, up to torsion and f or a regular scheme S , the 1 The full der iv ed formalism of ` -adic comple xes was fully established much later after [ A GV73 ] though, b y Ekedahl in [ Eke90 ]. xiii xiv Introduction e xt-groups between tw o T ate motiv es ov er S should coincide with Adams graded parts of Quillen algebraic K -theor y . 2 The ideas of Beilinson were v er y f ecund and not long after the publication of [ Be ˘ ı87 ], three candidates for a tr iangulated categor y of mix ed motiv es were proposed, respectiv ely by M. Hanamura [ Han95 , Han04 , Han99 ], M. Le vine [ Le v98 ], and V . V oev odsky [ V oe96 , V oe98 , VSF00 ]. As a matter of fact, V oev odsky introduced tw o variants: using the h -topology (obtained b y allo wing proper surjective maps as co v erings together with Zar iski co verings), he deﬁned a candidate f or a theory of étale motivic sheav es [ V oe96 ]. Inspired by his kno w ledg e of these and b y his work on r igidity results with Suslin [ S V96 ], he introduced a more Zariski local version [ V SF00 ] which is the one ﬁtting in his approach to the proof of the Milnor conjecture and of the Bloch-Kato conjecture. In this book, w e will f ocus on V oe v odsky’ s theories. A.2 V oe vodsky’ s motivic complex es As br ieﬂy alluded to abo v e, the ﬁrst attempt of V oev odsky in deﬁning the categor y of motivic comple xes, in his 1992 Harvard’ s thesis, introduces the fundamental process of A 1 -localization, which amounts to make the aﬃne line contractible in the category of mixed motives, by analogy with the topological case. It also in vol ves the use of the h -topology which was to become fundamental in the area of motives and cohomology . These tw o ingredients giv en, V oev odsky deﬁned the triangulated category of (eﬀectiv e) h -motiv es o v er any base in [ V oe96 ]. Ho w ev er , V oev odsky was a ware that his deﬁnition will give the correct an- sw er to Beilinson ’ s conjectural construction only with rational coeﬃcients (he was a ware that the torsion par t of the theory of h -motiv es would be closely related to Grothendiec k’ sétale cohomology 3 ). In [ V SF00 , c hap. 5], he introduces another deﬁnition of motivic complex es o ver a perfect ﬁeld with integral coeﬃcients, still using the A 1 -localization process but, this time, introducing the notion of Nisnevich shea v es with transfers and their derived categor y (see [ MVW06 ] for a detailed e x- position). At this stag e, all the proper ties f oreseen b y Beilinson are established f or this integ ral categor y ov er a per f ect ﬁeld, e xcept f or the construction of the motivic t -str ucture. 4 It remained to extend this deﬁnition to arbitrar y bases and to establish the Grothendieck six functors f ormalism. The path in this direction was laid down by V oe v odsky in [ V oe10a ] where he uses the theory of relative cycles introduced by Suslin and V oev odsky in [ VSF00 ] to extend the deﬁnition of sheaf with transfers. This deﬁnition was also exploited by 2 See page xxiii belo w f or the precise statement. 3 This is made precise by Suslin and V oev odsky [ S V96 ] o v er a ﬁeld, and we dev elopped this idea in full generality in [ CD16 ], using the main results and constructions of the present book. 4 This hoped for t -structure is described in [ V oe92 , Hyp. 0.0.21]. Moreov er, V oev odsky prov ed in [ V SF00 ] that such a t -structure does not e xist with integ ral coeﬃcients; how ev er , it should e xist with rational coeﬃcients, and, more generall y , for h -motives with integ ral coeﬃcients. A Historical background xv Iv or ra in [ Ivo07 ] to e xtend V oev odsky’ s constr uction of geometric motivic complex es o v er any base, av oiding the use of sheav es with transfers. Ne vertheless, constructing the Grothendieck six functors formalism for this deﬁnition remained untouched at this point. A.3 Morel and V oev odsky’s homotop y theory Soon after the introduction of V oe v odsky’ s motivic complex es, F . Morel and V o- ev odsky introduced the more general theor y of A 1 -homotop y of schemes [ MV99 ] whose design is to extend the framew ork of algebraic topology to algebraic g eome- try and is built around the A 1 -localization tool. It is within this theor y that another important tool in motivic homotopy theor y was introduced: the P 1 -stabilization pro- cess 5 . From the purel y motivic point of view , this amounts to in v er t the T ate motiv e Z ( 1 ) for the tensor product. From the homotopical point of view , this operation is much more inv ol ved and re v eals the theory of spectra, objects which incar nate coho- mology theories in algebraic topology . These two processes, of A 1 -localization and P 1 -stabilization, applied to the category of simplicial Nisnevich sheav es, led to the stable A 1 -homotop y category of schemes (see [ Jar00 ], or the last chapter of [ Jar15 ], f or instance, or [ R ob15 , Ho y17 ] f or more moder n approaches) a triangulated categor y with integral coeﬃcients, deﬁned o ver any base, which generalizes the categor y of motivic comple xes. 6 Ov er a perfect ﬁeld, and with rational coeﬃcients, the relation betw een A 1 - homotop y in variant shea ves and motiv es w as clariﬁed in an unpublished paper of Morel [ Mor06 ] (with precise statements but without proofs): the rational stable A 1 -homotop y category contains the stable ( i.e. P 1 -stable) version of the categor y of motivic comple xes as an explicit direct f actor , called the +-par t of the stable homotop y category (that is the par t where the alg ebraic Hopf ﬁbration acts as in oriented cohomology theor ies). 7 Then Morel introduces this + -par t as a good candidate for the rational v ersion of the triangulated categor y of motiv es ([ Mor06 , paragraph at the end of p.2]). W e will dub the objects of this category Morel motiv es . In the language of motivic stable homotopy theor y , as initiated by Spitzw eck in [ Spi01 ], a natural candidate f or the categor y of motivic sheav es is the homotop y category of modules o ver the motivic r ing spectr um which represents motivic coho- mology . With integ ral coeﬃcients, O. Röndigs and P .A. Øs tvær sho w ed that, o v er a ﬁeld of characteristic 0 , this categor y of modules is equiv alent to the P 1 -stable 5 At that time, with the impulse of V oev odsky’s theory , the general process of ⊗ -in verting an object such as a topological circle of P 1 in an homotopy -theoretic wa y quickl y was fully documented; see [ Ho v01 ]. 6 Heur isticall y , the essential diﬀerence between stable A 1 -homotop y and motivic complex es is the presence of transf ers in the later case. 7 One of the goals of this book is to provide a proof of the generalization to abitrar y base schemes of Morel’ s claim; see Theorem 11 in this introduction and its corollar y . xvi Introduction category of motivic complex es (see [ RØ08a ]). 8 This r ing spectrum was introduced b y V oev odsky [ V oe98 , §6.1], using the theory of relativ e cy cles. It is deﬁned o ver an y base and one is led to consider the categor y of modules ov er this r ing spectr um as a possible deﬁnition of the integ ral triangulated category of motiv es. A.4 V oe vodsky’ s cross functors and A y oub’s thesis The deﬁnitive step to wards the six functors formalism in motivic homotop y theor y was tak en up by V oev odsky in a ser ies of lectures were he laid do wn the theor y of cross functors . The main theorem of this theor y consists in giving a cr iter ion on a sy stem of triangulated categories index ed b y schemes, equipped with a basic functoriality , to be able to construct e x ceptional functors ( f ! , f ! ) satisfying the prop- erties required by Grothendieck six functors formalism. In par ticular , the system of tr iangulated categories must satisfy three notable proper ties: the A 1 -localization property , the P 1 -stability pr operty and the localization property . Unf or tunatel y , only an introductor y part on this theor y was released (see [ Del01 ]) in which the basic setup is established but which does not contain the proof of the main result. The wr iting of this theory was accomplished by J. A y oub in the ﬁrst half of his thesis (see [ A y o07a ]). A y oub uses the axioms laid do wn by V oev odsky: he calls a sys tem of tr iangulated categor ies satisfying the proper ties alluded to abo v e a ho- motopy stable functor . Moreo v er , he goes bey ond the original result of V oev odsky: apart from the complete theor y of cross functors (concer ned with f ! , f ! ), he also studied monoidal structures, constructibility proper ties and their stability under the six operations, homotopy t-structur es and specialization functor s such as the v an- ishing cycle functor . The main e xample of a s table homotopy functor is the stable A 1 -homotop y categor y . This was established independentl y b y R öndigs [ Rön05 ] and A y oub [ A yo07a ], who both also der iv ed two fundamental proper ties: the one of relativ e pur ity and the proper base change isomor phism. One readily deduces that the category of Morel motives is also a homotopy stable functor . Further more, A y oub’ s axiomatic approach allo ws a unif or m treatment which also applies to man y natural variations of the stable A 1 -homotop y categor y (as recalled in [ A y o07b ], we ma y v ar y the topology as w ell as the coeﬃcients in which sheav es take their v al- ues). Ho w ev er , despite its already great lev el of g enerality , A youb’ s w ork does not allo w us to reach all the constructions of interest. For instance, it only pro vides the construction of the functors f ! and f ! when f is quasi-projectiv e, and the ﬁniteness and duality theorems only apply under h ypothesises (such as absolute pur ity) which are far from being obvious in practice (A y oub only discusses this issue for schemes of ﬁnite type ov er a per f ect ﬁeld, in which case this f ollo ws from the proper ty of relativ e purity). Moreo v er , the tec hniques recalled in the third chapter of A youb’ s 8 See also Theorem 8 in this introduction for an extension of their result to arbitrar y base, at the price of working with rational coeﬃcients. For ﬁelds of characteristic p > 0 , this has been extended to Z [ 1 / p ] -linear coeﬃcients by Ho y ois, Kelly and Østvær in [ HKØ17 ]. Finall y , using the results of the present book, this has been e xtended to regular schemes of equal characteristic in [ CD15 ]. A Historical background xvii thesis do not explain ho w to construct e xamples out of sheav es equipped with e xtra structures, such as transfers, which are fundamental tools to understrand how alge- braic cycles play a role in A 1 -homotop y theor y . The e xtra technicalities related to this problem (such as having a derived tensor product as well as derived pull-back functors for suitable notions of P 1 -stable sheav es with transfers) where addressed in two approaches: the ﬁrst one, b y R öndigs and Østv ær [ RØ08a ], uses homotop y theory together with enriched category theory , while the second one, due to the authors [ CD09 ], uses abstract methods of homotopical algebras applied to cochain comple x es in Grothendieck abelian categor ies. A second kind of problems whic h is not addressed in the earl y w ork of A y oub is representability f or K -theory , or f or Cho w groups, according to Beilinson ’s vision. This is wh y , in order to discuss the original approach of V oev odsky to motivic shea v es alluded to abov e (using h -sheav es or Nisne vich sheav es with transf ers), as w ell as to pro ve the absolute pur ity theorem, we had to take ov er from scratch many of the basic constructions. This also lead us to reach a greater le v el of generality (a v oiding unnecessary quasi-projectivity h ypothesises) as well as more precise com- putations. T o be fair , we should mention that, after a ﬁrst v ersion of the present te xt has been made public in 2009, A y oub [ A yo14 ] reprov ed some of the representability results as w ell as the absolute pur ity theorem of this book in the par ticular case of the étale version of the motivic stable homotopy theor y with rational coeﬃcients. W e should also mention r ight aw ay that the integral v ersion of V oev odsky’s h -motiv es is only fully understood in a sequel of the present book [ CD16 ]. Finall y , w e would like to end this paragraph by recalling that the problem of constructing triangulated categor ies of motives related to Cho w g roups with integral coeﬃcients and which deﬁne a homotop y stable functor is still an open problem. For insance, it is b y no means obvious that the categor y of modules ov er the motivic homotop y r ing spectr um does meet the requirements of a homotopy stable functor . In fact, this latter proper ty can be seen to be equivalent to Conjecture 17 of V oe v odsky in [ V oe02b ], which s tates the stability of the motivic homotop y ring spectr um by pull-backs; this is made precise in this book in Prop. 11.4.7 , as an application of our main constructions. A.5 Gro thendieck six functors formalism A.5.1 W e no w give the precise f or mulation of the Grot hendiec k six functors formal- ism (although w e do not descr ibe all the coherences yet). As presented here, it is e xtracted from the proper ties of the der iv ed category of ` -adic shea ves. 9 A tr iangulated categor y T , ﬁbred o v er the category of schemes, satisﬁes the Gro thendieck six functors f ormalism if the f ollo wing conditions hold: 1. There e xists 3 pairs of adjoint functors as follo ws: 9 It also coincides with f ormulas gathered by Deligne in an unpublished note which he graciously shared with us. xviii Introduction f ∗ : T ( X ) / / o o T ( Y ) : f ∗ , f an y mor phism, f ! : T ( Y ) / / o o T ( X ) : f ! , f an y separated mor phism of ﬁnite type, ( ⊗ , Hom ) , symmetr ic closed monoidal structure on T ( X ) . The functors of type f ∗ are monoidal. 2. There e xists a structure of a cov ar iant (resp. contrav ar iant) 2 -functors on f  / / f ∗ , f  / / f ! (resp. f  / / f ∗ , f  / / f ! ). 3. There e xists a natural transf or mation α f : f ! / / f ∗ which is an isomor phism when f is proper . Moreo v er , α is a morphism of 2 -functors. 4. For any smooth separated mor phism f : X / / S in S of relativ e dimension d , there e xists a canonical natural isomor phism p 0 f : f ∗ ∼ / / f ! (− d )[− 2 d ] where ? (− d ) denotes the in v erse of the T ate twist iterated d -times. Moreo v er p 0 is an isomor phism of 2 -functors. 5. For any car tesian square in S : Y 0 f 0 / / g 0   ∆ X 0 g   Y f / / X , such that f is separated of ﬁnite type, there exis t natural isomor phisms g ∗ f ! ∼ / / f 0 ! g 0 ∗ , g 0 ∗ f 0 ! ∼ / / f ! g ∗ . 6. For an y separated mor phism of ﬁnite type f : Y / / X , there e xist natural isomorphisms ( f ! K ) ⊗ X L ∼ / / f ! ( K ⊗ Y f ∗ L ) , Hom X ( f ! ( L ) , K ) ∼ / / f ∗ Hom Y ( L , f ! ( K )) , f ! Hom X ( L , M ) ∼ / / Hom Y ( f ∗ ( L ) , f ! ( M )) . (Loc) For any closed immersion i : Z / / S with complementar y open immersion j , there exis ts a distinguished triangle of natural transf or mations as follo ws: j ! j ! α 0 j / / 1 α i / / i ∗ i ∗ ∂ i / / j ! j ! [ 1 ] A Historical background xix where α 0 ? (resp. α ? ) denotes the co-unit (resp. unit) of the relev ant adjunction. A.5.2 The ne xt part of Grothendieck six functors f or malism is concerned with duality . Historically , this is the initial motivation behind Grothendieck six functors f or malism, as it appears in the ﬁrs t account of this f ormalism, Har tshorne’ s notes of Grothendieck 1963/64 seminar , [ Har66 ]. It is considered more axiomatically , in the case of étale shea v es, in [ Gro77 , Exp. I]. 10 In loc. cit. , Grothendieck states the fundamental property of absolute pur ity and indicates its fundamental link with duality . W e state these proper ties as natural e xtensions of the properties giv en in the preceding paragraph; assume T satisﬁes the proper ties listed abov e: (7) Absolute purity .– For any closed immersion i : Z / / S of regular schemes of (constant) codimension c , there e xists a canonical isomor phism: 1 Z (− c )[− 2 c ] ∼ / / i ! ( 1 S ) where 1 denotes the unit object for the tensor product. (8) Duality .– Let S be regular scheme and K S be any inv er tible object of T ( S ) . For an y separated morphism f : X / / S of ﬁnite type, put K X = f ! ( K S ) . For an y object M of T ( X ) , put D X ( M ) = Hom ( M , K X ) . a. For any X / S as abo ve, K X is a dualizing object of T ( X ) : the canonical map M / / D X ( D X ( M )) is an isomor phism. b. For any X / S as abo v e, and any objects M , N of T ( X ) , w e ha v e a canonical isomorphism D X ( M ⊗ D X ( N )) ' Hom X ( M , N ) . c. For an y morphism between separated S -schemes of ﬁnite type f : Y / / X , w e hav e natural isomor phisms D Y ( f ∗ ( M )) ' f ! ( D X ( M )) f ∗ ( D X ( M )) ' D Y ( f ! ( M )) D X ( f ! ( N )) ' f ∗ ( D Y ( N )) f ! ( D Y ( N )) ' D X ( f ∗ ( N )) . A.5.3 The last proper ty we want to exhibit as a natural e xtension of Grothendieck six functors f or malism is the compatibility with projectiv e limits of schemes. The basis for the ne xt statement is [ A GV73 , Exp. VI] though it does not appear explicitl y . As in the case of the duality proper ty , it should inv olv e some ﬁniteness assumption (constructibility) on the objects of T . Note the formulation belo w is v alid f or an arbitrary tr iangulated monoidal categor y T ﬁbred ov er schemes. 10 The duality proper ties are stated in unpublished notes of Deligne, as par t of the complete f ormalism. xx Introduction (9) Continuity .– Let ( S α ) α ∈ A be an essentially aﬃne projective sy stem of schemes. Put S = lim o o α ∈ A S α . Then the canonical functor 2 - lim / / α T ( S α ) / / T ( S ) is an equiv alence of monoidal tr iangulated categor ies. The pur pose of this book is to discuss such a f or malism in various contexts of motivic shea ves. B V oev odsky’s motivic complex es The pr imary goal of this treatise is to dev elop the theor y of V oev odsky motives, integrally o v er any base scheme 11 , within the framew ork of sheav es with transfers. A ctually , we can deﬁne V oev odsky’ s motives with coeﬃcients in an arbitrar y r ing Λ and prov e all the results stated belo w in that case, but we restrict this presentation to integral coeﬃcients for simplicity . After reﬁning and completing Suslin- V oe v odsky’ s theory of relative cy cles, w e introduce the categor y S m c or Z , S of integ ral ﬁnite cor respondences o v er smooth S - schemes and the related notion of (Nisnevich) shea v es with transfers o ver a base scheme S (Def. 10.4.2 ) as in the usual case of a per f ect base ﬁeld. Follo wing the idea of s table homotop y , w e deﬁne the tr iangulated category DM ( X ) of stable motivic complexes (see Def. 11.1.1 ) as the P 1 -stabilization of the A 1 -localization of the derived categor y of the (Grothendieck) abelian category of sheav es with transf ers o v er S . One easily gets that the ﬁbred categor y DM is equipped with the basic functori- ality needed b y the cross-functor f or malism. The main diﬃculty is the localization property , labelled (Loc) in Paragraph A.5.1 . Unf or tunately , though all the functors in v olv ed in the f or mulation of (Loc) are w ell-deﬁned f or DM , we can onl y prov e this proper ty when S and Z are smooth o v er some base scheme (see Prop. 11.4.2 ). In particular, the f or malism of stable homotopy functors does not apply . How ev er, we are able to construct the six operations for DM using the method of Deligne, used in [ A GV73 , XVII], and par tiall y get the Grothendieck six functors f or malism: Theorem 1 (see Th. 11.4.5 ) The triangulated categor y DM , ﬁbred ov er the category of sc hemes, satisﬁes the follo wing part of the pr operties stated in P arag raph A.5.1 : • properties (1), (2), (3) (i.e. the construction of f ! and f ! in DM Λ f or any separat ed morphism of ﬁnite type f ), • property (4) when f is an open immersion or f is pr ojectiv e and smooth, • property (5) when g is smooth or f is projectiv e and smooth, 11 In this introduction, all schemes will be assumed to be noetherian of ﬁnite dimension. B V oev odsky’s motivic complex es xxi • property (6) when f is pr ojective and smooth, • Property (Loc) when S and Z ar e smooth ov er some common base sc heme. One of the applications of this theor y is that we get a w ell-deﬁned integral motivic cohomology theory f or an y scheme X : H n , m M ( X , Z ) = Hom DM ( X )  1 X , 1 X ( m )[ n ]  which enjoy s the f ollowing properties (see section 11.2 ): • it admits a ring structure, pullbac k maps associated with an y mor phism of schemes compatible with the r ing structure, • it admits push-f or w ard maps with respect to projectiv e morphisms between schemes smooth ov er some common base, or with respect to some ﬁnite mor - phisms (f or e xample ﬁnite ﬂat; see Paragraph 11.2.4 ), • it coincides with V oev odsky’ s motivic cohomology g roups when X is smooth o v er a per f ect ﬁeld (see Example 11.2.3 ); in par ticular one gets the f ollo wing identiﬁcation with higher Chow groups: H n , m M ( X , Z ) = C H m ( X , 2 m − n ) , • it admits Cher n classes and satisﬁes the projective bundle formula, • it admits a localization long ex act sequence associated with a closed immersion of schemes which are smooth ov er some common base. As in the classical case, any smooth S -scheme X admits a motive M S ( X ) in DM ( S ) . Moreov er, one deﬁnes the T ate motive 1 S ( 1 ) as the reduced motive of P 1 S . W e deﬁne the categor y of constructible motives DM c ( S ) as the thick tr iangulated subcategory of DM generated by the objects of the form M S ( X )( n ) for a smooth S -scheme X and an integer n ∈ Z , where ? ( n ) refers to the n -th T ate twist. One gets the follo wing generalization of the classical result obtained by V oe v odsky ov er a perfect ﬁeld: Theorem 2 (see Th. 11.1.13 ) A motiv e M in DM ( S ) is constructible if and only if it is compact. 12 The category DM c ( S ) is equivalent to the categor y obtained from the bounded homotopy categor y of the additiv e category S m c or Z , S in the f ollowing w ay : • take the V erdier quotient modulo the thick triangulated subcategory g enerat ed by : – f or any Nisnevich distinguished square W k / / g   V f   U j / / X of smooth S -sc hemes: [ W ] g ∗ − k ∗ / / [ U ] ⊕ [ V ] j ∗ + f ∗ / / [ X ] 12 Recall that M is compact if the functor Hom ( M , −) commutes with arbitrar y direct sums. xxii Introduction – f or any smooth S -scheme X , p : A 1 X / / X the canonical projection: [ A 1 X ] p ∗ / / [ X ] , • inv ert the T ate twist, • take the pseudo-abelian env elope. The triangulated categor y DM c ( X ) is s table under the operations f ∗ f or all f , and f ∗ , when f is smooth projective, as well as ⊗ , but we cannot pro v e the stability f or the other operations of DM and a fortiori do not g et the duality proper ties (7) and (8) of the Grothendieck six functors f ormalism. Ho w ev er , we are able to prov e the continuity proper ty (9) f or the categor y DM c : 2 - lim / / α DM c ( S α ) ' DM c ( S ) , with the restriction that the transition morphisms of ( X α ) are aﬃne and dominant (see Theorem 11.1.24 ). N ote this result allow s us to e xtend the compar ison of mo- tivic cohomology with higher Cho w groups to arbitrar y regular sc hemes of equal characteristics. Finall y , although we could not pro v e all the expected proper ties of the six opera- tions in DM Λ , w e pro v e that the six operations behav e as e xpected in DM Λ if and only if Conjecture 17 of V oev odsky in [ V oe02b ] is tr ue; see Prop. 11.4.7 . C Beilinson motiv es C.1 Deﬁnition and fundamental properties As anticipated b y Morel, the theor y of mixed motives with rational coeﬃcients is much simpler and we succeed in establishing a complete formalism f or them. Ho w- ev er , there are many candidates f or Q -linear motivic sheav es o v er a scheme X : there are V oev odsky’ s h -motiv es DM h , Q ( X ) , V oev odsky’ s motivic sheav es constructed out Q -linear sheav es with transfers DM Λ ( X , Q ) , Morel motiv es SH Q ( X ) + , Q -linear étale motiv es D A 1 , ´ e t ( X , Q ) ' SH ´ e t , Q ( X ) (also introduced by Morel, and used as length by A y oub). Our goal is not only to pro v e that the six operations act on each of these candidates, but also to compare all these v ersions of motivic sheav es with one another . In fact, our strategy consists in producing yet another candidate for Q -linear motivic sheav es, namely the one of Beilinson motiv es DM B ( S ) , f or which we can pro v e all the f eatures we w ant f or it, and use them to compare Beilinson motives with all the other versions of Q -linear motivic sheav es mentionned abov e. More precisely , we constr uct, out of the rational motivic stable homotop y cat- egory and the ring spectr um associated with rational Quillen K -theory a Q -linear triangulated categor y DM B ( X ) , which we call the triangulated category of Beilinson C Beilinson motiv es xxiii motiv es (see Def. 14.2.1 ). Essentially by construction, in the case where X is regular , w e hav e a natural identiﬁcation Hom DM B ( X ) ( Q X , Q X ( p )[ q ]) ' Gr p γ K 2 p − q ( X ) Q , where the right-hand side is the graded par t of the algebraic K -theor y of X with respect to the γ -ﬁltration. These groups were ﬁrst regarded b y Beilinson as the rational motivic cohomology groups. W e call them the Beilinson motivic cohomology gr oups . Part of the interest of our deﬁnition is that the localization property (Loc) can be easily deduced from its validity f or the stable homotopy categor y . Theref ore, the cross-functor f ormalism and more generall y , our generalization of the results of A y oub can be applied to DM B . Theorem 3 (see Cor . 14.2.11 and Th. 2.4.50 ) All the standar d Grot hendieck six functors formalism (see P arag raph A.5.1 ) is veriﬁed by the ﬁbred triangulated cate- gor y DM B . Concerning duality f or Beilinson motives, we ﬁrst deduce from Quillen ’s local- ization theorem in algebraic K -theor y the absolute pur ity theorem: Theorem 4 (see Th. 14.4.1 ) The absolute purity property (see A.5.2 (7)) holds f or DM B . As said bef ore, this result is not enough to establish duality f or Beilinson motives. W e ﬁrst ha v e to use descent theory and resolution of singular ities (as ﬁrst explained b y Grothendieck in [ Gro77 , I.3]). Using the exis tence of trace maps in algebraic K -theor y , we prov e the follo wing result: Theorem 5 ( h -descent, see Th. 14.3.3 and Th. 4.4.1 ) Consider a ﬁnite gr oup G and a pullback square of schemes T h / / g   Y f   Z i / / X in which Y is endow ed with an action of G ov er X . Put U = X − Z and assume the f ollowing thr ee conditions ar e satisﬁed. (a) The morphism f is proper and surjectiv e. (b) The induced morphism f − 1 ( U ) / / U is ﬁnite. (c) The morphism f − 1 ( U )/ G / / U is g enerically radicial. Put a = f ◦ h = i ◦ g . Then, for any object M of DM B ( X ) , we g et a canonical distinguished triang le in DM B ( X ) : M / / i ∗ i ∗ ( M ) ⊕ f ∗ f ∗ ( M ) G / / a ∗ a ∗ ( M ) G / / M [ 1 ] xxiv Introduction wher e ? G means the inv ariants under the action of G , and the ﬁrst (resp. second) map of the triang le is induced by the diﬀerence (resp. sum) of the obvious adjunction morphisms. In fact, we sho w that this apparently simple result implies a much stronger descent property f or the ﬁbred tr iangulated categor y DM B : descent for the h -topology , thus, in particular, étale descent ﬂat descent, as well as proper descent. The general fact that, in the presence of the six operations, the proper ty of Q -linear h -descent is essentially equivalent to the presence of a suitable theory trace maps is a ke y feature of this te xt; this is dev elopped sys temattically in Chapter 3 of this book. This will be at the hear t of our main compar ison results e xplained below . C.2 Constructible Beilinson motiv es The ne xt step to wards duality for Beilinson motiv es is the deﬁnition of a suitable ﬁniteness condition. As in the case of V oev odsky motiv es, we deﬁne the category of Beilinson constructible motives , denoted by DM B , c ( X ) , as the thick subcategor y of DM B ( X ) generated by the motiv es of the f or m M X ( Y )( p ) : = f ! f ! ( Q X )( p ) f or f : Y / / X separated smooth of ﬁnite type, and p ∈ Z . This categor y coincides with the full subcategor y of compact objects in DM B ( X ) . 13 The usefulness of this deﬁnition comes from the f ollo wing result, which is the analog of Gabber’ s ﬁniteness theorem in the ` -adic setting. Analogously , its proof relies on absolute pur ity , (a weak f or m of ) proper descent as w ell as Gabber’ s w eak unif or mization theorem. 14 Theorem 6 (ﬁniteness, see Th. 15.2.1 ) The subcategor y DM B , c is stable under the six operations of Grot hendieck when restrict ed to excellent sc hemes. The ﬁnal statement concer ning Grothendieck six functors formalism in the setting of Beilinson motives is that, when one res tr icts to constructible Beilinson motives and separated B -schemes of ﬁnite type for an e x cellent scheme B of dimension less than 2 , the complete formalism is a vailable: 15 Theorem 7 (see Th. 15.2.4 and Prop. 15.1.6 ) The ﬁbr ed category DM B , c ov er the category of schemes described abov e satisﬁes the complete Gro thendiec k six functors f ormalism described in section A.5 , in particular the duality proper ty A.5.2 (8) and the continuity property A.5.3 (9). 13 Note the striking analogy with per f ect complex es. 14 i.e. that, locally for the h -topology , any e x cellent scheme is regular , and any closed immersion betw een e xcellent schemes is the embedding of a strict normal crossing divisor into a regular scheme. 15 There is a wa y to av oid this extra hypothesis to get duality theorems (i.e. to w ork with quasi- e xcellent schemes ov er a regular base in full generality). How ev er, this comes at the price of higher coherence results (i.e. of promoting the construction f  / / f ! to ∞ -categories). See [ Cis18 ]. C Beilinson motiv es xxv C.3 Comparison theorems In the historical par t of this introduction, w e saw many approaches for the tr iangulated category of (rational) motiv es. W e succeed in compar ing them all with our deﬁnition of Beilinson motiv es. Denote b y KGL S the algebraic K -theory spectr um in Morel and V oev odsky’ s stable homotop y categor y SH ( S ) . By virtue of a result of Riou, the γ -ﬁltration on K -theor y induces a decomposition of KGL S , Q : K GL S , Q ' Ê n ∈ Z H B , S ( n )[ 2 n ] . The r ing spectr um H B , S represents Beilinson motivic cohomology . Almost by con- struction, the categor y DM B ( S ) is the full subcategor y of SH Q ( S ) which consists of objects E suc h that the unit map E / / H B , S ⊗ E is an isomor phism. In f act, our ﬁrst comparison result relates the theor y of Beilinson motives with the approach of Spitzw eck, Röndigs and Østv ær through modules ov er a r ing spectrum: Theorem 8 (see Th. 14.2.9 ) F or any scheme S , ther e is a canonical equivalence of categories DM B ( S ) ' Ho ( H B , S - mo d ) wher e the right hand side denotes the homotopy categor y of modules ov er the ring spectrum H B , S . The next comparison inv olv es the h -topology: we recall that this is the Grothen- dieck topology on the category of schemes, generated by étale surjective mor phisms and proper surjective mor phisms. The ﬁrst published work of V oe v odsky on tr iangu- lated categor ies of mixed motives [ V oe96 ], introduces the A 1 -homotop y category of the der iv ed categor y of h -sheav es. W e consider a Q -linear and P 1 -stable v ersion of it, which we denote by DM h , Q ( S ) . By construction, f or any S -scheme of ﬁnite type X , there is a h -motiv e M S ( X ) in DM h , Q ( S ) . W e deﬁne DM h , Q ( S ) as the smallest triangulated full subcategor y of DM h , Q ( S ) which is stable under (inﬁnite) direct sums, and which contains the objects M S ( X )( p ) , f or X / S smooth of ﬁnite type , and p ∈ Z . Using h -descent in DM B , w e get the f ollowing compar ison result. Theorem 9 (see Th. 16.1.2 ) If S is excellent, then we have canonical equivalences of categories DM B ( S ) ' DM h , Q ( S ) . In fact, we ﬁrst prov e this result for the variant of DM h , Q ( S ) obtained b y replacing ev erywhere the h -topology by the qfh -topology – in the later , also introduced by V oe v odsky , cov er ings are generated by étale cov ers and ﬁnite surjective morphisms. In par ticular , we g et an equivalence of categories: DM h , Q ( S ) ' DM qfh , Q ( S ) . This result allo ws us to link Beilinson motiv es with V oev odsky’s motivic comple xes. Let us denote b y DM Q the rationalization of the ﬁbred category of stable motivic comple x es alluded to in Paragraph B . Using the preceding result in the case of the qfh -topology , we prov e: xxvi Introduction Theorem 10 (see Th. 16.1.4 ) If S is excellent and g eometrically unibranc h, then ther e is a canonical equivalence of categories DM B ( S ) ' DM Q ( S ) . In particular, giv en such a scheme S , we get a descr iption of DM B , c ( S ) as in The- orem 2 cited abo v e. V oe v odsky’ s integral (resp. rational) motivic cohomology is represented in SH ( S ) by a r ing spectr um H M , S (resp. H Q M , S ). The preceding theo- rem immediately gives an isomor phism of r ing spectra: 16 H B , S ' H Q M , S . As Beilinson motivic cohomology r ing spectra ov er diﬀerent bases are compatible with pullbacks, w e easily deduce the f ollo wing corollar y which solv es aﬃr mativ ely conjecture 17 of [ V oe02b ] in some cases, and up to torsion: Corollary F or any morphism f : T / / S of excellent g eometrically unibr anc h sc hemes, the canonical map f ∗ H Q M , S / / H Q M , T is an isomorphism of ring spectra. The ne xt compar ison statement is concer ned with the approach of Morel, accord- ing to whom the categor y SH Q ( S ) can be decomposed into two factors, one of them being SH Q ( S ) + , that is the par t of SH Q ( S ) on which the map  : S 0 Q / / S 0 Q , induced b y the per mutation of the factors in G m ∧ G m , acts as − 1 . Let S 0 Q + be the unit object of SH Q ( S ) + . Using the presentation of Beilinson motiv es in ter ms of H B -modules (Theorem 8 cited abo v e) as well as Morel’ s computation of the motivic sphere spectr um in terms of Milnor - Witt K -theor y , we obtain a proof of a statement, which, in the case where S is the spectr um of a ﬁeld, was claimed b y Morel in [ Mor06 ]: Theorem 11 (see Th. 16.2.13 ) F or any sc heme S , the canonical map S 0 Q + / / H B , S is an isomorphism. In fact, we ev en get the follo wing corollar y : Corollary F or any scheme S , there is a canonical equiv alence of categories SH Q ( S ) + ' DM B ( S ) . 16 Note in par ticular that, when S is regular , we get an isomor phism: H p , q M ( S , Z ) ⊗ Q ' Gr p γ K 2 p − q ( S ) Q which e xtends the known isomorphism when S has equal characteristics. It is natural with respect to pullbacks, Gysin mor phisms, as well as compatible with products and Chern classes. C Beilinson motiv es xxvii Recall from Morel theor y that, when − 1 is a sum of squares in all the residue ﬁelds of S ,  is equal to − I d on the whole of SH Q ( S ) . Thus in that par ticular case (e.g. S is a scheme o v er an algebraicall y closed ﬁeld), the categor y of Beilinson motives coincide with the rational stable homotopy categor y . In general, we can introduce according to Morel the étale variant of SH Q ( S ) denoted by D A 1 , ´ e t ( S , Q ) . 17 As locally f or the étale topology , − 1 is alwa ys a square, and because DM B satisﬁes étale descent, w e get the f ollowing ﬁnal illuminating comparison statement. 18 Corollary F or any scheme S , there is a canonical equiv alence of categories D A 1 , ´ e t ( S , Q ) ' DM B ( S ) . Let us draw a conclusive picture which summar ize most of the comparison results w e obtained: Corollary Giv en any sc heme S , the category DM B ( S ) is a full subcategor y of the rational stable homotopy category SH Q ( S ) . Giv en a r ational spectr um E ov er S , the f ollowing conditions ar e equiv alent: (i) E is a Beilinson motiv e, (ii) E is an H B , S -module, (iii) E satisﬁes étale descent, (iii’) ( S excellent) E satisﬁes qfh -descent, (iii”) ( S excellent) E satisﬁes h -descent, (iv) ( S excellent g eometrically unibranc h) E admits transf ers, (v) the endomorphism  ∈ End ( S 0 Q ) acts by − I d on E i.e.  ⊗ 1 E = − 1 E . Remar k (see Corollary 14.2.16 ) Points (iv) and (v) are related to the orientation theory f or spectra (not onl y ring spectra). In fact, H B , S is the universal orientable rational ring spectrum o v er S . Let Q . Sm S be the Q -linear en v elop of the category Sm S . One obtains (see Example 5.3.43 in conjunction with Par . 5.3.35 ) that the full subcategor y of compact objects of SH Q ( S ) is equiv alent to the categor y obtained from the homotopy categor y K b ( Q . Sm S ) b y per f or ming the f ollo wing operations: • take the V erdier quotient modulo the thick tr iangulated subcategor y generated b y: 17 In br ief, this is the P 1 -stabilization of the A 1 -localization of the derived categor y of sheav es of Q -v ector spaces ov er the lisse-étale site of S . 18 In par ticular , the ﬁniteness theorem as well as the duality proper ty also hold f or D A 1 , ´ e t (− , Q ) . The ﬁniteness theorem and the duality theorem may be deduced from [ A yo07a ] (Scholie 2.2.34 and Theorem 2.3.73 respectiv ely) when one restricts to quasi-projective schemes ov er a ﬁeld or ov er a discrete valuation r ing. Ne v er theless, ev en if one is eager to accept such restrictions, ov er a discrete valuation r ing, the proof relies in an essential wa y on the absolute pur ity proper ty (Theorem 4 stated abov e) which is prov ed in the present text. xxviii Introduction – f or an y Nisnevic h distinguished square W k / / g   V f   U j / / X of smooth S -schemes: Q S ( W ) g ∗ − k ∗ / / Q S ( U ) ⊕ Q S ( V ) j ∗ + f ∗ / / Q S ( X ) – f or an y smooth S -scheme X , p : A 1 X / / X the canonical projection: Q S ( A 1 X ) p ∗ / / Q S ( X ) . • in v er t the T ate twist, • take the pseudo-abelian env elope. Let us denote b y D A 1 , c ( S , Q ) this category . W e ﬁnally obtain the follo wing concrete description of Beilinson constructible motives: Corollary Giv en any sc heme S , the category DM B , c ( S ) is equiv alent to the full subcategory of D A 1 , c ( S , Q ) spanned by the objects E which satisfy one the follo wing equiv alent conditions: (i) (Galois descent) giv en any smooth S -scheme X and any Galois S -cov er f : Y / / X of gr oup G , the canonical map E ⊗ Q S ( Y )/ G / / E ⊗ Q S ( X ) is an isomorphism, (ii) (Orientability)  acts by − I d on E , Recall again the f ollowing remarks: 1. When (− 1 ) is a sum of square in ev er y residue ﬁelds of S , conditions (i), (ii) are true for any rational spectrum E o ver S . 2. When S is ex cellent and g eometr icall y unibranch, the category DM B , c ( S ) is equiv alent to the category of rational geometric V oev odsky motiv es (same deﬁ- nition as in Theorem 2 but replacing Z by Q ). C.4 Realizations The last feature of Beilinson motives is that they are easily realizable in various cohomology theories. T o get this fact, we use the setting of modules o v er a strict ring spectr um. 19 Given such a r ing spectrum E in DM B ( S ) , one can deﬁne, for any S -scheme X , the tr iangulated category D ( X , E ) = Ho ( E X - mo d ) , where E X = f ∗ E , f or f : X / / S the structural map. 19 i.e. w e sa y a r ing spectr um is strict if it is a commutativ e monoid in the underl ying model category . C Beilinson motiv es xxix W e then ha v e realization functors DM B ( X ) / / D ( X , E ) , M  / / E X ⊗ X M which commute with the six operations of Grothendieck. Using A y oub’ s descr iption of the Betti realization, we obtain: Theorem 12 If S = Sp ec ( k ) with k a subﬁeld of C , and if E Betti r epresents Betti cohomology in DM B ( S ) , then, f or any k -sc heme of ﬁnite type, the full subcategory of compact objects of D ( X , E Betti ) is canonically equivalent to the derived category of constructible sheaves of g eometric origin D b c ( X ( C ) , Q ) . More generall y , if S is the spectr um of some ﬁeld k , given a mix ed W eil co- homology E , with coeﬃcient ﬁeld (of character istic zero) K , w e get realization functors DM B , c ( X ) / / D c ( X , E ) , M  / / E X ⊗ X M (where D c ( X , E ) stands for the categor y of compact objects of D ( X , E ) ), which commute with the six operations of Grothendieck (which preser v e compact objects on both sides). Moreo ver , the categor y D c ( S , E ) is then canonically equivalent to the bounded der iv ed category of the abelian categor y of ﬁnite dimensional K -vector spaces. As a b y-product, w e g et the f ollo wing concrete ﬁniteness result: f or an y k -scheme of ﬁnite type X , and f or any objects M and N in D c ( X , E ) , the K -vector space Hom D c ( X , E ) ( M , N [ n ]) is ﬁnite dimensional, and it is tr ivial f or all but a ﬁnite number of values of n . If the ﬁeld k is of character istic zero, this abstract construction gives essentially the usual categories of coeﬃcients (as seen abo v e in the case of Betti cohomology), and in a sequel of this work, we shall pro ve that one recov ers in this wa y the derived categories of constructible ` -adic sheav es (of geometric or igin) in any characteristic. But something new happens in positive characteristic: Theorem 13 Let V be a complet e discre te v aluation ring of mixed char acteristic, with ﬁeld of functions K , and residue ﬁeld k . Then rigid cohomology is a K -linear mixed W eil cohomology, and thus deﬁnes a ring spectrum E rig in DM B ( k ) . W e obtain a syst em of closed symmetric monoidal triangulated categories D rig ( X ) = D c ( X , E rig ) , f or any k -scheme of ﬁnite type X , suc h that Hom D rig ( X ) ( 1 X , 1 X ( p )[ q ]) ' H q rig ( X )( p ) , as w ell as realization functor s R rig : DM B , c ( X ) / / D rig ( X ) whic h preserve the six operations of Gr othendiec k . xxx Introduction D Detailed organization The book is organized in four par ts that we now revie w in more details. D.1 Gro thendieck six functors formalism (Part 1) The ﬁrst par t is concer ned with the f or malism described in section A.5 abov e. It is the f oundational par t of this w ork. W e use the language of ﬁbred categories (introduced in [ Gro03 , VI]), comple- mented b y that of 2 -functors (or pseudo-functors), in order to descr ibe the six functors f or malism. W e ﬁrst descr ibe axioms which allow one to derive the core formalism – i.e. the par t described in section A.5.1 – from simpler axioms. W e do not claim originality in this task: our main contr ibution is to giv e a synthesis of the approach of Deligne descr ibed in [ A GV73 , XVII] (see also [ Har66 , Appendix]) with that of V oe v odsky dev eloped by A y oub in [ A y o07a ]. Recall that a (cleav ed) ﬁbred categor y M o v er S can be seen as a famil y of categories M ( S ) f or e very object S of S together with a pullback functor f ∗ : M ( S ) / / M ( T ) for an y mor phism f : T / / S of S . 20 Giv en a suitable class P of mor phisms in S , w e set up a systematic study of a par ticular kind of ﬁbred categories, called P -ﬁbred categories (deﬁnition 1.1.10 ): one where f or an y f in P , the pullback functor f ∗ admits a left adjoint, generically denoted by f ] . The functor f ] has to be thought as a variant of the exceptional dir ect imag e functor . 21 In section 1 , we study basic properties of P -ﬁbred categories which will be the core of the six functors formalism, such as base chang e formulas and projection f ormulas when an additional monoidal str ucture is in v olv ed. These formulas are special cases of a compatibility relation between diﬀerent types of functors e xpressed through a canonical compar ison mor phism. This kind of compar ison morphisms are g ener icall y called exc hang e mor phisms . The y are v ery versatile and appears ev erywhere in the theor y (see Paragraphs 1.1.6 , 1.1.15 , 1.1.24 , 1.1.31 , 1.1.33 , 1.2.5 ). In f act, they appear fundamentall y in Grothendieck six functors f or malism: in the list of properties A.5.1 , the y are the isomor phisms of (5), (6) and ev en (4). In the direction of the full Grothendieck functor iality , we introduce a core axiomatic for P - ﬁbred categor ies that we consider minimal: the categories satisfying this axiomatic are called P -pr emotivic (section 1.4 ). P -premotivic categories will f or m the basic setting in all this w ork. They will appear in three diﬀerent ﬂav ors, depending on which par ticular kind of additional s tr ucture we consider on categories: abelian, triangulated and model categor ies. 20 These pullback functors are subject to the usual cocy cle condition ; see section 1 . 21 This kind of situation frequentl y happens: the analytic case (open immersions), sheav es on the small étale site (étale morphisms), Nisnevich sheav es on the smooth site (smooth mor phisms). D Detailed org anization xxxi In Section 2 , we restrict our attention to the triangulated and g eometric case, meaning that w e consider tr iangulated P -ﬁbred categor ies ov er a suitable categor y of schemes S . The aim of this section is to dev elop, and extend, Grothendieck six functors formalism in this basic setting. W e e xhibit man y proper ties of such ﬁbred categories which are index ed in the appendix. Let us concentrate in this introduc- tion on the two main properties which will cor respond respectivel y to Deligne and V oe v odsky’ s approach on the six functors f or malism. The ﬁrst one, called the support property and abbreviated by (Supp), asser ts that the adjoint functors of the kind f ∗ , for f proper , and j ] , for j an open immersion, satisfy a gluing proper ty that allow s to use the argument of Deligne to constr uct the e x ceptional direct image functor f ! . 22 Sev eral proper ties are derived from (Supp) and the basic axioms of P -ﬁbred categor ies. Eventuall y , it leads to a partial v ersion of the six functors formalism (see Theorem 2.2.14 ). The second proper ty , most fundamental in the motivic conte xt, is the localization property abbreviated by (Loc), which is in fact part of the six functors formalism (see Paragraph A.5.1 ). It has man y interesting consequences and ref or mulations that are deriv ed in Section 2.3.1 . N ote that (Loc) is also known in the literature as the “gluing f or malism”. Some proper ties that w e pro v e in loc.cit. are already classical (see [ BBD82 ]). The most interesting consequence of (Loc) was disco v ered by V oev odsky: together with the usual A 1 -localization and P 1 -stabilization proper ties of the motivic context, it implies the complete basic six functors f or malism as stated in P aragraph A.5.1 . This was prov ed by A youb in [ A y o07a ]. In section 2.4 , we revisit the proof of A y oub and giv e some impro v ement of his theorems (see Theorem 2.4.50 for the precise statement): • w e remo ve the quasi-projectivity assumption f or the exis tence of f ! , replacing it b y the assumption that f is separated of ﬁnite type; • w e introduce the orientation property which allo w s one to get a simpler, more usual, f or m of the purity isomorphism (the one actually stated in point (4) of A.5.1 ); • w e give another proof of the main theorem in the oriented case by sho wing that relative purity is equivalent to some (strong) duality proper ty in the smooth projectiv e case (see Theorem 2.4.42 ); • w e directly incorporate the monoidal structure whereas A y oub giv es a separate discussion f or this. Apart from these diﬀerences, the mater ial of section 2.4 is v er y similar to that of [ A y o07a ]. Moreov er , in the non or iented case, it should be clear that we rely on the original argument of A youb for the proof of Theorem 2.4.42 . Concerning ter minology , we hav e called motivic triangulated categor y (Deﬁnition 2.4.45 ) what A y oub calls a “monoidal stable homotopy functor” (ex cept that A youb only considers operations induced by quasi-projectiv e mor phisms). 22 In the context of torsion étale sheav es of [ A GV73 , XVII], proper ty (Supp) is a consequence of the proper base chang e theorem. xxxii Introduction The remaining of Part 1 is concer ned with extensions of Grothendieck six functors f or malism. In Section 3 , w e show how to use the setting of P -ﬁbred model categor ies as a frame w ork to formulate Deligne ’ s cohomological descent theor y . Ex cept in tr ivial cases, object of a der iv ed categor y are not local. 23 T o f ormulate descent theor y in der iv ed categor ies, the main idea of Deligne was to extend the derived category of a scheme by one relativ e to a simplicial sc heme, usually a h yperco v er with respect to a Grothendiec k topology (see [ A GV73 , Vbis]). The construction consists in ﬁrst e xtending the theory of sheav es to the case where the base is a simplicial scheme and then consider ing the associated der iv ed category . W e generalize this construction to the case of an arbitrary P -ﬁbred category equipped with a suitable model category structure. In fact, w e show in Section 3.1 ho w to e xtend a P -ﬁbred categor y ov er a categor y of schemes to the cor responding category of simplicial schemes and e ven of arbitrary diagrams of schemes. Most importantly , we show how to extend the ﬁbred model structure to the case of diagrams of schemes (see Prop. 3.1.11 ). 24 Concretel y , this means that we deﬁne a derived functor of the kind L ϕ ∗ (resp. R ϕ ∗ ) for an arbitrar y mor phism ϕ of diagrams of schemes. Let us underline that these der iv ed functors mingle two diﬀerent kinds of functor iality: the usual pullback f ∗ (resp. direct image f ∗ ) for a mor phism of schemes f together with homotop y colimits (resp. limits) of arbitrar y diagrams — see the discussion in Paragraph 3.1.12 till Proposition 3.1.16 . With that e xtension in hands, we can easily f or mulate (cohomological) descent theor y f or arbitrar y Grothendieck topologies on the category of schemes for the homotopy categor y of a P -ﬁbred model categor y : see Deﬁnition 3.2.5 . The end of Section 3 is dev oted to concrete ex amples of descent in P -ﬁbred model categor ies, and their relation with properties of the associated homotop y category , assuming it is tr iangulated, as introduced in Section 2 . The ﬁrst and most simple example cor responds to the case of a Grothendieck topology associated with a cd-structure in the sense of V oe v odsky (as the Nisne vich and the c dh -topology . See [ V oe10b ] or Paragraph 2.1.10 ). In that case, descent can be characterized as the exis tence of cer tain distinguished triangles (Ma y er - Vietoris for Zar iski topology , Bro wn-Gersten f or Nisnevic h topology): this is Theorem 3.3.2 which is in fact a ref or mulation of the results of V oev odsky . W e then proceed to the most fundamental case of descent in algebraic geometry , that f or proper surjective maps which allo ws in principle the use of resolution of singularities. In fact, the main result of the whole of Section 3 is a characterization of h -descent which allo ws us to reduce it, f or P -ﬁbred homotop y triangulated categories which are rational and motivic, to a simple proper ty easil y check ed in 23 The ﬁrst ex ample of this fact is the circle: any non-trivial connected open subset of S 1 is contractible whereas S 1 itself is not. 24 By restricting the mor phisms of diagrams of schemes to a cer tain class denoted by P c art , we also sho w how to get a P c art -ﬁbred model category ov er diag rams of schemes (Rem. 3.1.21 ) but this is not really needed in the descent theory . D Detailed org anization xxxiii practice 25 : this is Theorem 3.3.37 . Along the w ay , we also pro v ed the f ollo wing results, interesting on their o wn: • se veral characterizations of étale descent (Theorems 3.3.23 and 3.3.32 ); • a characterization of qfh -descent (Theorem 3.3.25 ) as if it was deﬁned b y a cd-structure. 26 In fact, the last point is the heart of the proof of the main result of this section (Theorem 3.3.37 ). Whereas the extension of ﬁbred homotopy categor ies to diag rams of schemes is not unprecedented (see [ A yo07b ]), our study of proper and h -descent seems to be completely new . In our opinion, it is one of the most impor tant technical inno vation of this book. In Section 4 , w e study the extension of Grothendieck six functors f or malism in rational motivic categories, mainly duality and continuity . As already mentioned, the general pr inciple is not ne w and follo ws mainl y the path laid b y Grothendieck in [ Gro77 ]. In the case of an abstract motivic tr iangulated category — whic h is f or the purposes of descent theory the homotopy categor y of an underlying ﬁbred model category — the ﬁrst task is to introduce a cor rect proper ty of ﬁniteness inherent to an y duality theorem. This is done f ollowing V oev odsky , as in the work of A y oub, by introducing the notion of constructibility in Deﬁnition 4.2.1 . The name is inspired by the étale case, but the notion of constructibility which w e consider here is deﬁned by a generation property which reall y cor responds to what V oev odsky called g eometric motiv es : constructible motives in our sense are generated b y twists of motives of smooth schemes and are stable b y cones, direct factors and ﬁnite sums. Let us mention that in good cases, the proper ty of being constructible coincides with that of being compact in a tr iangulated categor y , resounding with the theor y of perfect comple x es (in the conte xt of ` -adic sheav es, this corresponds to “constructible of geometric origin”). The main point on constructible motives is the study of their stability under the six operations that w e get from the axioms of a tr iangulated motivic category . This is done in Section 4.2 . As in the étale case, the crucial point is the stability with respect to the operation f ∗ , when f is a mor phism of ﬁnite type between e x cellent schemes. In Theorem 4.2.24 , we giv e conditions on a motivic tr iangulated categor y so that the stability f or f ∗ is guaranteed (then the stability b y the other operations f ollo ws easily , see 4.2.29 ). Our proof essentially f ollow s an argument of Gabber . The general principle, going back to [ A GV73 , XIX, 5.1], is to use resolution of singularities to reduce to an absolute pur ity statement which is among our assumptions. 27 In Section 4.3 , we introduce an impor tant proper ty of motivic tr iangulated cat- egories, called continuity , which allo ws reasoning that in vol v es projective limits of 25 This is the separation property deﬁned in 2.1.7 . Let us mention here it is a consequence of the e xistence of well-beha ved trace maps (see the proof of Theorem 14.3.3 ). 26 It is at the origin of the formulation of descent that we ga ve for DM B in Theorem 5 (b) abo v e. A sys tematic approach to such generalized cd-structures is dev elopped by Park in [ Par19 ]. 27 Absolute purity will be prov ed later f or Beilinson motiv es. xxxiv Introduction schemes. In fact, it is sho wn in Proposition 4.3.4 that this proper ty implies the prop- erty (9) of the (e xtended) Grothendieck six functors formalism (see Paragraph A.5.3 abo v e). W e also giv e a criter ion for continuity ( 4.3.6 ) which will be applied later in concrete cases and draw some interesting consequences. Finall y , Section 4.4 deals with duality itself f or constructible motiv es, that is property (8) of Paragraph A.5.2 . The main theorem 4.4.21 asser ts that, under the same condition as Theorem 4.2.24 , and if one restricts to schemes that are separated of ﬁnite type ov er an e x cellent base scheme B of dimension less or equal to 2, then the full duality proper ty holds (see also Corollary 4.4.24 ). The proof f ollow s the same lines as the analogous Theorem 2.3.73 of [ A y o07a ]. In par ticular the main point is the f act that constructible motiv es are generated by some nice motives adapted to the use of resolution of singularities: see Corollar y 4.4.3 . The main diﬀerence with op. cit. is that w e implement De Jong’ s equiv ar iant resolution of singularities [ dJ97 ], so that our assumptions are much weak er . 28 D.2 The constructiv e part (Part 2) The pur pose of this par t is to giv e a method of construction of tr iangulated categories that satisﬁes the f or malism descr ibed in P ar t 1. W e ha ve chosen to mainly use the setting of der iv ed categories. Also, we use our notion of P -ﬁbred categor ies ( P - premotivic with a good monoidal str ucture). R ecall this means the pullback functor f ∗ admits a left adjoint f ] when f ∈ P . Essentially , P will be either the class of smooth mor phisms of ﬁnite type or the class of all morphisms of ﬁnite type (ev entually separated). In Section 5.1 , starting from a P -premotivic abelian categor y A , w e ﬁrst show ho w to prov e that the associated derived categor y D ( A ) is also a P -premotivic category . This consists in deriving the structural functors of a P -premotivic cat- egory , which is done by building a suitable underl ying P -ﬁbred model categor y in Proposition 5.1.12 . A ctuall y , the proof of the axioms of a model categor y has already appeared in our previous w ork [ CD09 ]. Let us mention the ﬂav or of this model structure: we can descr ibe e xplicitly coﬁbrations as w ell as ﬁbrations, by the use of an appropr iate Grothendieck topology t . This model structure is linked with cohomological t -descent (as sho wn later in Proposition 5.2.10 ). The adv antage of our frame w ork is to easily obtain the functoriality of this construction (Paragraph 5.1.23 ), as well as other homotopical constr uctions (dg-structure: Rem. 5.1.19 , e xtension to diagrams of schemes: Par . 5.1.20 ). In paragraph 5.1.c , w e also describe in suitable cases the constructible objects of the derived categor y b y a presentation similar to that of V oev odsky’ s geometric motiv es o v er a per f ect ﬁeld. In Section 5.2 (resp. Section 5.3 ) w e sho w how to descr ibe the A 1 -localization (resp. P 1 -stabilization) process in P -premotivic derived categor ies: to an y P - premotivic abelian categor y A is associated an A 1 -derived category D eﬀ A 1 ( A ) (resp. 28 See also footnote 15 page xxiv , which applies to this more general setting as well. D Detailed org anization xxxv P 1 -stable and A 1 -derived category D A 1 ( A ) ) in Deﬁnition 5.2.16 (resp. 5.3.22 ). From the model categor y obtained in Section 5.1 , the construction uses the classical tools of motivic homotopy theor y as introduced by Morel and V oev odsky . Ag ain, our frame w ork allo ws us to get the same homotopical constructions as in the simple derived case as well as some nice universal proper ties. W e also get a descr iption of constructible objects under suitable assumptions: Section 5.2.d (resp. 5.3.e ). These sections are ﬁlled with concrete examples. In Section 6 , we f ocus on the main (in fact univ ersal) e xample of motivic der iv ed categories, the A 1 -derived categor y of Morel, obtained by the process descr ibed abo v e from the abelian premotivic category of abelian sheav es ov er the smooth Nisnevic h site. The main point here is that one gets the localization proper ty for this categor y by a theorem of Morel and V oev odsky . W e giv e tw o new contr ibutions on this topic. First we show in Section 6.1 that the A 1 -derived categor y can be embedded in a larg er categor y which naturally contains objects that we can call motiv es of singular sc hemes. This is useful to state descent properties and will be essential to study h -motiv es. Second, we sho w in Section 6.3 ho w one can use the A 1 -derived category to obtain good proper ties of another premotivic derived category satisfying suitable assumptions. This will be applied to motivic comple xes. In Section 7 , we go bac k to the case of an arbitrary monoidal P -ﬁbred model category M and e xplain ho w to use the setting of ring spectra and modules ov er ring spectra in the premotivic conte xt. The main construction associates to a suitable collection of (commutativ e) ring spectra R in M a P -ﬁbred monoidal category denoted by Ho ( R - mo d ) : Proposition 7.2.13 . This construction will be used sev eral times: • in the study of alg ebraic K -theor y (Section 13 ): the categor y of modules ov er K -theor y is the fundamental technical tool to get motivic proper descent as w ell as motivic absolute pur ity ; • in the study of Beilinson motives when w e will relate them with modules o v er motivic cohomology (Theorem 14.2.9 ); • in the study of realizations associated with a mixed W eil cohomology (Section 17 ). D.3 Motivic complex es (Part 3) This par t is concer ned with the constructions descr ibed abov e, in Section B . Our aim is to extend the deﬁnition of V oev odsky’ s integ ral motivic complex es to an y base, then study their functoriality and introduce their non-eﬀectiv e, or rather P 1 -stable, counter -par t. Our ﬁrst task, in Section 8 , is to revisit Suslin- V oev odsky’s theor y of relativ e cycles e xposed in [ SV00b ]. Indeed, the y will be at the heart of the general construction. Our presentation is made to prepare the theor y of ﬁnite correspondences , a par ticular case of relative cy cles. Especially , w e want to give a meaning to the f ollo wing picture representing the composition of ﬁnite cor respondences α from X to Y and β from Y xxxvi Introduction to Z : β ⊗ Y α / /   β / /   Z . α / /   Y X (see also ( 9.1.4.1 )). More precisely , we want to inter pret this as a diagram of cycles. Thus w e are led to consider cy cles (with their suppor t) as objects of a category . Con- cretely , a cy cle is considered as a multi-pointed scheme, each point being endo wed with some multiplicity (an integ ral or rational number). This conceptual shift has the advantag e of allowing a treatment of cycles anal- ogous to that of alg ebraic varieties, or rather schemes, promoted b y Grothendiec k via studying mor phisms. Thus, we replace the various g roups of relative cycles in- troduced by Suslin and V oev odsky in op. cit. by properties of mor phisms of cy cles. Here is a list of the pr incipal ones: • pseudo-dominant ( 8.1.2 ), equidimensional ( 8.1.3 and 8.3.18 ), • pre-special ( 8.1.20 ), • special ( 8.1.28 ), • Λ -univ ersal ( 8.1.49 ). The most intr iguing one, being pr e-special , has no counter -par t in op. cit. Its idea comes from a mistake (fortunately insigniﬁcant) in the con v ention of Suslin and V oe v odsky . Indeed, Lemma 3.2.4 of op. cit. is f alse whenev er the base S is non reduced and ir reducible: then any fat point ( x 0 , x 1 ) and any ﬂat S -scheme give a counter -ex ample. 29 The e xplanation is that the operation of specialization along a fat point does not take into account the geometric multiplicities of the base. On the contrary , when X is ﬂat ov er an ir reducible scheme S , the geometric multiplicity of any ir reducible component of X is a multiple of the g eometr ic multiplicity of S . This leads us to the deﬁnition of a pre-special mor phism of cycles β / α , where a divisibility condition appears in the multiplicities of β with respect to that of α . 30 The main achie v ement of Suslin and V oev odsky’ s theor y is the construction of a pullback operation f or relative cycles. In our language, it cor responds to a kind of tensor product, more precisely a product of cycles relative to a common base cycle (as f or ex ample the cy cle β ⊗ Y α of the preceding picture). Despite our diﬀerent presentation, the method to deﬁne this operation f ollow s closely the original idea of Suslin and V oev odsky: use the ﬂatiﬁcation theor em of Gruson and Raynaud to reduce to the case of ﬂat base chang e of cycles. Recall that the ke y point is to ﬁnd the cor rect condition on cy cles – or rather morphisms of cy cles in our language – so 29 Explicitl y , tak e S = Z = Sp ec  k [ t ]/( t 2 )  = { η } , R =  k [ t ]  ( t ) . The left-hand side of the equality of 3.2.4 is 2 .η while the r ight-hand side is η . 30 T o anticipate the rest of the construction, given a non reduced scheme S , this will allow for the operation of pull-back along the immersion S r e d / / S associated with the reduction of S : it simply corresponds to dividing by the geometric multiplicities of S , as the base change to S r e d does f or ﬂat S -schemes. D Detailed org anization xxxvii that one obtains a uniquely deﬁned operation independent of the chosen ﬂatiﬁcation. This is measured b y a specialization procedure (Deﬁnition 8.1.25 ) associated with fat points (Deﬁnition 8.1.22 ) and leads to the central notion of special mor phisms of cy cles (Deﬁnition 8.1.28 ). An inno vation that we introduce in the theory is to giv e, as soon as possible, local deﬁnitions at a point in the sty le of EGA. This is in particular the case f or the proper ty of being special. Once this notion is in place, one deﬁnes f or a base cy cle α , a special α -cy cle β and an y morphism φ : α 0 / / α the relativ e product denoted by β ⊗ α α 0 . Equivalentl y , it corresponds to the base chang e of β / α along φ (Deﬁnition 8.1.40 ). This notion is close to the cor respondence mor phisms of Section 3.2 of op. cit. In par ticular , it usually in v olv es denominators. The last impor tant notion, being Λ -univ ersal , cor responds to cycles β / α with coeﬃcients in a r ing Λ ⊂ Q , which keeps their coeﬃcients in Λ after an y base change. One sees that our language is especiall y con venient when it is time to consider the stability of cer tain proper ties of mor phisms of cy cles under composition (Cor . 8.2.6 ) or base c hange (Cor . 8.1.46 ). Then the usual statements of intersection the- ory are prov en in Section 8.2 , still f ollo wing or e xtending Suslin and V oe v odsky: commutativity , associativity , projection f or mulas. This makes our relative product a good e xtension of the classical notion of e xter ior product of cy cles (o v er a ﬁeld). The focal point of intersection theor y is the study of multiplicities. Thus we in- troduce Suslin- V oevodsky’ s multiplicities , as the ones appear ing as a corollary of the e xistence of the relativ e cycle β ⊗ α α 0 (Deﬁnition 8.1.43 ). A very impor tant result in the theory , already enlightened by Suslin and V oe v odsky , is the fact these multi- plicities can be expressed in ter ms of Samuel multiplicities . 31 In fact, independently of Suslin and V oev odsky , we prov e a new cr iterion for the proper ty of being special at a point in vol ving Samuel multiplicities at the branc hes of the point: see Corol- lary 8.3.25 . R oughly speaking, the multiplicities arising from Samuel’ s deﬁnition at each branches of the point must coincide: then this common v alue is simpl y the Suslin- V oev odsky’ s multiplicity . Finall y , still follo wing the treatment of algebraic geometry by Grothendieck, we add to the theor y of Suslin and V oev odsky the study of constructibility properties f or mor phisms of cy cles (special and Λ -univ ersal). Here, our categorical point of vie w is plainly justiﬁed. Explicitly , we prov e that giv en a relative cycle β / α , when α is the cycle associated with a scheme S , the locus where β is special (resp. Λ - univ ersal) is an ind-constructible subset of S (Lemma 8.3.4 ). This allo ws to pro ve the good behavior of these notions with respect to projectiv e limits of schemes (see in par ticular 8.3.9 ). This will be the key point when proving the continuity proper ty — (9) of A.5.3 — of the ﬁbred categor y DM . The rest of Part 3, consists in extending the theor y of shea ves with transfers introduced by V oev odsky , originally o v er a perfect ﬁeld, to the case of an arbitrary base and apply to it the g eneral procedures studied in Part 2 to get the ﬁbred categor y DM . 31 When a cor rect regular ity assumption is added, one reduces to the usual Ser re ’ s T or -intersection f ormula: see 8.3.31 and 8.3.32 ). xxxviii Introduction In Section 9 , we w ork out the theor y of ﬁnite cor respondences using the f or malism of relativ e cycles. The construction is summar ized in Corollar y 9.4.1 : giv en a class of mor phisms P contained in the class of separated mor phisms of ﬁnite type and a ring of coeﬃcients Λ , w e produce a monoidal P -ﬁbred category , denoted by P c or Λ , whose ﬁber ov er a noether ian scheme S (ev entually singular) is the categor y of P -schemes o v er S with mor phisms the ﬁnite cor respondences. In Section 10 , w e dev elop the theor y of sheav es with transfers along the v er y same line as the original treatment of V oev odsky . This time, the outcome can be summarized by Corollaries 10.3.11 and 10.3.15 : given a class P of morphisms as abo v e and a suitable Grothendieck topology t , we construct an abelian premotivic category Sh t ( P , Λ ) which is compatible with the topology t (cf P ar t 2); its ﬁber o v er a scheme S is given b y t -shea v es of Λ -modules with transfers (in par ticular preshea v es on P c or Λ , S ). 32 The section is closed with an impor tant comparison result, essentially due to V oev odsky , between Nisne vich shea ves with transf ers and sheav es f or the qfh -topology (with rational coeﬃcients ov er g eometr icall y unibranch bases): see Theorem 10.5.15 . Finall y , Section 11 is dev oted to gathering the w ork done previousl y and deﬁne the stable der iv ed categor y of motivic complex es DM Λ , giv en an arbitrar y ring of coeﬃcients Λ . The out-come has already been described in Section B abov e. D.4 Beilinson motiv es (Part 4) This par t contains the construction of Beilinson motiv es as well as the proof of all the proper ties stated bef ore. It is based on the ﬁrst and second par ts but independent of the third one — ex cept in the comparison statements of Section 16.1 . Section 12 contains a shor t revie w of the stable homotopy categor y and the notion of oriented ring spectra. Section 13 is the hear t of our construction. It contains a detailed study of the K -theor y r ing spectr um KGL and the associated notion of KGL -modules in the homotopical sense (based on the formalism introduced in Section 7 ). Using the works of sev eral authors (most notably : Riou, Naumann, Spitzw eck, Østv ær), w e show ho w the central results of Quillen on algebraic K -theor y giv e important properties of K GL -modules: absolute pur ity (Th. 13.6.3 ) and trace maps (Def. 13.7.4 ). In Section 14 , w e ﬁnall y introduce the deﬁnition of Beilinson motives. Let us describe it in detail no w . It is based on the process of Bousﬁeld localization of the stable homotopy categor y with respect to a cohomology . This operation is fundamen- tal in moder n algebraic topology . W e apply it in algebraic geometry to the rational stable homotopy categor y (or , what amount to the same, to the rational stable A 1 - derived category of Morel, Section 6 ) and to the rational K -theor y spectr um KGL Q : the Bousﬁeld localization of D A 1 ( S , Q ) with respect to K GL Q , S is the categor y of Beilinson motiv es DM B ( S ) ov er S (Deﬁnition 14.2.1 ). Using the preceding study of 32 The most notable topologies t that ﬁt in this result are the Nisnevic h and the c dh ones. See Section 10.4 . E Dev elopments since the ﬁrst arXiv v ersion xxxix K GL Q together with the decomposition of Riou recalled in the beginning of Section C.3 , we get the main proper ties of the premotivic category DM B : the h -descent theorem ( 14.3.4 ) and the absolute pur ity theorem ( 14.4.1 ). Then the theoretical background laid down in Part 1 is applied to DM B , giv en in par ticular the complete Grothendieck six functors f or malism for constructible Beilinson motiv es (Section 15 ). Our work closed with the two main subjects descr ibed abo v e on Beilinson motives: the comparison statements (Section 16 ) and the study of motivic realizations (Section 17 ). E De v elopments since the ﬁrst arXiv version The ﬁrst version of this work has ﬁrst appeared on arXiv on December 2009. 33 During almost ten y ears, until the actual publication b y Springer Edition, it has been used in sev eral works, as w ell as completed by sev eral other mathematicians, solving questions left open in the present te xt. For completeness, it appears to us beneﬁcial to the reader to give an account of some of these dev elopments which are the most directly related with the present contribution. Mathematics is indeed a collective w ork, each part of which is destined to be used, completed, renew ed or superseded. E.1 Nisne vich motiv es with integral coeﬃcients E.1.1 c dh -motives .– One aim of the present work was to work out the theory of ﬁnite correspondences in the spir it of [ V SF00 ], whose original aim is to obtain an integral theory of motivic complex es related to Chow groups. The theor y of c dh -sheav es with transf ers (see Proposition 10.4.8 ) was introduced with this motivation in mind. The theor y of c dh -motiv es and motivic complex es w as successfull y dev eloped in the equal characteristic case in [ CD15 ], provided one in v er ts the residue characteristic. In this latter w ork, the cr ucial proper ty of localization f or c dh -motives is sho wn, as well as all the expected results: constructibility of the six operations, duality , continuity , compar ison with modules o v er the c dh -local v ersion of V oev odsky’s motivic cohomology , relation with higher Cho w groups. T o get these results, ke y points are the continuity proper ty of motivic comple xes which is pro v ed in this book (Theorem 11.1.24 ), as a result of our reinf orcement of Suslin- V oev odsky’ s theor y of relativ e cycles (see in particular Section 8.3.a on constructibility f or proper ties of relative cycles) together with Kell y’ s new motivic descent results [ Kel17 ] which allo w to use Gabber’ s impro v ements of de Jong’ s alteration theorems [ ILO14 ]. Note also that [ CD15 , 3.6 and 5.1] generalizes our result on V oev odsky’s conjecture on base chang e of the motivic Eilenberg-MacLane spectr um (Corollary 16.1.7 ). 33 A new version was uploaded in 2012, containing more or less the actual introduction which was written in order to clear-up the contributions and histor y on mixed motiv es and more speciﬁcally motivic homotop y theor y . xl Introduction E.1.2 Spitzw eck’ s motivic cohomology spectr um .– One of the problems with deﬁn- ing mix ed motiv es as modules o v er V oev odsky’ s motivic cohomology spectr um is the compatibility of this spectr um with base chang e. 34 The idea of Spitzwec k’ s paper [ Spi18 ] is to build a spectr um which satisﬁes compatibility b y base chang e; equiv- alently , one has to build a ring spectrum H Z ov er S = Spec ( Z ) (or more g enerally o v er a Dedekind r ing) which pullbacks to V oev odsky’s motivic cohomology spec- trum o ver the residue ﬁelds of S . This is what M. Spitzw eck achiev es with virtuosity in loc. cit. , theref ore obtaining a conv enient category of H Z -modules which coin- cides with V oev odsky’ s original tr iangulated categor y ov er the residue ﬁelds of S ; in fact, it also coincides with DM c dh o v er any k -scheme, after in v er ting the residue characteristic of k . But the construction of Spitzw eck w orks integrally . Moreov er , b y its very construction, the cohomology represented by H Z coincides with Bloch ’ s higher Cho w groups f or smooth S -schemes. A question left open is a possible com- parison with V oev odsky’ s motivic cohomology spectrum (which is again equiv alent to V oev odsky’ s base chang e conjecture). E.2 Étale motiv es with integral coeﬃcients and ` -adic realization E.2.1 V oevodsky’ s motiv es in the étale topology and rigidity theor ems .– With rational coeﬃcients, the comparison theorems obtained in this book (Section 16 ) show that varying the underlying topology is beneﬁcial. In par ticular , with rational coeﬃcients, w e are not able to g et the localization proper ty f or Nisnevic h motivic comple xes f or all base schemes, but w e do get that property when replacing Nisnevich topology with the qfh -topology , or the h -topology . This lets one believ e that the transfers will be better beha v ed with integral coeﬃcients with respect to strong er topologies. In [ CD16 ], we do prov e that the localization property holds f or motivic complex es with torsion coeﬃcients locally for the étale topology ([ CD16 , Theorem 4.3.1]). It f ollo ws that the same proper ty holds with integ ral coeﬃcients for geometricall y unibranch schemes. Moreo ver , we pro v e in loc. cit. that, locally f or the h -topology , motivic comple x es with integral coeﬃcients are perfectl y well-beha ved and satisfy all the e xpected proper ties, as listed in Section A.5 of this introduction. 35 It is remarkable that we were able to g et the complete Grothendiec k six functors formalism for V oe v odsky’ s or iginal construction of étale motiv es, as deﬁned in his Ph. D. thesis [ V oe96 ], and sho w the visionar y pow er of V oev odsky once more time. Besides, we also show that one recov ers the theor y of ` -adic complex es out of h -motives b y the categorical process of ` -adic completion (see [ CD16 , §7.2]). This gives a ne w insight on ` -adic realization of motiv es. 34 Recall again this compatibility was conjectured by V oev odsky . See Conjecture 11.2.22 for an e xplicit f or mulation. Note also that w e pro v e the latter conjecture is actually equivalent to the localization property for (Nisnevich) motivic complex es: see Proposition 11.4.7 . 35 Based on the results of this book, we only get Grothendieck - V erdier duality f or schemes of ﬁnite type ov er a regular 2 -dimensional ex cellent scheme, but this extra hypothesis has been remov ed by the ﬁrst author in [ Cis18 ]. E Dev elopments since the ﬁrst arXiv v ersion xli E.2.2 Rigidity theor ems without transf ers .– Another extension of Suslin and V o- ev odsky’ s r igidity theorem to arbitrar y bases is due to A y oub, [ A y o14 ]. In this latter w ork, A youb s tudies the categor y introduced in the present book under the nota- tion D A 1 , ´ e t ( S , Λ ) (follo wing Morel), while he uses the notation DA ( S , Λ ) . The ﬁrst result of loc. cit. , inspired b y an earlier work of R öndigs and Østvær [ RØ08b ], is indeed a variation on the r igidity theorem, identifying the category D A 1 , ´ e t ( S , Λ ) for a Λ = Z / N Z with N inv er tible on S with the der iv ed category of the category of shea v es of Λ -modules on the small étale site of S (under suitable hypothesises on S and N ). From there, one can e xtend the results prov ed in this book f or D A 1 , ´ e t ( S , Q ) (see Theorem 16.2.18 ) to the case of arbitrar y coeﬃcients: absolute purity , con- structibility of the six operations, duality . Note how ev er that, despite what is claimed in the appendix of A y oub’ s ar ticle, the par ticular case of 2 -torsion f or base schemes S of mix ed or positiv e characteristic is problematic in his approach (see [ CD16 , Rem. 5.5.8]). Since then, Bachmann [ Bac18b ] has e xtended by f ar the preceding rigidity the- orems to torsion P 1 -stable motivic étale sheav es of spectra . This result also solv es the af orementioned issues left open in [ A yo14 ]. E.3 Motivic st able homotop y theory with rational coeﬃcients E.3.1 Witt sheaves .– In this book, f ollo wing Morel’ s insights, we hav e splitted the rational motivic stable homotop y categor y SH ( X ) Q ' D A 1 ( X , Q ) into tw o factors ( D A 1 ( X , Q ) + ) × ( D A 1 ( X , Q ) − ) ' D A 1 ( X , Q ) and w e ha ve identiﬁed the oriented part D A 1 ( X , Q ) + with Beilinson ’ s motiv es DM B ( X ) . On the other hand, in the case where X = Sp ec ( k ) is the spectrum of a ﬁeld, Anan y ev skiy , Levine and Panin [ ALP17 ] ha v e identiﬁed the non-or iented part D A 1 ( k , Q ) − with a suitable categor y of Witt shea v es. The conjunction of their results with ours may be seen as a motivic analog of (a tr ivial consequence of ) a theorem of Ser re that the stable homotopy g roups of spheres are ﬁnite in degree > 0 ; see the introduction of loc. cit. The results of Anany ev skiy , Levine and Panin hav e been improv ed b y Bachmann [ Bac18a ], where the comparison of SH ( k ) − with Witt shea v es is promote d to Z [ 1 / 2 ] -linear coeﬃcients. Bachmann ’ s results f ollo w from a nice analog of the r igidity theorem o v er a general base f or the real étale topology . E.3.2 Rational absolute purity .– Déglise, Fasel, Jin and Khan [ DFJK19 ] hav e pro ved absolute purity proper ty f or the motivic sphere spectr um with rational coeﬃcients. xlii Introduction E.4 Duality , w eights and traces E.4.1 W eight complexes .– Bondarko ’ s theor y of weight complex es [ Bon09 ] has been sho w ed to be compatible with the six operations with rational coeﬃcients in [ Héb11 , Bon14 ]. In the setting of c dh -shea v es [ CD15 ], this has been extended by Bondarko and Ivano v [ BI15 ] to Z [ 1 / p ] -linear coeﬃcients in equal characteristic, where p ≥ 1 denotes the exponent characteristic of the ground ﬁeld. Such weight complex es hav e been used by Wildeshaus [ Wil17 , Wil18 ], in order to giv e inconditional constructions of motivic intersection complex es of certain Shimura varieties. They also play a role, together with realization functors associated to mixed W eil cohomologies, in geometric representation theor y , in the w ork of Soergel and his collaborators [ SW18 , SVW18 ]. E.4.2 Motivic Lefsc hetz- V erdier trace formula .– An obvious application of the the- ory of motivic sheav es and their realizations is the proof of independence of ` results f or a w ealth of trace-like constructions. A Q -linear version of suc h kind of results is provided by Olsson [ Ols15 , Ols16 ], where some versions of the Motivic Lefsc hetz- V erdier trace formula and of characteristic classes are discussed. A slight impro v ement, allo wing torsion, may be f ound in [ Cis18 ]. But a full account on inte- gral formulas, including f or characteristic classes, is settled in the recent work of Jin and Y ang [ JY19 ]. E.5 Enriched realizations E.5.1 Structur ed mixed W eil cohomologies .– In his thesis [ Dre13 ], Drew extends the formalism of mix ed W eil cohomologies to cohomologies with values in a T an- nakian category . He also deﬁnes the realization functor into algebraic D -modules f or schemes of ﬁnite type o ver a ﬁeld of characteristic zero and prov es that, f or an y separated smooth scheme X o v er a ﬁeld of character istic zero, cons tr uctible modules ov er de Rham cohomology in SH ( X ) embedd fully faithfull y in algebraic D -modules. Drew deduces from this embedding a ne w purely algebraic proof of the Riemann-Hilbert correspondence, using motivic shea ves, as predicted in Example 17.2.22 in the present book. His w ork is also a wa y to deﬁne Hodge realizations of mix ed motivic sheav es; see [ Dre18 ]. E.5.2 Arakelo v motivic cohomology .– Holmstrom and Scholbac h [ HS15 ] hav e e x- tended the representability of algebraic de Rham cohomology to the ﬁltered de Rham comple x, and used it to deﬁne a motivic version of Arak elov cohomology . The rela- tion with more classical versions of Arakelo v cohomology and with height pairings is discussed in [ Sch15 ]. E Dev elopments since the ﬁrst arXiv v ersion xliii Thanks The authors want to thank hear tily the follo wing people for help, motiv ation, cor - rections or sugges tions during the elaboration of this te xt: Joseph A y oub, Ale xan- der Beilinson, Pier re Deligne, Brad Drew , Da vid Héber t, Jens Hor nbostel, Annette Huber -Klawitter , Bruno Kahn, Shane Kell y , Marc Levine, Georg es Maltsiniotis, F a- bien Morel, Paul Ar ne Østvær , Joël Riou, Oliv er R öndigs, V alentina Sala, Markus Spitzw eck, Vladimir V oev odsky , V adim V ologodsky , Chuck W eibel, and ﬁnally , as w ell as par ticularl y , Jörg Wildeshaus. W e w ould like to thank our editors of Springer for simpl y making the publication of this book possible. D.-C. C is partially suppor ted by the SFB 1085 “Higher Inv ar iants ” funded b y the Deutsche Forschungsg emeinschaft (DFG), and F . D. by the French “In v estissements d’ A v enir” prog ram, project ISITE-BFC (contract ANR -lS-IDEX -OOOB). No tations and conv entions In e very section, we will ﬁx a categor y denoted by S which will contain our geometric objects. Most of the time, S will be a category of schemes which are suitable for our needs; the required hypothesis on S are giv en at the head of each section. In the text, when no details are giv en, any scheme will be assumed to be an object of S . When A is an additiv e categor y , we denote by A \ the pseudo-abelian env elope of A . W e denote b y C ( A ) the categor y of comple xes of A . W e consider K ( A ) (resp. K b ( A ) ) the categor y of complex es (resp. bounded comple xes) of A modulo the chain homotopy equivalences and when A is abelian, we let D ( A ) be the der iv ed category of A . If M is a model categor y , Ho ( M ) will denote its homotop y categor y . W e will use the notation α : C / / o o D : β to mean a pair of functors such that α is left adjoint to β . Similarly , when we speak of an adjoint pair of functors ( α, β ) , α will alwa ys be the left adjoint. W e will denote b y a d ( α , β ) : 1 / / β α (resp. a d 0 ( α , β ) : α β / / 1 ) the unit (resp. counit) of the adjunction ( α, β ) . Considering a natural tranf ormation η : F / / G of functors, w e usually denote by the same letter η — when the context is clear — the induced natural transf or mation A F B / / AG B obtained when con- sidering functors A and B composed on the left and r ight with F and G respectivel y . In section 8 , w e will assume that equidimensional mor phisms ha ve constant relativ e dimension. P art I Fibred categories and the six functors f ormalism 1 General deﬁnitions and axiomatic 3 1 General deﬁnitions and axiomatic 1.0.1 W e assume that S is an arbitrar y category . W e shall say that a class P of mor phisms of S is admissible if it is has the f ollowing proper ties. (Pa) Any isomor phism is in P . (Pb) The class P is stable by composition. (Pc) The class P is stable by pullbacks: f or any mor phism f : X / / Y in P and an y mor phism Y 0 / / Y , the pullback X 0 = Y 0 × Y X is representable in S , and the projection f 0 : X 0 / / Y 0 is in P . The mor phisms which are in P will be called the P -mor phisms . 36 In what f ollo ws, we assume that an admissible class of mor phisms P is ﬁxed. 1.1 P -ﬁbred categories 1.1.a Deﬁnitions Let C at be the 2 -category of categor ies. 1.1.1 Let M be a ﬁbred categor y o v er S , seen as a 2 -functor M : S op / / C at ; see [ Gro03 , Exp. VI] Giv en a mor phism f : T / / S in S , w e shall denote by f ∗ : M ( S ) / / M ( T ) the cor responding pullback functor betw een the cor responding ﬁbers. W e shall al- wa ys assume that ( 1 S ) ∗ = 1 M ( S ) , and that for any mor phisms W g / / T f / / S in S , we hav e str uctural isomorphisms: (1.1.1.1) g ∗ f ∗ ∼ / / ( f g ) ∗ which are subject to the usual cocycle condition with respect to composition of morphisms. Giv en a mor phism f : T / / S in S , if the cor responding inv erse image functor f ∗ has a left adjoint, we shall denote it by f ] : M ( T ) / / M ( S ) . 36 In practice, S will be an adequate subcategor y of the category of noether ian schemes and P will be the class of smooth mor phisms (resp. étale mor phisms, mor phisms of ﬁnite type, separated or not necessarily separated) in S . 4 Fibred categor ies and the six functors f or malism For an y morphisms W g / / T f / / S in S such that f ∗ and g ∗ ha v e a left adjoint, w e hav e an isomor phism obtained by transposition from the isomor phism ( 1.1.1.1 ): (1.1.1.2) ( f g ) ] ∼ / / f ] g ] . Deﬁnition 1.1.2 A pr e- P -ﬁbr ed cat egor y M o ver S is a ﬁbred category M o v er S such that, f or any mor phism p : T / / S in P , the pullbac k functor p ∗ : M ( S ) / / M ( T ) has a left adjoint p ] : M ( T ) / / M ( S ) . Conv ention 1.1.3 Usually , we will consider that ( 1.1.1.1 ) and ( 1.1.1.2 ) are identities. Similarl y , we consider that f or any object S of S , ( 1 S ) ∗ = 1 M ( S ) and ( 1 S ) ] = 1 M ( S ) . 37 Example 1.1.4 Let S be an object of S . W e let P / S be the full subcategor y of the comma categor y S / S made of objects o v er S whose structural mor phism is in P . W e will usually call the objects of P / S the P -objects ov er S . Giv en a mor phism f : T / / S in S and a P -mor phism π : X / / S , w e put f ∗ ( π ) = π × S T using the proper ty (Pc) of P (see Paragraph 1.0.1 ). This deﬁnes a functor f ∗ : P / S / / P / T . Giv en tw o P -mor phisms f : T / / S and π : Y / / T , w e put f ] ( π ) = f ◦ π using the property (Pb) of P . this deﬁnes a functor f ] : P / T / / P / S . A ccording to the proper ty of pullbacks, f ] is left adjoint to f ∗ . W e thus get a pre- P -ﬁbred category P / ? : S  / / P / S . Example 1.1.5 Assume S is the categor y of noether ian schemes of ﬁnite dimension, and P = Sm . For a scheme S of S , let H • ( S ) be the pointed homotopy category of schemes o ver S deﬁned by Morel and V oev odsky in [ MV99 ]. Then according to op. cit. , H • is a pre- Sm -ﬁbred categor y o ver S . 1.1.6 Exc hang e structur es I .– Suppose given a pre- P -ﬁbred categor y M . Consider a commutativ e square of S Y q / / g   ∆ X f   T p / / S such that p hence q are P -mor phisms, we get using the identiﬁcation of con v ention 1.1.3 a canonical natural transformation E x ( ∆ ∗ ] ) : q ] g ∗ a d ( p ] , p ∗ ) / / q ] g ∗ p ∗ p ] = q ] q ∗ f ∗ p ] a d 0 ( q ] , q ∗ ) / / f ∗ p ] 37 W e can alw ay s s trictify globall y the ﬁbred category structure so that g ∗ f ∗ = ( f g ) ∗ f or any composable mor phisms f and g , and so that ( 1 S ) ∗ = 1 M ( S ) f or any object S of S ; moreo v er, for a morphism h of S such that a left adjoint of h ∗ e xists, and we can choose the left adjoint functor h ] which we f eel as the most conv enient f or us, depending on the situation we deal with. For instance, if h = 1 S , we can choose h ] to be 1 M ( S ) , and if h = f g , with f ∗ and g ∗ ha ving left adjoints, w e can choose h ] to be f ] g ] (with the unit and counit naturally induced by composition). 1 General deﬁnitions and axiomatic 5 called the exchang e transf ormation betw een q ] and g ∗ . Remar k 1.1.7 These ex change transf or mations satisfy a coherence condition with respect to the relations ( f g ) ∗ = g ∗ f ∗ and ( f g ) ] = f ] g ] . As an e xample, consider tw o commutativ e squares in S : Z q 0 / / h   Θ Y q / / g   ∆ X f   W p 0 / / T p / / S and let ∆ ◦ Θ be the commutativ e square made by the exterior maps — it is usually called the hor izontal composition of the squares. Then, the f ollo wing diagram of 2 -morphisms is commutativ e: ( q q 0 ) ] h ∗ E x ( ∆ ◦ Θ ) ∗ ] / / f ∗ ( p p 0 ) ] q ] q 0 ] h ∗ E x ( Θ ∗ ] ) / / q ] g ∗ p 0 ] E x ( ∆ ∗ ] ) / / f ∗ p ] p 0 ] T o see this, one proceeds as f ollo ws. Firs t, w e observe that, since a d 0 q is a natural transf or mation, f or each object M of M ( T ) , the square q 0 ] q 0∗ g ∗ ( M ) q 0 ] q 0∗ g ∗ ( a d q ( M )) / / a d 0 q ( g ∗ ( M ))   q 0 ] q 0∗ g ∗ p ∗ p ] ( M ) a d 0 q ( g ∗ p ∗ p ] ( M ))   g ∗ ( M ) g ∗ ( a d q ( M )) / / g ∗ p ∗ p ] ( M ) commutes. In other w ords, with a slight abuse of notations, we hav e the f ollo wing commutativ e square of functors. q 0 ] q 0∗ g ∗ a d q / / a d 0 q   q 0 ] q 0∗ g ∗ p ∗ p ] a d 0 q   g ∗ ( M ) a d q / / g ∗ p ∗ p ] W e then consider the diagram belo w , in which a d r (resp. a d 0 r ) indicates the mor phism obtained from the ob vious unit morphism (resp. counit mor phism) of the adjunction ( r ] , r ∗ ) by e v entually adding functors on the left side or on the right side, and w e can chec k easily that each cell below is commutative, proving our claim. 6 Fibred categor ies and the six functors formalism ( q q 0 ) ] h ∗ a d ( p p 0 ) / / ( q q 0 ) ] h ∗ ( p p 0 ) ] ( p p 0 ) ∗ ( q q 0 ) ] ( q q 0 ) ∗ f ∗ ( p p 0 ) ∗ a d 0 ( q q 0 ) / / f ∗ ( p p 0 ) ] q ] q 0 ] h ∗ p 0∗ p 0 ] a d p / / q ] q 0 ] h ∗ p 0∗ p ∗ p ] p 0 ] q ] q 0 ] q 0∗ q ∗ f ∗ p ] p 0 ] a d 0 q 0 / / q ] q ∗ f ∗ p ] p 0 ] q ] q 0 ] q 0∗ g ∗ p 0 ] a d p / / q ] q 0 ] q 0∗ g ∗ p ∗ p ] p 0 ] a d 0 q 0 / / q ] g ∗ p ∗ p ] p 0 ] q ] q 0 ] h ∗ a d p 0 / / q ] q 0 ] h ∗ p 0∗ p 0 ] q ] q 0 ] q 0∗ g ∗ p 0 ] a d 0 q 0 / / q ] g ∗ p 0 ] a d p / / q ] g ∗ p ∗ p ] p 0 ] q ] q ∗ f ∗ p ] p 0 ] a d 0 q / / f ∗ p ] p 0 ] 1 General deﬁnitions and axiomatic 7 Theref ore, according to our abuse of notation f or natural transf ormations, E x beha v es as a contrav ar iant functor with respect to the hor izontal composition of squares. The same is tr ue f or vertical composition of commutative squares. Remar k 1.1.8 In the sequel, we will introduce sev eral e x chang e transf or mation be- tw een v ar ious functor . W e speak of an exchang e isomorphism when the transforma- tion is an isomor phism. When onl y tw o kind of functors are inv olv ed, sa y of type a and b, w e sa y that functors of type a and functors of type b commute when the e x chang e transf ormation is an isomor phism. As an e xample (see also ne xt deﬁnition), when the e xc hange transformation E x ( ∆ ∗ ] ) is an isomorphism, we simply say that f ∗ and p ] commute — or also that f ∗ commutes with p ] . 1.1.9 U nder the setting of 1.1.6 , w e will consider the follo wing proper ty: ( P -BC) P -base chang e .– For any Cartesian square Y q / / g   ∆ X f   T p / / S such that p is a P -mor phism, the ex chang e transf ormation E x ( ∆ ∗ ] ) : q ] g ∗ / / f ∗ p ] is an isomor phism. 38 Deﬁnition 1.1.10 A P -ﬁbred categor y o v er S is a pre- P -ﬁbred category M ov er S which satisﬁes the proper ty of P -base chang e. Example 1.1.11 Consider the notations of Example 1.1.4 . Then the transitivity proper ty of pullbacks of mor phisms in P amounts to sa y that the categor y P / ? satisﬁes the P -base c hange property . Thus, P / ? is in fact a P -ﬁbred categor y , called the canonical P -ﬁbred categor y . Deﬁnition 1.1.12 A P -ﬁbred category M o ver S is complete if, f or an y mor phism f : T / / S , the pullback functor f ∗ : M ( S ) / / M ( T ) admits a right adjoint f ∗ : M ( S ) / / M ( T ) . Remar k 1.1.13 In the case where P is the class of isomor phisms, a P -ﬁbred categor y is what we usually call a biﬁbred categor y o ver S . Example 1.1.14 The pre- Sm -ﬁbred categor y H • of Example 1.1.5 is a complete Sm - ﬁbred category according to [ MV99 , p. 102-105, 108-110]. 38 In other words, f ∗ commutes with p ] . 8 Fibred categor ies and the six functors f or malism 1.1.15 Exc hang e structures II .– Let M be a complete P -ﬁbred categor y . Consider a commutativ e square Y q / / g   ∆ X f   T p / / S . W e obtain an e x chang e transf ormation: E x ( ∆ ∗ ∗ ) : p ∗ f ∗ a d ( g ∗ , g ∗ ) / / g ∗ g ∗ p ∗ f ∗ = g ∗ q ∗ f ∗ f ∗ a d 0 ( f ∗ , f ∗ ) / / g ∗ q ∗ . Assume moreo v er that p and q are P -morphism. Then we can check that E x ( ∆ ∗ ∗ ) is the transpose of the ex chang e E x ( ∆ ∗ ] ) . Thus, when ∆ is Car tesian and p is a P -mor phism, E x ( ∆ ∗ ∗ ) is an isomor phism according to ( P -BC). W e can also deﬁne an ex chang e transf ormation: E x ( ∆ ] ∗ ) : p ] g ∗ a d ( f ∗ , f ∗ ) / / f ∗ f ∗ p ] g ∗ E x ( ∆ ∗ ] ) − 1 / / f ∗ q ] g ∗ g ∗ a d 0 ( g ∗ , g ∗ ) / / f ∗ q ] . Remar k 1.1.16 As in remark 1.1.7 , we obtain coherence results for these e xc hange transf or mations. Firs t with respect to the identiﬁcations of the kind f ∗ g ∗ = ( g f ) ∗ , ( f g ) ∗ = f ∗ g ∗ , ( f g ) ] = f ] g ] . Second when sev eral ex chang e transf ormations of diﬀerent kinds are in v olv ed. As an ex ample, we consider the f ollowing commutativ e diag ram in S : Y q + + Γ 0 Z h   q 0 3 3 q 0 + + Θ X f   ∆ Y q 3 3 g   T p + + Γ Q p 0 3 3 p 0 + + S T p 3 3 Then the f ollo wing diagram of natural transf ormations is commutativ e: q ] g ∗ p 0 ∗ E x ( ∆ ∗ ] ) / / E x ( Θ ∗ ∗ )   f ∗ p ] p 0 ∗ E x ( Γ ] ∗ ) ( ( q ] q 0 ∗ h ∗ E x ( Γ 0 ] ∗ ) ( ( f ∗ p ∗ p 0 ] E x ( ∆ ∗ ∗ )   q ∗ q 0 ] h ∗ E x ( Θ ∗ ] ) / / q ∗ g ∗ p 0 ] 1 General deﬁnitions and axiomatic 9 W e leav e the veriﬁcation to the reader (it is analogous to that of R emark 1.1.7 ex cept that it inv olv es also to the compatibility of the unit and counit of an adjunction). Deﬁnition 1.1.17 Let M be a complete P -ﬁbred categor y . Consider a commutativ e square in S Y q / / g   ∆ X f   T p / / S . W e will sa y that ∆ is M -transv ersal if the e x chang e transf ormation E x ( ∆ ∗ ∗ ) : p ∗ f ∗ / / g ∗ q ∗ of 1.1.15 is an isomor phism. Giv en an admissible class of morphisms Q in S , we say that M has the transv er - sality (resp. cotr ansversality ) pr operty with respect to Q -morphisms , if, f or an y Cartesian square ∆ as abov e such that f is in Q (resp. p is in Q ), ∆ is M -transversal. Remar k 1.1.18 Assume S is a sub-categor y of the categor y of schemes. When Q is the class of smooth morphisms (resp. proper mor phisms), the cotransv ersality (resp. transv ersality) proper ty with respect to Q is usually called the smooth base chang e property (resp. proper base chang e property ). See also Deﬁnition 2.2.13 . A ccording to Paragraph 1.1.15 , we derive the f ollo wing consequence of our axioms: Proposition 1.1.19 Any complete P -ﬁbred category has the cotr ansver sality prop- erty with respect to P . Let us note f or future reference the follo wing corollar y : Corollary 1.1.20 If M is a P -ﬁbr ed categor y, then, for any monomor phism j : U / / S in P , the functor j ] is fully fait hful. If mor eov er M is complete, then the functor j ∗ is fully fait hful as well. Proof Because j is a monomor phism, we get a Car tesian square in S : U ∆ U j   U j / / S . Remark that E x ( ∆ ∗ ] ) : 1 / / j ∗ j ] is the unit of the adjunction ( j ] , j ∗ ) . Thus the P -base change proper ty sho ws that j ] is fully faithful. Assume M is complete. W e remark similar ly that E x ( ∆ ∗ ∗ ) : j ∗ j ∗ / / 1 is the counit of the adjunction ( j ∗ , j ∗ ) . Thus, the abov e proposition show s readil y that j ∗ is fully faithful.  10 Fibred categor ies and the six functors f or malism 1.1.b Monoidal structures Let C at ⊗ be the sub- 2 -category of C at made of symmetr ic monoidal categories whose 1 -mor phisms are (strong) symmetric monoidal functors and 2 -mor phisms are symmetric monoidal transf ormations. Deﬁnition 1.1.21 A monoidal pre- P -ﬁbr ed category ov er S is a 2 -functor M : S / / C at ⊗ such that M is a pre- P -ﬁbred category . In other w ords, M is a pre- P -ﬁbred categor y such that each of its ﬁbers M ( S ) is endo w ed with a structure of a monoidal category , and any pullback mor phism f ∗ is monoidal, with the obvious coherent s tructures. For an object S of S , w e will usually denote by ⊗ S (resp. 1 S ) the tensor product (resp. unit) of M ( S ) . In par ticular , we then ha v e the f ollo wing natural isomor phisms: • f or a mor phism f : T / / S in S , and objects M , N of M ( S ) , f ∗ ( M ) ⊗ T f ∗ ( N ) ∼ / / f ∗ ( M ⊗ S N ) ; • f or a mor phism f : T / / S in S , f ∗ ( 1 S ) ∼ / / 1 T . Conv ention 1.1.22 As in conv ention 1.1.3 , we will write formula as though these structural isomor phisms are identities. Example 1.1.23 Consider the notations of Example 1.1.4 . Using the proper ties (Pb) and (Pc) of P (see Paragraph 1.0.1 ), f or two S -objects X and Y in P / S , the Car tesian product X × S Y is an object of P / S . This deﬁnes a symmetric monoidal structure on P / S with unit the tr ivial S -object S . Moreov er , the functor f ∗ deﬁned in Example 1.1.4 is monoidal. Thus, the pre- P -ﬁbred categor y P / ? is in fact monoidal. 1.1.24 Monoidal exchang e structur es I . Let M be a monoidal pre- P -ﬁbred categor y o v er S . Consider a P -mor phism f : T / / S , and M (resp. N ) an object of M ( T ) (resp. M ( S ) ). W e get a mor phism in M ( S ) E x ( f ∗ ] , ⊗ ) : f ] ( M ⊗ T f ∗ ( N )) / / f ] ( M ) ⊗ S N as the composition f ] ( M ⊗ T f ∗ ( N )) / / f ] ( f ∗ f ] ( M ) ⊗ T f ∗ ( N )) ' f ] f ∗ ( f ] ( M ) ⊗ S N ) / / f ] ( M ) ⊗ S N . 1 General deﬁnitions and axiomatic 11 This map is natural in M and N . It will be called the exc hang e tr ansformation betw een f ] and ⊗ T . Remark also that the functor f ] , as a left adjoint of a symmetric monoidal functor , is colax symmetr ic monoidal: f or any objects M and N of M ( T ) , there is a canonical morphism (1.1.24.1) f ] ( M ) ⊗ S f ] ( N ) / / f ] ( M ⊗ T N ) natural in M and N , as w ell as a natural map (1.1.24.2) f ] ( 1 T ) / / 1 S . Remar k 1.1.25 As in remark 1.1.7 , the preceding ex chang e transf or mations satisfy a coherence condition for composable mor phisms W g / / T f / / S . W e get in f act a commutativ e diagram: ( f g ) ]  M ⊗ S ( f g ) ∗ ( N )  E x (( f g ) ∗ ] , ⊗ ) / /  ( f g ) ] ( M )  ⊗ W N f ] g ]  M ⊗ S g ∗ f ∗ ( N )  E x ( g ∗ ] , ⊗ ) / / f ]  g ] ( M ) ⊗ T f ∗ ( N )  E x ( f ∗ ] , ⊗ ) / /  f ] g ] ( M )  ⊗ W N As in remark 1.1.16 , there is also a coherence relation when diﬀerent kinds of e x chang e transf or mations are inv olv ed. Consider a commutative square in S Y q / / g   ∆ X f   T p / / S such that p and q are P -mor phisms. Then the f ollo wing diagram is commutative: q ] g ∗ ( M ⊗ T p ∗ N ) E x ( ∆ ∗ ] ) / / f ∗ p ] ( M ⊗ T p ∗ N ) E x ( p ∗ ] , ⊗ ) / / f ∗ ( p ] M ⊗ S N ) q ] ( g ∗ M ⊗ Y q ∗ f ∗ N ) E x ( q ∗ ] , ⊗ ) / / ( q ] g ∗ M ) ⊗ X f ∗ N E x ( ∆ ∗ ] ) / / ( f ∗ p ] M ) ⊗ X f ∗ N W e leav e the v eriﬁcation to the reader . 1.1.26 U nder the assumptions of 1.1.24 , w e will consider the follo wing proper ty: ( P -PF) P -projection f ormula .– For any P -mor phism f : T / / S the e xc hange transf or mation E x ( f ] , ⊗ T ) : f ] ( M ⊗ T f ∗ ( N )) / / f ] ( M ) ⊗ S N is an isomor phism f or all M and N . 12 Fibred categor ies and the six functors f ormalism Deﬁnition 1.1.27 A monoidal P -ﬁbred category o v er S is a monoidal pre- P - ﬁbred category M : S op / / C at ⊗ o v er S whic h satisﬁes the P -projection for - mula. Example 1.1.28 Consider the canonical monoidal weak P -ﬁbred categor y P / ? (see Example 1.1.23 ). The transitivity proper ty of pullbacks implies readily that P / ? satisﬁes the proper ty ( P -PF). Thus, P / ? is in fact a monoidal P -ﬁbred categor y called canonical . Deﬁnition 1.1.29 A monoidal P -ﬁbred categor y M o ver S is complete if it satisﬁes the f ollo wing conditions: 1. M is complete as a P -ﬁbred categor y . 2. For an y object S of S , the monoidal category M ( S ) is closed (i.e. has an internal Hom). In this case, w e will usually denote b y Hom S the inter nal Hom in M ( S ) , so that we ha v e natural bijections Hom M ( S ) ( A ⊗ S B , C ) ' Hom M ( S ) ( A , Hom S ( B , C )) . Example 1.1.30 The P -ﬁbred category H • of Example 1.1.14 is in fact a complete monoidal P -ﬁbred category . The tensor product is giv en by the smash product (see [ MV99 ]). 1.1.31 Monoidal exc hang e structures II .– Let M be a complete monoidal P -ﬁbred category . Consider a mor phism f : T / / S in S . Then w e obtain an ex chang e transf orma- tion: E x ( f ∗ ∗ , ⊗ S ) : ( f ∗ M ) ⊗ S N a d ( f ∗ , f ∗ ) / / f ∗ f ∗  ( f ∗ M ) ⊗ S N  = f ∗  ( f ∗ f ∗ M ) ⊗ T f ∗ N  a d 0 ( f ∗ , f ∗ ) / / f ∗ ( M ⊗ T f ∗ N ) . Remar k 1.1.32 As in remark 1.1.25 , these e xc hange transf or mations are compatible with the identiﬁcations ( f g ) ∗ = f ∗ g ∗ and ( f g ) ∗ = g ∗ f ∗ . Moreov er , there is a coher- ence relation when composing the ex chang e transf or mations of the kind E x ( f ∗ ∗ , ⊗ ) with e xc hange transf or mations of the kind E x ( ∆ ∗ ∗ ) as in loc. cit. Finall y , note that there is another kind of coherence relations inv olving E x ( f ∗ ∗ , ⊗ ) , E x ( ∆ ∗ ] ) (resp. E x ( f ∗ ] , ⊗ ) ) and E x ( ∆ ] ∗ ) . W e lea v e the f or mulation of these coherence relations to the reader , on the model of the preceding ones. 1.1.33 Monoidal exchang e structures III .– Let M be a complete monoidal P -ﬁbred category and f : T / / S be a mor phism in S . Because f ∗ is monoidal, we get b y adjunction a canonical isomor phism Hom S ( M , f ∗ N ) / / f ∗ Hom T ( f ∗ M , N ) . 1 General deﬁnitions and axiomatic 13 Assume that f is a P -mor phism. Then from the P -projection f ormula, w e get by adjunction tw o canonical isomor phisms: f ∗ Hom S ( M , N ) / / Hom T ( f ∗ M , f ∗ N ) , Hom S ( f ] M , N ) / / f ∗ Hom T ( M , f ∗ N ) These isomor phisms are genericall y called exchang e isomorphisms . 1.1.c Geometric sections 1.1.34 Consider a complete monoidal P -ﬁbred category M . Let S be a scheme. For an y P -mor phism p : X / / S , w e put M S ( X ) : = p ] ( 1 X ) . A ccording to our conv entions, this object is identiﬁed with p ] p ∗ ( 1 S ) . As the P - ﬁbred categor y M is complete, the functor p ] p ∗ is left adjoint to p ∗ p ∗ . Consider a commutativ e diagram of schemes in S : Y q   f / / X p   S such that p and q are in P . In other w ords, f is a mor phism in the category P / S of Example 1.1.4 . Then w e get a natural transf ormation of functors: (1.1.34.1) p ∗ p ∗ a d ( f ∗ , f ∗ ) / / p ∗ f ∗ f ∗ p ∗ = q ∗ q ∗ . By adjunction, one deduces a natural transformation: q ] q ∗ / / p ] p ∗ which gives a mor phism M S ( Y ) f ∗ / / M S ( X ) . One can check that the relation f ∗ g ∗ = ( f g ) ∗ holds — by reducing to the same asser tion for the map ( 1.1.34.1 ) which f ollo ws b y a s tandard 2 -functor iality argument. Therefore, one has obtained a co variant functor M S : P / S / / M . Consider a Car tesian square in S Y g / / q   ∆ X p   T f / / S such that p is a P -mor phism. With the notations of Example 1.1.4 , Y = f ∗ ( X ) . Then w e get a natural e x chang e transf ormation 14 Fibred categor ies and the six functors formalism E x ( M T , f ∗ ) : M T ( f ∗ ( X )) = q ] ( 1 Y ) = q ] g ∗ ( 1 X ) E x ( ∆ ∗ ] ) / / f ∗ p ] ( 1 X ) = f ∗ M S ( X ) . In other w ords, M deﬁnes a lax natural transf ormation P / ? / / M . Consider P -mor phisms p : X / / S , q : Y / / S . Let Z = X × S Y be the Car tesian product and consider the Car tesian square: Z p 0 / / q 0   Θ Y q   X p / / S . Using the e xc hange transf or mations of the preceding paragraph, we g et a canonical morphism E x ( M S , ⊗ S ) : M S ( X × S Y ) / / M S ( X ) ⊗ S M S ( Y ) as the composition M S ( X × S Y ) = p ] q 0 ] p 0∗ ( 1 Y ) E x ( Θ ∗ ] ) / / p ] p ∗ q ] ( 1 Y ) = p ] ( 1 X ⊗ X p ∗ q ] ( 1 Y )) E x ( p ] , ⊗ X ) / / p ] ( 1 X ) ⊗ S q ] ( 1 Y ) = M S ( X ) ⊗ S M S ( Y ) . In other w ords, the functor M S is symmetric colax monoidal. Remark ﬁnally that for any P -morphism p : T / / S , and any P -object Y ov er T , w e obtain according to conv ention an identiﬁcation p ] M T ( Y ) = M S ( Y ) . Deﬁnition 1.1.35 Giv en a complete monoidal P -ﬁbred category M o ver S , the lax natural transf or mation M : P / ? / / M constructed abov e will be called the g eometric sections of M . The f ollo wing lemma is obvious from the deﬁnitions abo v e: Lemma 1.1.36 let M be a complete monoidal P -ﬁbred category. Let M : P / ? / / M be the g eometric sections of M . Then: (i) F or any mor phism f : T / / S in S , the exc hang e E x ( M T , f ∗ ) deﬁned abov e is an isomorphism. (ii) F or any scheme S , the exchang e E x ( M S , ⊗ S ) deﬁned abov e is an isomorphism. In other wor ds, M is a Cartesian functor and M S is a (strong) symmetric monoidal functor . 1.1.37 In the situation of the lemma we thus obtain the f ollo wing isomor phisms: • f ∗ M S ( X ) ' M T ( X × S T ) , • p ] M T ( Y ) ' M S ( Y ) , • M S ( X × S Y ) ' M S ( X ) ⊗ S M S ( Y ) , whenev er it makes sense. 1 General deﬁnitions and axiomatic 15 1.1.d T wists 1.1.38 Let M be a pre- P -ﬁbred category of S . Recall that a Car tesian section of M ( i.e. a Car tesian functor A : S / / M ) is the data of an object A S of M ( S ) for each object S of S and of isomor phisms f ∗ ( A S ) ∼ / / A T f or each morphism f : T / / S , subject to coherence identities; see [ Gro03 , Exp. VI]. If M is monoidal, the tensor product of two Car tesian sections is deﬁned termwise. Deﬁnition 1.1.39 let M be a monoidal pre- P -ﬁbred categor y . A set of twists τ f or M is a set of Car tesian sections of M which is stable by tensor product (up to isomorphism), and contains the unit 1 . For shor t, when M is endo wed with a set of twists τ , we say also that M is τ -twisted . 1.1.40 Let M be a monoidal pre- P -ﬁbred categor y endow ed with a set of twists τ . The tensor product on τ induces a monoid str ucture that we will denote by + (the unit object of τ will be wr itten 0 ). Consider an object i ∈ τ . For any object S of S , we thus obtain an object t ( i ) S in M ( S ) associated with i . Giv en an y object M of M ( S ) , w e simply put: M { i } = M ⊗ S i S and call this object the twist of M b y i . W e also deﬁne M { 0 } = M . For an y i , j ∈ τ , and an y object M of M ( S ) , we deﬁne M { i + j } = ( M { i } ) { j } . Giv en a morphism f : T / / S , an object M of M ( S ) and a twist i ∈ τ , w e also obtain f ∗ ( M { i } ) = ( f ∗ M ) { i } . If f is a P -mor phism, f or any object M of M ( T ) , the e x chang e transf ormation E x ( f ∗ ] , ⊗ T ) of paragraph 1.1.6 induces a canonical mor - phism E x ( f ] , { i } ) : f ] ( M { i } ) / / ( f ] M ) { i } . W e will sa y that f ] commutes with τ -twists (or simply twists when τ is clear) if f or an y i ∈ τ , the natural transf ormation E x ( f ] , { i } ) is an isomor phism. Deﬁnition 1.1.41 Let M be a complete monoidal P -ﬁbred categor y with a set of twists τ and M : P / ? / / M be the geometric sections of M . W e say M is τ -g enerat ed if f or any object S of S , the famil y of functors Hom M ( S ) ( M S ( X ){ i } , −) : M ( S ) / / S et inde x ed by a P -object X / S and an element i ∈ τ is conser v ativ e. Of course, we do not e x clude the case where τ is tr ivial, but then, w e shall simply sa y that M is geome trically g enerated . W e shall frequentl y use the f ollowing proposition to character ize complete monoidal P -ﬁbred categor ies o ver S : 16 Fibred categories and the six functors formalism Proposition 1.1.42 Let M : S / / C at ⊗ be a 2 -functor suc h that : 1. F or any P -morphism f : T / / S , the pullbac k functor f ∗ : M ( S ) / / M ( T ) is monoidal and admits a lef t adjoint f ] in C . 2. F or any morphism f : T / / S , the pullback functor f ∗ : M ( S ) / / M ( T ) admits a right adjoint f ∗ in C . W e consider M as a complet e monoidal P -ﬁbred categor y and denot e by M : P / ? / / M its associated g eometric sections. Suppose given a set of twists τ suc h that M is τ -g enerat ed. Then, the f ollowing assertions are equivalent : (i) M satisﬁes properties ( P -BC) and ( P -PF) ( i.e. M is a complete monoidal P -ﬁbred categor y.) (ii) a. M is a Cartesian functor . b. F or any object S of S , M S is (str ong ly) monoidal. c. F or any P -morphism f , f ] commutes with τ -twists. Proof ( i ) ⇒ ( ii ) : This is obvious (see Lemma 1.1.36 ). ( ii ) ⇒ ( i ) : W e use the f ollo wing easy lemma: Lemma 1.1.43 Let C 1 and C 2 be categories, F , G : C 1 / / C 2 be two lef t adjoint functors, and η : F / / G be a natural transf ormation. Let G be a class of objects of C 1 whic h is g enerating in the sense that the family of functors Hom C 1 ( X , −) for X in G is conser v ative. Then the f ollowing conditions ar e equiv alent: 1. η is an isomor phism. 2. F or all X in G , η X is an isomorphism.  Giv en this lemma, to pro v e property ( P -BC), w e are reduced to check that the e x chang e transf or mation E x ( ∆ ∗ ] ) is an isomor phism when ev aluated on an object M T ( U ){ i } f or an object U of P / T and a twist i ∈ τ . Then it f ollow s from (ii), 1.1.40 and Example 1.1.11 . 39 T o prov e property ( P -PF), w e proceed in tw o steps ﬁrs t pro ving the case M = M T ( U ){ i } and N an y object of M ( S ) using the same argument as abov e with the help of 1.1.28 . Then, w e can prov e the general case b y another application of the same argument.  Suppose giv en a complete monoidal P -ﬁbred categor y M with a set of twis ts τ . Let f : T / / S be a morphism of S . Then the e x chang e transf or mation 1.1.31 induces f or an y i ∈ τ an ex chang e transf or mation E x ( f ∗ , { i } ) : ( f ∗ M ) { i } / / f ∗ ( M { i } ) . 39 The cautious reader will use remark 1.1.7 to chec k that the corresponding map M X ( U × T Y ) { i } / / M X ( U × T Y ) { i } is the identity . 1 General deﬁnitions and axiomatic 17 Deﬁnition 1.1.44 In the situation abo v e, we say that f ∗ commutes with τ -twists (or simply with twists when τ is clear) if, f or an y i ∈ τ , the ex chang e transf or mation E x ( f ∗ , { i } ) is an isomor phism. It will frequentl y happen that twists are ⊗ -in v er tible. Then f ∗ commutes with twists as its r ight adjoint does. 1.2 Morphisms of P -ﬁbred categories 1.2.a General case 1.2.1 Consider tw o P -ﬁbred categories M and M 0 o v er S , as well as a Car tesian functor ϕ ∗ : M / / M 0 betw een the underl ying ﬁbred categor ies: f or any object S of S , we hav e a functor ϕ ∗ S : M ( S ) / / M 0 ( S ) , and f or an y map f : T / / S in S , w e hav e an isomor phism of functors c f M ( S )   c f ϕ ∗ S / / f ∗   M 0 ( S ) f ∗   M ( T ) ϕ ∗ T / / M 0 ( T ) c f : f ∗ ϕ ∗ S ∼ / / ϕ ∗ T f ∗ (1.2.1.1) satisfying some cocy cle condition with respect to composition in S . For any P -mor phism p : T / / S , w e construct an e x chang e mor phism E x ( p ] , ϕ ∗ ) : p ] ϕ ∗ T / / ϕ ∗ S p ] as the composition p ] ϕ ∗ T a d ( p ] , p ∗ ) / / p ] ϕ ∗ T p ∗ p ] c − 1 p / / p ] p ∗ ϕ ∗ S p ] a d 0 ( p ] , p ∗ ) / / ϕ ∗ S p ] . Deﬁnition 1.2.2 Consider the situation abov e. W e say that the Car tesian functor ϕ ∗ : M / / M 0 is a mor phism of P -ﬁbr ed categories if, f or any P -morphism p , the e xc hange transf or mation E x ( p ] , ϕ ∗ ) is an isomor phism. Example 1.2.3 If M is a monoidal P -ﬁbred category , then the g eometr ic sections M : P / ? / / M is a mor phism of P -ﬁbred categories ( 1.1.36 ). Deﬁnition 1.2.4 Let M and M 0 be two complete P -ﬁbred categor ies. A morphism of complet e P -ﬁbr ed categories is a mor phism of P -ﬁbred categor ies 18 Fibred categor ies and the six functors f or malism ϕ ∗ : M / / M 0 such that, for an y object S of S , the functor ϕ ∗ S : M ( S ) / / M 0 ( S ) has a r ight adjoint ϕ ∗ , S : M 0 ( S ) / / M ( S ) . When we want to indicate a notation f or the r ight adjoint of a mor phism as abo v e, w e use the wr iting ϕ ∗ : M / / o o N : ϕ ∗ the left adjoint being in the left hand side. 1.2.5 Exc hang e structur es III . Consider a mor phism ϕ ∗ : M / / M 0 of complete P -ﬁbred categor ies. Then f or an y mor phism f : T / / S in S , w e deﬁne ex chang e transf ormations E x ( ϕ ∗ , f ∗ ) : ϕ ∗ S f ∗ / / f ∗ ϕ ∗ T , (1.2.5.1) E x ( f ∗ , ϕ ∗ ) : f ∗ ϕ ∗ , S / / ϕ ∗ , T f ∗ , (1.2.5.2) as the respectiv e compositions ϕ ∗ S f ∗ a d ( f ∗ , f ∗ ) / / f ∗ f ∗ ϕ ∗ S f ∗ ' f ∗ ϕ ∗ T f ∗ f ∗ a d 0 ( f ∗ , f ∗ ) / / f ∗ ϕ ∗ T , f ∗ ϕ ∗ , S a d ( f ∗ , f ∗ ) / / f ∗ ϕ ∗ , S f ∗ f ∗ ' f ∗ f ∗ ϕ ∗ , T f ∗ a d 0 ( f ∗ , f ∗ ) / / ϕ ∗ , T f ∗ . Remar k 1.2.6 W e warn the reader that ϕ ∗ : M 0 / / M is not a Car tesian functor in general, meaning that the e x chang e transf or mation E x ( f ∗ , ϕ ∗ ) is not necessar ily an isomorphism, ev en when f is a P -mor phism. 1.2.b Monoidal case Deﬁnition 1.2.7 Let M and M 0 be monoidal P -ﬁbred categories. A morphism of monoidal P -ﬁbred categories is a mor phism ϕ ∗ : M / / M 0 of P -ﬁbred categor ies such that f or any object S of S , the functor ϕ ∗ S : M ( X ) / / N ( S ) has the structure of a (strong) symmetric monoidal functor, and such that the structural isomorphisms ( 1.2.1.1 ) are isomor phisms of symmetr ic monoidal functors. In the case where M and M 0 are complete monoidal P -ﬁbred categor ies, we shall sa y that suc h a mor phism ϕ ∗ is a mor phism of complet e monoidal P -ﬁbred categories if ϕ ∗ is also a mor phism of complete P -ﬁbred categories. Remar k 1.2.8 If we denote by M (− , M ) and M (− , M 0 ) the g eometr ic sections of M and M 0 respectiv ely , w e hav e a natural identiﬁcation: ϕ ∗ S ( M S ( X , M )) ' M S ( X , M 0 ) . 1 General deﬁnitions and axiomatic 19 1.2.9 Monoidal exc hang e structures IV . Consider a a mor phism ϕ ∗ : M / / M 0 of complete monoidal P -ﬁbred categories. For objects M (resp. N ) of M ( S ) (resp. M 0 ( S ) ), we deﬁne an e x chang e transf ormation E x ( ϕ ∗ , ⊗ , ϕ ∗ ) : ( ϕ ∗ , S M ) ⊗ S N / / ϕ ∗ , S ( M ⊗ T ϕ ∗ S N ) , natural in M and N , as the f ollo wing composite ( ϕ ∗ , S M ) ⊗ S N a d ( ϕ ∗ , ϕ ∗ ) / / ϕ ∗ , S ϕ ∗ S (( ϕ ∗ , S M ) ⊗ S N ) = ϕ ∗ , S (( ϕ ∗ S ϕ ∗ , S M ) ⊗ T ϕ ∗ S N ) a d 0 ( ϕ ∗ , ϕ ∗ ) / / ϕ ∗ , S ( M ⊗ T ϕ ∗ S N ) . As in remark 1.1.32 , we g et coherence relations between the various e x chang e transf or mations associated with a mor phism of monoidal P -ﬁbred categor ies. W e left the f ormulation to the reader . Note also that, because ϕ ∗ is monoidal, we get by adjunction a canonical isomor - phism: Hom M ( S ) ( M , ϕ ∗ , S M 0 ) ∼ / / ϕ ∗ , S Hom M 0 ( S ) ( ϕ ∗ S M , M 0 ) . 1.2.10 Consider tw o monoidal P -ﬁbred categories M , M 0 and a Car tesian functor ϕ ∗ : M / / M 0 such that, f or an y scheme S , ϕ ∗ S : M ( S ) / / M 0 ( S ) is monoidal. Giv en a Car tesian section K = ( K S ) S ∈ S of M , we obtain f or any mor phism f : T / / S in S a canonical map f ∗ ϕ ∗ S ( K S ) = ϕ ∗ T ( f ∗ ( K S )) / / ϕ ∗ T ( K T ) which deﬁnes a Car tesian section of M 0 , which w e denote by ϕ ∗ ( K ) . Deﬁnition 1.2.11 Let ( M , τ ) and ( M 0 , τ 0 ) be twisted monoidal P -ﬁbred categor ies. Let ϕ ∗ : M / / M 0 be a Car tesian functor as abo v e (resp. a mor phism of monoidal P -ﬁbred categor ies). W e say that ϕ ∗ : ( M , τ ) / / ( M 0 , τ 0 ) is compatible with twists if f or an y i ∈ τ , the Car tesian section ϕ ∗ ( i ) is in τ 0 (up to isomor phism in M 0 ). Remar k 1.2.12 In par ticular , ϕ ∗ induces a map τ / / τ 0 (if we consider the isomor - phism classes of objects). Moreov er , for an y object K of M ( S ) and any twist i ∈ τ , w e get an identiﬁcation: ϕ ∗ S ( K { i } ) ' ( ϕ ∗ S K ) { ϕ ∗ ( i ) } . Moreo v er , the e x chang e transf or mation E x ( ϕ ∗ , ⊗ ) induces an e x chang e: E x ( ϕ ∗ , { i } ) : ϕ ∗ , S ( K ) { i } / / ϕ ∗ , S  K { ϕ ∗ ( i ) }  . When this transf or mation is an isomor phism for any twist i ∈ τ , we say that ϕ ∗ commutes with twists. Note ﬁnally that Lemma 1.1.43 allo ws to pro ve, as f or Proposition 1.1.42 , the f ollowing useful lemma: 20 Fibred categories and the six functors formalism Lemma 1.2.13 Consider two complet e monoidal P -ﬁbr ed categories M , M 0 and denot e by M (− , M ) and M (− , M 0 ) their r espective g eometric sections. Let ϕ ∗ : M / / M 0 be a Cartesian functor suc h that 1. F or any scheme S , ϕ ∗ S : M ( S ) / / M 0 ( S ) is monoidal. 2. F or any scheme S , ϕ ∗ S admits a right adjoint ϕ ∗ , S . Assume M (r esp. M 0 ) is τ -g enerated (resp. τ 0 -twisted) and that ϕ ∗ induces a sur - jectiv e map from the set of isomorphism classes of τ -twists to the set of isomorphism classes of τ 0 -twists. Then the f ollowing conditions are equivalent : (i) ϕ ∗ is a morphism of complete monoidal P -ﬁbr ed categories. (ii) F or any object X of P / S , the exchang e transf ormation (cf. 1.2.1 ) ϕ ∗ M S ( X , M ) / / M S ( X , M 0 ) is an isomorphism. 1.3 Structures on P -ﬁbred categories 1.3.a Abstract deﬁnition 1.3.1 W e ﬁx a sub- 2 -categor y C of C a t with the f ollowing proper ties 40 : (1) the 2 -functor C at / / C at 0 , A  / / A op sends C to C 0 , where C 0 denotes the 2 -categor y whose objects and maps are those of C and whose 2 -mor phisms are the 2 -mor phisms of C , put in the rev erse direction. (2) C is closed under adjunction: for an y functor u : A / / B in C , if a functor v : B / / A is a right adjoint or a left adjoint to u , then v is in C . (3) the 2 -mor phisms of C are closed by transposition: if u : A / / o o B : v and u 0 : A / / o o B : v 0 are tw o adjunctions in C (with the left adjoints on the left-hand side), a nat- ural transf ormation u / / u 0 is in C if and only if the cor responding natural transf or mation v 0 / / v is in C . W e can then deﬁne and manipulate C -structured P -ﬁbred categories as follo ws. Deﬁnition 1.3.2 A C -structured P -ﬁbred category (resp. C -structured complet e P -ﬁbred category ) M ov er S is simply a P -ﬁbred categor y (resp. a complete 40 See the follo wing sections for examples. 1 General deﬁnitions and axiomatic 21 P -ﬁbred categor y) whose underl ying 2 -functor M : S op / / C at f actors through C . If M and M 0 are C -structured ﬁbred categor ies o ver S , a Car tesian functor M / / M 0 is C -structured if the functors M ( S ) / / M 0 ( S ) are in C f or any object S of S , and if all the structural 2 -mor phisms ( 1.2.1.1 ) are in C as well. Deﬁnition 1.3.3 A mor phism of C -structured P -ﬁbred categor ies (resp. C -struct- ured complete P -ﬁbred categor ies) is a mor phism of P -ﬁbred categories (resp. of complete P -ﬁbred categor ies) which is C -structured as a Car tesian functor . 1.3.4 Consider a 2 -categor y C as in the paragraph 1.3.1 . In order to deal with the monoidal case, we will consider also a sub- 2 -categor y C ⊗ of C such that: 1. The objects of C ⊗ are objects of C equipped with a symmetr ic monoidal structure; 2. the 1 -mor phisms of C ⊗ are ex actly the 1 -mor phisms of C which are symmetric monoidal as functors; 3. the 2 -mor phisms of C ⊗ are ex actly the 2 -mor phisms of C which are symmetric monoidal as natural transf or mations. Note that C ⊗ satisﬁes condition (1) of 1.3.1 , but it does not satisfy conditions (2) and (3) in general. Instead, we get the f ollowing proper ties: (2 0 ) If u : A / / B is a functor in C ⊗ , a r ight (resp. left) adjoint v is a lax 41 (resp. colax) monoidal functor in C . (3 0 ) Consider adjunctions u : A / / o o B : v and u 0 : A / / o o B : v 0 in C (with the left adjoints on the left-hand side). If u / / u 0 (resp. v / / v 0 ) is a 2 -mor phism in C ⊗ then v / / v 0 (resp. u / / u 0 ) is a 2 -mor phism in C which is a symmetr ic monoidal transf or mation of lax (resp. colax) monoidal functors. W e thus adopt the f ollo wing deﬁnition: Deﬁnition 1.3.5 A ( C , C ⊗ ) -structur ed monoidal P -ﬁbred category (resp. a ( C , C ⊗ ) - structur ed comple te monoidal P -ﬁbred categor y ) is simpl y a monoidal P -ﬁbred category (resp. a complete monoidal P -ﬁbred categor y) whose underl ying 2 -functor M : S op / / C at ⊗ factors through C ⊗ . Mor phisms of such objects are deﬁned in the same wa y . Note that, with the hypothesis made on C , all the ex chang e natural transformations deﬁned in the preceding parag raphs lie in C and satisfy the appropr iate coherence property with respect to the monoidal str ucture. 41 For any object a , a 0 in A , F is lax if there e xists a structural map F ( a ) ⊗ F ( a 0 ) ( 1 ) / / F ( a ⊗ a 0 ) satisfying coherence relations (see [ Mac98 , XI. 2]). Colax is deﬁned by rev ersing the ar ro w ( 1 ) . 22 Fibred categories and the six functors formalism 1.3.b The abelian case 1.3.6 Let A b be the sub- 2 -categor y of C at made of the abelian categories, with the additiv e functors as 1 -mor phisms, and the natural transformations as 2 -mor phisms. Obviousl y , it satisﬁes proper ties of 1.3.1 . When we will apply one of the deﬁnitions 1.3.2 , 1.3.3 to the case C = A b , w e will use the simple adjectiv e abelian f or A b - structured. This allow s speaking of morphisms of abelian P -ﬁbred categories. Let A b ⊗ be the sub- 2 -categor y of A b made of the abelian monoidal categories, with 1 -mor phisms the symmetr ic monoidal additiv e functors and 2 -morphisms the symmetric monoidal natural transf or mations. It satisﬁes the hypothesis of paragraph 1.3.4 . When we will apply deﬁnition 1.3.5 to the case of ( A b , A b ⊗ ) , we will use the simple expression abelian monoidal f or ( A b , A b ⊗ ) -structured monoidal. This allo ws speaking of mor phisms of abelian monoidal P -ﬁbr ed categories. Lemma 1.3.7 Consider an abelian P -ﬁbred category A such that f or any object S of S , A ( S ) is a Grot hendiec k abelian category. Then the follo wing conditions are equiv alent : (i) A is complete. (ii) F or any mor phism f : T / / S in S , f ∗ commutes with sums. If in addition, A is monoidal, the f ollowing conditions ar e equivalent : (i 0 ) A is monoidal complete. (ii 0 ) (a) F or any mor phism f : T / / S in S , f ∗ is right exact. (b) F or any object S of S , the bifunctor ⊗ S is right exact. In vie w of this lemma, w e adopt the follo wing deﬁnition: Deﬁnition 1.3.8 A Grothendieck abelian (resp. Grothendieck abelian monoidal) P - ﬁbred categor y A ov er S is an abelian P -ﬁbred categor y which is complete (resp. complete monoidal) and such that f or an y scheme S , A ( S ) is a Grothendieck abelian category . Remar k 1.3.9 Let A be a Grothendieck abelian monoidal P -ﬁbred categor y . Con- v entionally , we will denote b y M S (− , A ) its g eometr ic sections. N ote that if A is τ -twisted, then an y object of A is a quotient of a direct sum of objects of shape M S ( X , A ) { i } f or a P -object X / S and a twist i ∈ τ . 1.3.10 Consider an abelian categor y A which admits small sums. Recall the f ollow - ing deﬁnition: An object X of T is ﬁnitely present ed if the functor Hom T ( X , −) commutes with small ﬁlter ing colimits. A essentially small G of objects of A is called generating if f or an y object A of A there e xists an epimor phism of the f orm: Ê i ∈ I G i / / A where ( G i ) i ∈ I is a famil y of objects if G . 1 General deﬁnitions and axiomatic 23 Deﬁnition 1.3.11 Let A be an abelian P -ﬁbred categor y ov er S . Giv en a set of twists τ of A , we sa y A is ﬁnitely τ -present ed if f or any object S of S , f or any P -object X / S and any twist i ∈ τ , the object M S ( X ){ i } is ﬁnitely presented and the class of such objects f or m an essentiall y small g enerating famil y of A ( S ) . 1.3.c The triangulated case 1.3.12 Let T r i be the sub- 2 -categor y of C a t made of the triangulated categor ies, with the tr iangulated functors as 1 -mor phisms, and the tr iangulated natural trans- f or mations as 2 -mor phisms. Then T r i satisﬁes the proper ties of 1.3.1 (proper ty (2) can be f ound f or instance in [ A y o07a , Lemma 2.1.23], and we leav e proper ty (3) as an ex ercise for the reader). When we will apply one of the deﬁnitions 1.3.2 , 1.3.3 to the case C = T r i , w e will use the simple adjectiv e triangulated f or T r i -structured. This allo ws speaking of morphisms of triangulated P -ﬁbr ed categories . Let T r i ⊗ be the sub- 2 -categor y of T r i made of the triangulated monoidal categories, with 1 -mor phisms the symmetr ic monoidal tr iangulated functors and 2 -morphisms the symmetr ic monoidal natural transf or mations. It satisﬁes the hy - pothesis of paragraph 1.3.4 . When w e will apply deﬁnition 1.3.5 to the case of ( T r i , T r i ⊗ ) , we will use the expression triangulated monoidal for ( T r i , T r i ⊗ ) - structured monoidal. This allow s speaking of morphisms of triangulated monoidal P -ﬁbred categories . Conv ention 1.3.13 The set of twists of a triangulated monoidal P -ﬁbred categor y T will alw ay s be of the f or m Z × τ , b y which w e mean that τ is a set of twists, while Z × τ is the closure of τ b y suspension functors [ n ] , n ∈ Z . In the notation, w e shall often make the abuse of only indicating τ . In par ticular , the expression T is τ -generated will mean conv entionally that T is ( Z × τ ) -generated in the sense of deﬁnition 1.1.41 . 1.3.14 Consider a tr iangulated categor y T which admits small sums. R ecall the f ollowing deﬁnitions: An object X of T is called compact if the functor Hom T ( X , −) commutes with small sums. A class G of objects of T is called generating if the famil y of functor Hom T ( X [ n ] , −) , X ∈ G , n ∈ Z , is conser vativ e. The triangulated categor y T is called compactly gener ated if there e xists a generating set G of compact objects of T . This proper ty of being compact has been generalized b y A. Neeman to the proper ty of being α -small for some cardinal α ( cf. [ Nee01 , 4.1.1]) — recall compact= ℵ 0 -small. Then the proper ty of being compactly generated has been generalized by Neeman to the proper ty of being w ell g enerated ; see [ Kra01 ] f or a conv enient characterization of well g enerated tr iangulated categories. Deﬁnition 1.3.15 Let T be a tr iangulated P -ﬁbred category o v er S . W e sa y that T is compactly g enerated (resp. well g enerat ed ) if for any object S of S , T ( S ) admits small sums and is compactly g enerated (resp. well g enerated). 24 Fibred categories and the six functors formalism Giv en a set of twists τ f or T , w e sa y T is compactly τ -g enerat ed if it is compactly generated in the abov e sense and for any P -object X / S , any twist i ∈ τ , M S ( X ){ i } is compact. 1.3.16 For a triangulated categor y T which has small sums, giv en a f amily G of objects of T , we denote by h G i the localizing subcategory of T generated by G , i.e. h G i is the smallest triangulated full subcategory of T which is stable b y small sums and which contains all the objects in G . Recall that, in the case T is well generated (e.g. if T compactly g enerated), then the famil y G g enerates T (in the sense that the famil y of functors { Hom T ( X , −) } X ∈ G is conser vativ e) if and only if T = h G i . The f ollo wing lemma is a consequence of [ Nee01 ]: Lemma 1.3.17 Let T be a triangulated monoidal P -ﬁbr ed category ov er S wit h g eometric sections M . Assume T is τ -g enerat ed. If T is well g enerat ed, then f or any object S of S , T ( S ) = h M S ( X ){ i } ; X / S a P -object , i ∈ τ i Mor eov er , there exists a regular car dinal α suc h that all the objects of shape M S ( X ){ i } are α -compact. Note ﬁnally that the Brown representability theorem of Neeman ( cf. [ N ee01 ]) giv es the f ollo wing lemma (analog of 1.3.7 ): Lemma 1.3.18 Consider a well g enerated triangulated P -ﬁbr ed categor y T . Then the f ollowing conditions ar e equivalent : (i) T is complete. (ii) F or any mor phism f : T / / S in S , f ∗ commutes with sums. If in addition, T is monoidal, the follo wing conditions are equivalent : (i 0 ) T is monoidal complete. (ii 0 ) (a) F or any mor phism f : T / / S in S , f ∗ is right exact. (b) F or any object S of S , the bifunctor ⊗ S is right exact. W e ﬁnish this section with a proposition which will constitute a useful tr ick: Proposition 1.3.19 Consider an adjunction of triangulated categories a : T / / o o T 0 : b . Assume that T admits a set of compact g enerat ors G such that any object in a ( G ) is compact in T 0 . Then b commutes with direct sums. If in addition T 0 is well g enerated then b admits a right adjoint. Proof The second assertion f ollow s from the ﬁrs t one according to a corollar y of the Bro wn representability theorem of Neeman ( cf. [ Nee01 , 8.4.4]). For the ﬁrst one, we consider a famil y ( X i ) i ∈ I of objects of T 0 and prov e that the canonical mor phism 1 General deﬁnitions and axiomatic 25 ⊕ i ∈ I b ( X i ) / / b ( ⊕ i ∈ I X i ) is an isomorphism in T . T o pro ve this, it is suﬃcient to appl y the functor Hom T ( G , −) f or any object G of G . Then the result is ob vious from the assump- tions.  W e shall often use the follo wing standard argument to produce equivalences of triangulated categor ies. Corollary 1.3.20 Let a : T / / T 0 be a triangulated functor betw een triangulated categories. Assume that the functor a pr eser v es small sums, and that T admits a small set of compact g enerat ors G , such that a ( G ) form a f amily of compact objects in T 0 . Then a is fully f aithful if and only if, f or any couple of objects G and G 0 in G , the map Hom T ( G , G 0 [ n ]) / / Hom T 0 ( a ( G ) , a ( G 0 )[ n ]) is bijectiv e for any integ er n . If a is fully fait hful, then a is an equiv alence of categories if and only if a ( G ) is a gener ating family in T 0 . Proof Let us pro v e that this is a suﬃcient condition. As T is in particular well generated, b y the Brown representability theorem, the functor b admits a r ight adjoint b : T 0 / / T . By vir tue of the preceding proposition, the functor b preserves small sums. Let us pro v e that a is full y faithful. W e hav e to chec k that, f or any object M of T , the map M / / b ( a ( M )) is inv er tible. As a and b are triangulated and preser v e small sums, it is suﬃcient to chec k this when M r uns ov er a generating f amily of objects of T (e.g. G ). As G is generating, it is suﬃcient to prov e that the map Hom T ( G , M [ n ]) / / Hom T 0 ( a ( G ) , a ( M )[ n ]) = Hom T 0 ( a ( G ) , b ( a ( M ))[ n ]) is bi jectiv e f or any integer n , which hold then by assumption. The functor a thus identiﬁes T with the localizing subcategor y of T 0 generated by a ( G ) ; if moreo v er a ( G ) is a generating famil y in T 0 , then T 0 = h a ( G )i , which also pro v es the las t assertion.  1.3.d The model category case 1.3.21 W e shall use Ho ve y’ s book [ Ho v99 ] for a general ref erence to the theor y of model categor ies. Note that, f ollo wing loc. cit. , all the model categor ies w e shall consider will hav e small limits and small colimits. Let M be the sub- 2 -categor y of C at made of the model categories, with 1 - morphisms the left Quillen functors and 2 -mor phisms the natural transformations. When we will apply deﬁnition 1.3.2 (resp. 1.3.3 ) to C = M , w e will speak of a P - ﬁbr ed model categor y f or a M -structured P -ﬁbred categor y M (resp. morphism of P -ﬁbred model categor ies). Note that according to the deﬁnition of left Quillen functors, M is then automatically complete. Giv en a property ( P ) of model categor ies (like being coﬁbrantl y generated, left and/or right proper , combinator ial, stable, etc), w e will sa y that a P -ﬁbred model 26 Fibred categor ies and the six functors f or malism category M ov er S has the proper ty ( P ) if, f or an y object S of S , the model category M ( S ) has the proper ty ( P ) . For the monoidal case, w e let M ⊗ be the sub- 2 -categor ies of M made of the symmetr ic monoidal model categor ies (see [ Hov99 , Deﬁnition 4.2.6]), with 1 -morphisms the symmetr ic monoidal left Quillen functors and 2 -morphisms the symmetric monoidal natural transf ormations, f ollowing the conditions of 1.3.4 . When w e will apply deﬁnition 1.3.5 to the case of ( M , M ⊗ ) , w e will speak simply of a monoidal P -ﬁbr ed model cat egor y (resp. morphism of monoidal P -ﬁbred model categories ) for a (resp. mor phism of ) ( M , M ⊗ ) -structured monoidal P -ﬁbred cat- egory M . Ag ain, M is then monoidal complete. Remar k 1.3.22 Let M be a P -ﬁbred model category o ver S . Then f or any P - morphism p : X / / Y , the inv erse image functor p ∗ : M ( Y ) / / M ( X ) has very strong e xactness properties: it preser v es small limits and colimits (having both a left and a r ight adjoint), and it preser v es weak equiv alences, coﬁbrations, and ﬁbrations. The only non (completel y) tr ivial asser tion here is about the preservation of weak equiv alences. For this, one notices ﬁrst that it preserves trivial coﬁbrations and trivial ﬁbrations (being both a left Quillen functor and a right Quillen functor). In particular, by vir tue of Ken Brown Lemma [ Hov99 , Lemma 1.1.12], it preserves weak equiv alences betw een coﬁbrant (resp. ﬁbrant) objects. Giv en a weak equivalence u : M / / N in M ( Y ) , we can ﬁnd a commutative square M 0 u 0 / /   N 0   M u / / N in which the two vertical maps are tr ivial ﬁbrations, and where u 0 is a weak equiv - alence between coﬁbrant objects, from which w e deduce easily that p ∗ ( u ) is a weak equiv alence in M ( X ) . 1.3.23 Consider a P -ﬁbred model categor y M o v er S . By assumption, we get the f ollowing pairs of adjoint functors: (a) For any mor phism f : X / / S of S , L f ∗ : Ho ( M ( S )) / / o o Ho ( M ( X )) : R f ∗ (b) For any P -mor phism p : T / / S , the pullback functor L p ] : Ho ( M ( S )) / / o o Ho ( M ( T )) : L p ∗ = p ∗ = R p ∗ Moreo v er , the canonical isomorphism of shape ( f g ) ∗ ' g ∗ f ∗ induces a canonical isomorphism R ( f g ) ∗ ' R g ∗ R f ∗ . In the situation of the P -base chang e f or mula 1.1.9 , w e obtain also that the base chang e map L q ] L g ∗ / / L f ∗ L p ] 1 General deﬁnitions and axiomatic 27 is an isomor phism from the equivalent proper ty of M . Thus, we hav e deﬁned a complete P -ﬁbred categor y whose ﬁber ov er S is Ho ( M ( S )) . Deﬁnition 1.3.24 Giv en a P -ﬁbred model category M as abov e, the complete P - ﬁbred categor y deﬁned abo ve will be denoted by Ho ( M ) and called the homotopy P -ﬁbred categor y associated with M . 1.3.25 Assume that M is a monoidal P -ﬁbred model categor y ov er S . Then, f or an y object S of S , Ho ( M )( S ) has the structure of a symmetr ic closed monoidal category; see [ Ho v99 , Theorem 4.3.2]. The (der iv ed) tensor product of Ho ( M )( S ) will be denoted by M ⊗ L S N , and the (der iv ed) internal Hom will be wr itten R Hom S ( M , N ) , while the unit object will be wr itten 1 S . For any morphism f : T / / S in S , the der iv ed functor L f ∗ is symmetric monoidal as f ollo ws from the equiv alent proper ty of its counter part f ∗ . Moreo v er , f or an y P -mor phism p : T / / S and f or an y object M in Ho ( M )( T ) and an y object N in Ho ( M )( S ) , the e x chang e map of 1.1.24 L p ] ( M ⊗ L p ∗ ( N )) / / L p ] ( M ) ⊗ L N is an isomor phism. Deﬁnition 1.3.26 Giv en a monoidal P -ﬁbred model categor y M as abov e, the complete monoidal P -ﬁbred category deﬁned abov e will be denoted by Ho ( M ) and called the homotopy monoidal P -ﬁbred categor y associated with M . 1.4 Premotivic categories In the present ar ticle, we will focus on a par ticular type of P -ﬁbred categor y . 1.4.1 Let S be a scheme. Assume S is a full subcategor y of the category of S - schemes. In most of this work, we will denote by S f t the class of morphisms of ﬁnite type in S and b y Sm be the class of smooth mor phisms of ﬁnite type in S . There is an e xception to this rule: throughout Part 3, S f t will be the class of separated mor phisms of ﬁnite type in S and Sm will be the class of separated smooth morphisms of ﬁnite type in S . Ho we v er , the axiomatic which we will present in the sequel can be applied identically in each cases so that the reader can freely use the restriction that all mor phisms of Sm and S f t are separated. In any case, the classes Sm and S f t are admissible in S in the sense of Paragraph 1.0.1 (this is automatic, for instance, if S is stable by pullbacks). Deﬁnition 1.4.2 Let P be an admissible class of mor phisms in S . A P -premotivic category ov er S — or simply P -premo tivic categor y when S is clear — is a complete monoidal P -ﬁbred category o ver S . A mor phism of P -premo tivic categories is a mor phism of complete monoidal P -ﬁbred categor ies o v er S . 28 Fibred categor ies and the six functors f or malism As a par ticular case, when C is the 2 -categor y T r i of tr iangulated categor ies (resp. A b of abelian categor ies), a P -premotivic triangulated (r esp. abelian) cat- egor y ov er S is a ( C , C ⊗ ) -structured complete monoidal P -ﬁbred categor y ov er S (def. 1.3.5 ). Mor phisms of P -premotivic triangulated (resp. abelian) categor ies are deﬁned accordingly . W e will also sa y: premo tivic for Sm -premotivic and g eneralized pr emotivic f or S f t -premotivic. The sections of a P -premotivic category will be called premo tives . Example 1.4.3 Let S be the categor y of noetherian schemes of ﬁnite dimension. For such a scheme S , recall H • ( S ) is the pointed homotopy categor y of Morel and V oe v odsky; cf. ex amples 1.1.5 , 1.1.14 , 1.1.30 . Then, according to the fact recalled in these examples the 2 -functor H • is a geometrically generated premotivic category (recall Deﬁnition 1.1.41 ). For such a scheme S , consider the stable homotop y category SH ( S ) of Morel and V oe v odsky (see [ Jar00 , A y o07b ]). A ccording to [ A yo07b ], it deﬁnes a tr iangulated premotivic category denoted by SH . Moreov er , it is compactl y ( Z × Z ) -g enerated in the sense of deﬁnition 1.1.41 where the ﬁrst factor refers to the suspension and the second one refers to the T ate twist ( i.e. as a tr iangulated premotivic categor y , it is compactly generated by the T ate twists). 1.4.4 Let T be a P -premotivic triangulated category with geometric sections M and τ be a set of twists f or T (Deﬁnition 1.1.39 ). Recall from Con vention 1.3.13 (resp. and Deﬁnition 1.3.15 ) that T is said to be τ -g enerated (resp. compactl y τ -generated) if f or an y scheme S , the famil y of isomorphism of classes of premotives of the form M S ( X ){ i } f or a P -scheme X o v er S and a twist i ∈ τ is a set of generators (resp. compact generators) f or the triangulated categor y T ( S ) (in the respective case, w e also assume T ( S ) admits small sums). Let E be a premotiv e o ver S and X be a P -scheme o v er S . For an y ( n , i ) ∈ Z × τ , w e deﬁne the cohomology of X in deg ree n and twist i with coeﬃcients in E as: H n , i T ( X , E ) = Hom T ( S )  M S ( X ) , E { i } ( n )  . The fact T is τ -generated amounts to say that any such premotive E is deter mined b y its cohomology . Example 1.4.5 All the kno wn triangulated premotivic categor ies are τ -generated f or a giv en set of twist τ . In fact, one deﬁnes as usual the T ate twist 1 S ( 1 ) in such a premotivic triangulated category T b y the formula: M S ( P 1 S ) = 1 ⊕ 1 ( 1 )[ 2 ] . Then 1 ( 1 ) = ( 1 S ( 1 )) S ∈ S is a car tesian section of T . W e will say that T is g enerated by T ate twists if it is Z -generated where Z ref ers to the set of twists ( 1 ( n )) n ∈ Z . The premotivic tr iangulated category S H of the previous example is compactly generated b y T ate twis ts. Similar ly , the stable A 1 -derived categor y D A 1 , Λ (cf. Ex- ample 5.3.31 ), the category of V oev odsky motives DM (cf. Deﬁnition 11.1.1 ), the 1 General deﬁnitions and axiomatic 29 category of K GL -modules (cf. Deﬁnition 13.3.3 ) and the category of Beilinson motiv es DM B (cf. Deﬁnition 14.2.1 ) are all compactly g enerated by T ate twists. Deﬁnition 1.4.6 Let M and M 0 be P -premotivic categor ies. A mor phism of P -premotivic categories (or simply a premo tivic morphism ) is a mor phism ϕ ∗ : M / / M 0 of complete monoidal P -ﬁbred categor ies. W e shall also sa y that ϕ ∗ : M / / o o M 0 : ϕ ∗ is a pr emotivic adjunction . When moreov er M and M 0 are P -premotivic tr iangu- lated (resp. abelian) categories, we will ask ϕ ∗ is a compatible with the tr iangulated (resp. additiv e) structure – as in Deﬁnition 1.3.3 . If w e assume that M (resp. M 0 ) is τ -twisted (resp. τ 0 -twisted), we will sa y as in Deﬁnition 1.2.11 that ϕ ∗ is compatible with twists if for any i ∈ τ , ϕ ∗ ( i ) belongs up to isomorphism to τ 0 . W e say ϕ ∗ is strictly compatible with twists if it is compatible with twists and if an y element of τ 0 is isomor phic to the image of an element of τ . Usually , premotivic categor ies comes equip with canonical twists (especially the T ate twist, see the abo v e example) and premotivic mor phisms are compatible with twists. Example 1.4.7 With the hypothesis and notations of 1.4.3 , w e get a premotivic ad- junction Σ ∞ : H • / / o o SH : Ω ∞ induced b y the inﬁnite suspension functor according to [ Jar00 ]. 1.4.8 Let T (resp. A ) be a tr iangulated P -premotivic categor y with geometric sections M and a set of twists τ . For an y scheme S , we let T τ, c ( S ) be the small- est tr iangulated thick 42 subcategor y of T ( S ) which contains premotives of shape M S ( S ) { i } (resp. M S ( X , A ) { i } ) f or a P -scheme X / S and a twist i ∈ τ . This sub- category is stable b y the operations f ∗ , p ] and ⊗ . In par ticular , T τ, c deﬁnes a not necessarily complete tr iangulated (resp. abelian) P -ﬁbred categor y ov er S . W e also obtain a mor phism of triangulated (resp. abelian) monoidal P -ﬁbred categor ies, fully faithful as a functor , ι : T τ, c / / T Deﬁnition 1.4.9 Consider the notations introduced abo v e. W e will call T τ, c the τ - constructible part of T . For an y scheme S , the objects of T τ, c ( S ) will be called τ -constructible . When τ is clear from the conte xt, we will put T c : = T τ, c and use the ter minology constructible . Remar k 1.4.10 The condition of τ -constructibility is a good categor ical notion of ﬁniteness which extends the notion of g eometric motiv es as introduced by V oev odsky . In the tr iangulated motivic case, it will be studied thoroughly in section 4 . 42 i.e. stable by direct factors. 30 Fibred categor ies and the six functors formalism Proposition 1.4.11 Let T be a τ -twisted P -pr emotivic triangulated category. Let S be a scheme suc h that : 1. The category T ( S ) admits small sums. 2. F or any P -scheme X o ver S , and any twist i ∈ τ , the premotiv e M S ( X ){ i } is compact. Then, a premo tiv e M o v er S is τ -constructible if and only if it is compact. Proof In any triangulated category D , one easily obtains that the proper ty of being compact is stable under extensions and retracts. In par ticular , the thick triangulated subcategory of D generated by compact objects consists precisel y of the compact objects of D . Moreo ver , if D admits small sums and is g enerated by a f amily of compact objects G , then the thick tr iangulated subcategor y of D generated b y G contains all compact objects, and is therefore equal to the full subcategor y of compact objects (see [ Nee92 , Lem. 2.2]). Coming back to the deﬁnition of being τ -constructible, this general fact ﬁnishes the proof.  Thus, when the conditions of this proposition are fulﬁlled, the categor y T τ, c ( S ) does not depend on the par ticular choice of τ . This will often be the case in practice (see 5.1.33 , 5.2.39 , 5.3.42 ). Remar k 1.4.12 The notion of compact objects in a tr iangulated categor y w as heav - ily dev eloped b y A. N eeman. Its relation with ﬁniteness conditions is particularl y emphasized when consider ing the der iv ed categor y of complex es of quasi-coherent shea v es o ver a quasi-compact separated scheme: in this tr iangulated categor y , being compact is equiv alent to being per f ect ([ Nee96 , Cor . 4.3]). Deﬁnition 1.4.13 Consider a τ -generated premotivic categor y M . An enlarg ement of M is the data of a τ 0 -twisted generalized premotivic categor y M together with a premotivic adjunction ρ ] : M / / M : ρ ∗ (where M is considered as a premotivic categor y in the obvious w ay), satisfying the f ollo wing proper ties: (a) For any scheme S in S , the functor ρ ] , S : M ( S ) / / M ( S ) is fully faithful and its right adjoint ρ ∗ S : M ( S ) / / M ( S ) commutes with sums. (b) ρ ] is strictly compatible with twists. Ag ain, this notion is deﬁned similarl y f or a C -structured P -premotivic category . Note that f or any smooth S -scheme X , we get in the conte xt of an enlarg ement as abo v e the f ollo wing identiﬁcations: ρ ] , S ( M S ( X )) ' M S ( X ) , ρ ∗ S ( M S ( X )) ' M S ( X ) 2 T r iangulated P -ﬁbred categories in algebraic geometry 31 where M (resp. M ) denote the geometric sections of M (resp. M ). Remember also that f or an y mor phism of schemes f and any smooth mor phism p , ρ ] commutes with f ∗ and p ] , while ρ ∗ commutes with f ∗ and p ∗ . 2 T riangulated P -ﬁbred categories in algebraic geome try 2.0.1 In this entire section, w e ﬁx a base scheme S , assumed to be noether ian, and a full subcategor y S of the categor y of noether ian S -schemes satisfying the f ollowing proper ties: (a) S is closed under ﬁnite sums and pullback along mor phisms of ﬁnite type. (b) For any scheme S in S , any quasi-projectiv e S -scheme belongs to S . In sections 2.2 and 2.4 , we will add the f ollo wing assumption on S : (c) An y separated mor phism f : Y / / X in S , admits a compactiﬁcation in S in the sense of [ A GV73 , 3.2.5], i.e. admits a factorization of the form Y j / / ¯ Y p / / X where j is an open immersion, p is proper , and ¯ Y belongs to S . Fur thermore, if f is quasi-projective, then p can be chosen to be projective. (d) Cho w’ s lemma holds in S (i.e., f or any proper mor phism Y / / X in S , there e xists a projectiv e birational morphism p : Y 0 / / Y in S such that f p is projectiv e as well). A categor y S satisfying all these proper ties will be called adequate f or future ref erences. 43 W e also ﬁx an admissible class P of mor phisms in S and a complete tr iangulated P -ﬁbred categor y T . W e will add the f ollo wing assumptions: (d) In section 2.2 and 2.3 , P contains the open immersions. (e) In section 2.4 , P contains the smooth mor phisms of S . In the case T is monoidal, we denote b y M : P / ? / / T its geometric sections. A ccording to the conv ention of 1.4.2 , we will speak of the pr emotivic case when P is the class of smooth morphisms of ﬁnite type 44 in S and T is a premotivic triangulated categor y . 43 For instance, the scheme S can be the spectrum of a prime ﬁeld or of a Dedekind domain. The category S might be the category of all noetherian S -schemes of ﬁnite dimension or simply the categor y of quasi-projectiv e S -schemes. In all these cases, property (c) is ensured by Nag ata ’ s theorem (see [ Con07 ]) and property (d) by Chow’ s lemma (see [ GD61 , 5.6.1]). 44 or smooth separated morphisms of ﬁnite type when applying this section in Par t 3 32 Fibred categor ies and the six functors f or malism 2.1 Elementary properties Deﬁnition 2.1.1 W e say that T is additiv e, if for an y ﬁnite famil y ( S i ) i ∈ I of schemes in S , the canonical map T Þ i S i ! / / Ö i T ( S i ) is an equiv alence. Recall this proper ty implies in par ticular that T ( ∅ ) = 0 . Lemma 2.1.2 Let S be a sc heme, p : A 1 S / / S be the canonical projection. The f ollowing conditions ar e equiv alent: (i) The functor p ∗ : T ( S ) / / T ( A 1 S ) is fully faithful. (ii) The counit adjunction mor phism 1 / / p ∗ p ∗ is an isomorphism. In the pr emotivic case, these conditions ar e equivalent to the f ollowing ones: (iii) The unit adjunction mor phism p ] p ∗ / / 1 is an isomor phism. (iv) The morphism M S ( A 1 S ) p ∗ / / 1 S induced by p is an isomorphism. (iv’) F or any smooth S -sc heme X , the mor phism M S ( A 1 X ) ( 1 X × p ) ∗ / / M S ( X ) is an iso- morphism. The onl y thing to recall is that in the premotivic case, p ] p ∗ ( M ) = M S ( A 1 S ) ⊗ M and p ∗ p ∗ ( M ) = Hom S ( M S ( A 1 S ) , M ) . Deﬁnition 2.1.3 The equivalent conditions of the previous lemma will be called the homotopy property f or T , denoted by (Htp). 2.1.4 Recall that a sieve R of a scheme X is a class of mor phisms in S / X which is stable b y composition on the right b y any mor phism of schemes (see [ A GV73 , I.4]). Giv en such a siev e R , we will sa y that T is R -separ ated if the class of functors f ∗ f or f ∈ R is conser vativ e. Given tw o siev es R , R 0 of X , the f ollo wing proper ties are immediate: (a) If R ⊂ R 0 then T is R -separated implies T is R 0 -separated. (b) If T is R -separated and is R 0 -separated then T is ( R ∪ R 0 ) -separated. A famil y of mor phisms ( f i : X i / / X ) i ∈ I of schemes deﬁnes a siev e R = h f i , i ∈ I i such that f is in R if and only if there e xists i ∈ I such that f can be factored through f i . Obviousl y , (c) T is R -separated if and only if the famil y of functors ( f ∗ i ) i ∈ I is conservativ e. Recall that a topology on S is the data f or an y sc heme X of a set of sie v es of X satisfying cer tain stability conditions ( cf. [ A GV73 , II, 1.1]), called t -co v er ing siev es. A pre-topology t 0 on S is the data for an y scheme X of a set of families of morphisms of shape ( f i : X i / / X ) i ∈ I satisfying certain stability conditions ( cf. [ A GV73 , II, 1.3]), called t 0 -co v ers. A pre-topology t 0 generated a unique topology t . 2 T r iangulated P -ﬁbred categories in algebraic geometry 33 Deﬁnition 2.1.5 Let t be a Grothendiec k topology on S . W e sa y that T is t - separat ed if the f ollo wing proper ty holds: (t-sep) For an y t -cov er ing siev e R , T is R -separated in the sense deﬁned abov e. Obviousl y , given tw o topologies t and t 0 on S such that t 0 is ﬁner than t , if T is t -separated then it is t 0 -separated. If the topology t on S is generated by a pre-topology t 0 then T is t -separated if and only if f or any t 0 -co v ers ( f i ) i ∈ I , the famil y of functors ( f ∗ i ) i ∈ I is conser v ativ e – use [ A GV73 , 1.4] and 2.1.4 (a)+(c). 2.1.6 Recall that a mor phism of schemes f : T / / S is radicial if it is injective and f or any point t of T , the residual extension induced by f at t is radicial ( cf. [ GD60 , 3.5.4, 3.5.8]) 45 The follo wing deﬁnition is inspired by [ A y o07a , Def. 2.1.160]. Deﬁnition 2.1.7 W e say that T is separat ed (resp. semi-separated ) if T is separated f or the topology generated by sur jectiv e families of mor phisms of ﬁnite type (resp. ﬁnite radicial mor phisms) in S . W e also denote b y (Sep) (resp. (sSep)) this property . Remar k 2.1.8 If T is additive, proper ty (Sep) (resp. (sSep)) is equivalent to ask that f or any surjective mor phism of ﬁnite type (resp. ﬁnite surjective radicial mor phism) f : T / / S in S , the functor f ∗ is conservativ e. Proposition 2.1.9 Assume T is semi-separat ed and satisﬁes the transv ersality prop- erty with respect to ﬁnite surjectiv e radicial mor phisms. Then f or any ﬁnite surjective radicial morphism f : Y / / X , the functor f ∗ : T ( X ) / / T ( Y ) is an equiv alence of categories. Proof W e ﬁrst consider the case when f = i is in addition a closed immersion. In this case, we can consider the pullback square below . Y Y i   Y i / / Z Using the transversality proper ty with respect to i , we see that the counit i ∗ i ∗ / / 1 is an isomor phism. It thus remains to prov e that the unit map 1 / / i ∗ i ∗ is an isomor - phism. As i ∗ is conservativ e b y semi-separability , it is suﬃcient to chec k that i ∗ / / i ∗ i ∗ i ∗ ( M ) is an isomor phism. But this is a section of the map i ∗ i ∗ i ∗ ( M ) / / i ∗ ( M ) , which is already kno wn to be an isomor phism. 45 It is equivalent to ask that f is universall y injective. When f is surjective, this is equiv alent to ask that f is a universal homeomor phism. 34 F ibred categor ies and the six functors formalism Consider no w the general case of a ﬁnite radicial e xtension f . W e introduce the pullback square Y × X Y p / / q   Y f   Y f / / X Consider the diagonal immersion i : Y / / Y × X Y . Because Y is noether ian and p is separable, i is ﬁnite ( cf. [ GD61 , 6.1.5]) thus a closed immersion. As p is a univ ersal homeomorphism, the same is tr ue f or its section i . The preceding case thus implies that i ∗ is an equiv alence of categor ies. Moreo v er , as pi = qi = 1 Y , w e see that p ∗ and q ∗ are both quasi-in verses to i ∗ , which implies that they are isomor phic equiv alences of categories. More precisely , we get canonical isomorphisms of functors i ∗ ' p ∗ ' q ∗ and i ∗ ' p ∗ ' q ∗ . W e check that the unit map 1 / / f ∗ f ∗ is an isomor phism. Indeed, by semi- separability , it is suﬃcient to prov e this after applying the functor f ∗ , and w e g et, using the transv ersality proper ty f or f : f ∗ ' i ∗ p ∗ f ∗ ' q ∗ p ∗ f ∗ ' f ∗ f ∗ f ∗ . W e then c heck that the counit map f ∗ f ∗ / / 1 is an isomorphism as w ell. In f act, using again the transv ersality proper ty f or f , we ha v e isomor phisms f ∗ f ∗ ( M ) ' q ∗ p ∗ ( M ) ' i ∗ i ∗ ( M ) ' M . 2.1.10 Recall from [ V oe10b ] that a cd-structure on S is a collection P of commu- tativ e squares of schemes B / /   Q Y f   A e / / X which is closed under isomorphisms. W e will say that a sq uare Q in P is P - distinguished. V oe v odsky associates to P a topology t P , the smallest topology such that: • f or any P -distinguished square Q as abo v e, the sie ve generated by { f : A / / X , e : Y / / X } is t P -co v ering on X . • the empty siev e cov ers the empty scheme. Example 2.1.11 A Nisnevic h distinguished square is a square Q as abov e such that Q is car tesian, f is étale, e is an open embedding with reduced complement Z and the induced map f − 1 ( Z ) / / Z is an isomorphism. The cor responding cd-str ucture is called the upper cd-structure (see section 2 of [ V oe10c ]). Because we work with noetherian schemes, the cor responding topology is the Nisnevich topology (see proposition 2.16 of loc.cit. ). 2 T r iangulated P -ﬁbred categories in algebraic geometry 35 A proper c dh -distinguished square is a square Q as abov e such that Q is Cartesian, f is proper , e is a closed embedding with open complement U and the induced map f − 1 ( U ) / / U is an isomor phism. The cor responding cd-structure is called the low er cd-structur e . The topology associated with the low er cd-structure is called the proper cdh-topology . The topology generated by the lo w er and upper cd-structures is b y deﬁnition (according to the preceding remark on Nisnevich topology) the cdh-topology . All these three e xamples are complete cd-structures in the sense of [ V oe10b , 2.3]. Lemma 2.1.12 Let P be a complete cd-structure (see [ V oe10b , def 2.3]) on S and t P be the associated topology. The f ollowing conditions ar e equivalent : (i) T is t P -separat ed. (ii) F or any distinguished squar e Q for P of the abov e form, the pair of functors ( e ∗ , f ∗ ) is conser vativ e. Proof This f ollo ws from the deﬁnition of a complete cd-str ucture and 2.1.4 (a).  Remar k 2.1.13 If we assume that S is stable by arbitrary pullback then an y cd- structure P on S such that P -distinguished squares are stable b y pullback is complete (see [ V oe10b , 2.4]). 2.2 Ex ceptional functors, f ollo wing Deligne 2.2.a The support axiom 2.2.1 Consider an open immersion j : U / / S . Appl ying 1.1.15 to the car tesian square U U j   U j / / S w e get a canonical natural transf or mation γ j : j ] = j ] 1 ∗ E x ( ∆ ] ∗ ) / / j ∗ 1 ] = j ∗ . Recall that the functors j ] and j ∗ are fully faithful (see Corollary 1.1.20 ). Note that according to remark 1.1.7 , this natural transformation is compatible with identiﬁcations of the kind ( j k ) ] = j ] k ] and ( j k ) ∗ = j ∗ k ∗ . Lemma 2.2.2 Let S be a scheme, U and V be subschemes suc h that S = U t V . W e let h : U / / S (resp. k : V / / S ) be the canonical open immersions. Assume that the functor ( h ∗ , k ∗ ) : T ( S ) / / T ( U ) × T ( V ) is conser vativ e and that T ( ∅ ) = 0 . Then the natural transf ormation γ h (r esp. γ k ) is an isomor phism. Mor eov er , the functor ( h ∗ , k ∗ ) is then an equiv alence of categories. 36 Fibred categories and the six functors formalism Proof As h ] and h ∗ are full y f aithful, w e ha ve h ∗ h ] ' h ∗ h ∗ . By P -base chang e, w e also get k ∗ h ] ' k ∗ h ∗ ' 0 . It remains to pro v e the last assertion. The functor R = ( h ∗ , k ∗ ) has a left adjoint L deﬁned by L= h ] ⊕ k ] : L ( M , N ) = h ] ( M ) ⊕ k ] ( N ) . The natural transf ormation L R / / 1 is an isomor phism: to see this, is it suﬃcient to ev aluate at h ∗ and k ∗ , which gives an isomor phism in T ( U ) and T ( V ) respectiv ely . The natural transformation 1 / / R L is also an isomor phism because h ] and k ] are fully faithful.  Remar k 2.2.3 Assume T is Zariski separated (deﬁnition 2.1.5 ). Then, as a corollar y of this lemma, T is additive (deﬁnition 2.1.1 ) if and only if T ( ∅ ) = 0 . 2.2.4 Exc hang e structures V .– Assume T is additive. W e consider a commutativ e square of schemes V k / / q   ∆ T p   U j / / S (2.2.4.1) such that j , k are an open immersions and p , q are proper mor phisms. This diagram can be factored into the follo wing commutative diagram: V k % % q ( ( l & & U × S T j 0 / / p 0   Θ T p   U j / / S . Then l is an open and closed immersion so that the previous lemma implies the canonical mor phism γ l : l ] / / l ∗ is an isomorphism. As a consequence, w e get a natural e x chang e transf ormation E x ( ∆ ] ∗ ) : j ] q ∗ = j ] p 0 ∗ l ∗ E x ( Θ ] ∗ ) / / p ∗ j 0 ] l ∗ γ − 1 l / / p ∗ j 0 ] l ] = p ∗ k ] using the ex chang e of 1.1.15 . Note that, with the notations introduced in 2.2.1 , the f ollowing diag ram is commutativ e. j ] q ∗ E x ( ∆ ] ∗ ) / / γ j q ∗   p ∗ k ] p ∗ γ k   j ∗ q ∗ ∼ / / ( j q ) ∗ = ( p k ) ∗ p ∗ k ∗ ∼ o o (2.2.4.2) 2 T r iangulated P -ﬁbred categories in algebraic geometry 37 Indeed one sees ﬁrs t that it is suﬃcient to treat the case where ∆ is cartesian. Then, as j ] is a fully faithful left adjoint to j ∗ it is suﬃcient to chec k that ( 2.2.4.2 ) commutes after having applied j ∗ . Using the cotransversality proper ty with respect to open immersions, one sees then that this consists of v erifying the commutativity of ( 2.2.4.2 ) when j is the identity , in which case it is tr ivial. Deﬁnition 2.2.5 Let p : T / / S be a proper mor phism in S . W e sa y that the tr iangulated P -ﬁbred category T satisﬁes the support property with respect to p , denoted by (Supp p ), if it is additiv e and f or any commutative square of shape ( 2.2.4.1 ) the e x chang e transf or mation E x ( ∆ ] ∗ ) : j ] q ∗ / / p ∗ k ] deﬁned abo v e is an isomor phism. W e sa y that T satisﬁes the suppor t property , also denoted b y (Supp), if it satisﬁes (Supp p ) f or all proper mor phism p in S . By deﬁnition, it is suﬃcient to chec k the last proper ty of proper ty (Supp) in the case where ∆ is car tesian. 2.2.b Ex ceptional direct imag e 2.2.6 W e denote b y S s e p (resp. S o p e n , S pr o p ) the sub-categor y of the category S with the same objects but mor phisms are separated morphisms of ﬁnite type (resp. open immersions, proper mor phisms). W e denote by T ∗ : S / / T r i ⊗ resp. T ] : S o p e n / / T r i ⊗ the 2 -functor deﬁned respectiv ely by mor phisms of type f ∗ and j ] ( f any mor phism of schemes). The proposition belo w is essentially based on a result of Deligne [ A GV73 , XVII, 3.3.2]: Proposition 2.2.7 Assume T is a monoidal P -ﬁbred categor y and satisﬁes prop- erty (Supp). Then ther e exists a unique 2 -functor T ! : S s e p / / T r i ⊗ with the property that T ! | S p r o p = T ∗ | S p r o p , T ! | S o p e n = T ] and for any commutative squar e ∆ of shape ( 2.2.4.1 ) with p and q proper , the composition of the structural isomor phisms j ] q ∗ = j ! q ! ' ( j q ) ! = ( p k ) ! ' p ! k ! = p ∗ k ] is equal to the exchang e transf or mation E x ( ∆ ] ∗ ) . 38 Fibred categor ies and the six functors f or malism 2.2.8 U nder the assumptions of the proposition, f or an y separated morphism of ﬁnite type f : Y / / X , we will denote by f ! : T ( Y ) / / T ( X ) the functor T ! ( f ) . The functor f ! is called the dir ect imag e functor wit h compact support or the lef t exceptional functor associated with f . Proof W e recall the principle of the proof of Deligne. Let f : Y / / X be a separated morphism of ﬁnite type in S . Let C f be the categor y of compactiﬁcations of f in S , i.e. of factorizations of f of the f orm (2.2.8.1) Y j / / ¯ Y p / / X where j is an open immersion, p is proper, and ¯ Y belongs to S . Mor phisms of C f are giv en by commutative diag rams of the f orm ¯ Y 0 p 0 + + π   Y j 0 3 3 j + + X . ¯ Y p 3 3 (2.2.8.2) in S . T o any compactiﬁcation of f of shape ( 2.2.8.1 ), w e associate the functor p ∗ j ] . T o any mor phism of compactiﬁcations ( 2.2.8.2 ), w e associate a natural isomor phism p 0 ∗ j 0 ] = p ∗ π ∗ j 0 ] E x ( ∆ ] ∗ ) − 1 / / p ∗ j ] 1 ∗ = p ∗ j ] . where ∆ stands f or the commutativ e square made by remo ving π in the diag ram ( 2.2.8.2 ), and E x ( ∆ ] ∗ ) is the cor responding natural transf ormation (see 2.2.4 ). The compatibility of E x ( ∆ ] ∗ ) with composition of mor phisms of schemes sho ws that w e ha v e deﬁned a functor Γ f : C op f / / Hom ( T ( Y ) , T ( X )) which sends all the maps of C f to isomor phisms (b y the suppor t proper ty). The categor y C f is non-empty b y the assumption 2.0.1 (c) on S , and it is in fact left ﬁlter ing; see [ A GV73 , XVII, 3.2.6(ii)]. This deﬁnes a canonical functor f ! : T ( Y ) / / T ( X ) , independent of any choice compactiﬁcation of f , deﬁned in the category of functors Hom ( T ( Y ) , T ( X )) by the formula f ! = lim / / C op f Γ f . If f = p is proper , then the compactiﬁcation Y = / / Y p / / X is an initial object of C f , which giv es a canonical identiﬁcation p ! = p ∗ . Similarl y , if f = j is an open immersion, then the compactiﬁcation 2 T r iangulated P -ﬁbred categories in algebraic geometry 39 Y j / / X = / / X is a ter minal object of C j , so that we get a canonical identiﬁcation j ! = j ] . This construction is compatible with composition of mor phisms. Let g : Z / / Y and f : Y / / X be two separated mor phisms of ﬁnite type in S . For any a couple of compactiﬁcations Z k / / ¯ Z q / / Y and Y j / / ¯ Y p / / X of f and g respectivel y , w e can choose a compactiﬁcation ¯ Z h / / T r / / Y of j q , and w e get a canonical isomor phism f ! g ! ' p ∗ j ] q ∗ k ] ' p ∗ r ∗ h ] k ] ' ( pr ) ∗ ( h k ) ] ' ( f g ) ! . The independence of these isomor phisms with respect to the choices of compact- iﬁcation f ollo ws from [ A GV73 , XVII, 3.2.6(iii)]. The cocycle conditions (i.e. the associativity) also f ollo ws formally from [ A GV73 , XVII, 3.2.6]. The uniqueness statement is obvious.  2.2.9 This construction is functor ial in the f ollo wing sense. Deﬁne a 2 -functor with support on T to be a triple ( D , a , b ) , where: (i) D : S s e p / / T r i is a 2 -functor (we shall wr ite the structural coherence iso- morphisms as c g , f : D ( g f ) ∼ / / D ( g ) D ( f ) for composable ar ro ws f and g in S s e p ); (ii) a : T ∗ | S p r o p / / D | S p r o p and b : T ] / / D | S o p e n are mor phisms of 2 - functors which agree on objects, i.e. suc h that for an y sc heme S in S , we ha v e ψ S = a S = b S : T ( S ) / / D ( S ) ; (iii) f or any commutativ e square of shape ( 2.2.4.1 ) in which j and k are open immersions, while p and q are proper mor phisms, the diagram belo w commutes. ψ S j ] q ∗ ψ S Ex ( ∆ ] ∗ ) / / b q ∗   ψ S p ∗ k ] a k ]   D ( j ) ψ U q ∗ D ( j ) a   D ( p ) ψ T k ] D ( p ) b   D ( j ) D ( q ) ψ V c − 1 j , q / / D ( j q ) = D ( p k ) ψ V D ( p ) D ( k ) ψ V c − 1 p , k o o Morphisms of 2 -functors with suppor t on T ( D , a , b ) / / ( D 0 , a 0 , b 0 ) 40 Fibred categor ies and the six functors f ormalism are deﬁned in the obvious w a y: these are mor phisms of 2 -functors D / / D 0 which preserve all the structure on the nose. Using the arguments of the proof of 2.2.7 , one chec ks easily that the category of 2 -functors with suppor t has an initial object, which is nothing else but the 2 -functor T ! together with the identities of T ∗ | S p r o p and of T ] respectiv ely . In par ticular , f or an y 2 -functor D : S s e p / / T r i , a mor phism of 2 -functors T ! / / D is completely determined b y its res tr ictions to S pr o p and S o p e n , and by its compatibility with the ex chang e isomorphisms of type E x ( ∆ ] ∗ ) in the sense descr ibed in condition (iii) abo v e. Proposition 2.2.10 Assume that T satisﬁes the suppor t property and consider the notations of Proposition 2.2.7 . F or any separated mor phism of ﬁnite type f in S , ther e exists a canonical natural transf or mation α f : f ! / / f ∗ . The collection of maps α f deﬁnes a morphism of 2 -functors α : T ! / / T ∗ | S s e p , f  / / ( α f : f ! / / f ∗ ) whose res trictions to S pr o p and S o p e n ar e respectiv ely the identity and the mor - phism of 2 -functors γ : T ] / / T ∗ | S o p e n deﬁned in 2.2.1 . Proof The identities f ∗ = f ∗ f or f proper (resp. projective) and the ex chang e natural transf or mations of type E x ( ∆ ] ∗ ) tur ns T ∗ | S s e p into a 2 -functor with suppor t (resp. restricted suppor t) on T (proper ty (iii) of 2.2.9 is e xpressed by the commutative square ( 2.2.4.2 )).  Proposition 2.2.11 Let T 0 be another triangulated complete P -ﬁbr ed categor y ov er S . Assume that T and T 0 both have the support property, and consider given a triangulated morphism of P -ﬁbred categories ϕ ∗ : T / / T 0 (r ecall deﬁnition 1.2.2 ). Then, ther e is a canonical family of natur al transf or mations E x ( ϕ ∗ , f ! ) : ϕ ∗ X f ! / / f ! ϕ ∗ Y f or eac h separ ated morphism of ﬁnit e type f : Y / / X in S , whic h is functorial with respect to composition in S (i.e. deﬁnes a mor phism of 2 -functors) and such that, the following conditions are veriﬁed: (a) if f is proper , t hen, under the identiﬁcation f ! = f ∗ , the map E x ( ϕ ∗ , f ! ) is t he exc hang e transf ormation E x ( ϕ ∗ , f ∗ ) : ϕ ∗ X f ∗ / / f ∗ ϕ ∗ Y deﬁned in 1.2.5 ; (b) if f is an open immersion, then, under the identiﬁcation f ! = f ] , the map E x ( ϕ ∗ , f ! ) is the inver se of the exc hang e isomor phism E x ( f ] , ϕ ∗ ) : f ] ϕ ∗ Y / / ϕ ∗ X f ] deﬁned in 1.2.1 . Proof The ex chang e maps of type E x ( ϕ ∗ , f ∗ ) deﬁne a mor phism of 2 -functors a : T ∗ | S p r o p / / T 0 ∗ | S p r o p = T 0 ! | S p r o p 2 T r iangulated P -ﬁbred categories in algebraic geometry 41 while the inv erse of the e x chang e isomor phisms of type E x ( f ] , ϕ ∗ ) deﬁne a mor phism of 2 -functors b : T ] / / T 0 ] = T 0 ! | S o p e n , in such a w ay that the tr iple ( T 0 ! , a , b ) is a 2 -functor with suppor t on T .  Corollary 2.2.12 Suppose T satisﬁes the suppor t property and consider the nota- tions of proposition 2.2.7 . 1. F or any car tesian squar e Y 0 f 0 / / g 0   ∆ X 0 g   Y f / / X , suc h that f is separated of ﬁnite type, ther e exists a canonical natural tr ansf or - mation E x ( ∆ ∗ ! ) : g ∗ f ! / / f 0 ! g 0 ∗ compatible with horizontal and v ertical compositions of squar es, and satisfying the f ollowing identiﬁcations in T ( X 0 ) ( a ) f pr oper: ( b ) f open immersion: g ∗ f ! E x ( ∆ ∗ ! ) / / f 0 ! g 0 ∗ g ∗ f ! E x ( ∆ ∗ ! ) / / f ! g 0 ∗ g ∗ f ∗ E x ( ∆ ∗ ∗ ) / / f 0 ∗ g 0 ∗ , g ∗ f ] E x ( ∆ ∗ ] ) − 1 / / f 0 ] g 0 ∗ . Mor eov er , when g is a P -mor phism, E x ( ∆ ∗ ! ) is an isomor phism. 2. F or any cartesian squar e ∆ as in (1), assuming f is separated of ﬁnite type and g is a P -morphism, there exists a canonical natural transf ormation E x ( ∆ ] ! ) : g ] f 0 ! / / f ! g 0 ] compatible with horizontal and v ertical compositions of squar es, and satisfying the f ollowing identiﬁcations in T ( X 0 ) ( a ) f pr oper: ( b ) f open immersion: g ] f 0 ! E x ( ∆ ] ! ) / / f ! g 0 ] g ] f 0 ! E x ( ∆ ] ! ) / / f ! g 0 ] g ] f 0∗ E x ( ∆ ] ∗ ) / / f ∗ g 0 ] , g ] f 0 ] f ] g 0 ] . 3. If furthermore T is monoidal then for any separated mor phism of ﬁnite type f : Y / / X , there is a natural tr ansformation 42 Fibred categories and the six functors formalism E x ( f ∗ ! , ⊗ ) : ( f ! K ) ⊗ L / / f ! ( K ⊗ f ∗ L ) whic h is compatible with respect to composition in S , and such that, in each of the f ollowing cases, w e have the follo wing identiﬁcations: ( a ) f pr oper: ( b ) f open immersion: ( f ! K ) ⊗ L E x ( f ∗ ! , ⊗ ) / / f ! ( K ⊗ f ∗ L ) ( f ! K ) ⊗ L E x ( f ∗ ! , ⊗ ) / / f ! ( K ⊗ f ∗ L ) ( f ∗ K ) ⊗ L E x ( f ∗ ∗ , ⊗ ) / / f ∗ ( K ⊗ f ∗ L ) , ( f ] K ) ⊗ L E x ( f ∗ ] , ⊗ ) − 1 / / f ] ( K ⊗ f ∗ L ) . As in the previous analogous cases, the natural transf or mations E x ( ∆ ∗ ! ) , E x ( ∆ ] , ! ) and E x ( f ∗ ! , ⊗ ) will be called exchang e transf ormations . Proof T o prov e (1), consider a ﬁx ed map g : X 0 / / X in S . W e consider the triangulated P / X -ﬁbred categor ies T 0 and T 00 o v er S / X deﬁned b y T 0 ( Y ) = T ( Y ) and T 00 ( Y ) = T ( Y 0 ) for any X -scheme Y (in S ), with g 0 : Y 0 = Y × X X 0 / / Y the map obtained from Y / / X b y pullback along g . The collection of functors g 0 ∗ : T ( Y ) / / T ( Y 0 ) deﬁne an ex act morphism of triangulated P / X -ﬁbred categor ies ov er S / X (b y the P -base change f or mula): ϕ ∗ : T 0 / / T 00 . Applying the preceding proposition to the latter giv es (1). The fact that w e g et an isomorphism whenev er g is a P -mor phism f ollo ws from the P -base chang e formula and from paragraph 1.1.15 . For point (2), we consider the notations abo ve assuming that g is a P -mor phism. The collection of functors g 0 ] : T ( Y 0 ) / / T ( Y ) associated with an X -scheme Y , g 0 : Y 0 = Y × X X 0 / / Y obtained from g as abo v e, deﬁne an e xact mor phism of tr iangulated P / X -ﬁbred categor ies ov er S / X (applying again the P -base change f or mula): ϕ ∗ : T 00 / / T 0 . Applying the preceding proposition to the latter gives (2). The proof of (3) is similar: ﬁx a scheme X in S , as well as an object L in T ( X ) . Let T 0 be the restriction of T to S / X as abov e. W e can consider L as a car tesian section of T 0 , and b y the P -projection f or mula, we then hav e an ex act mor phism of triangulated P / X -ﬁbred categor ies o v er S / X : L ⊗ (−) : T 0 / / T 0 . 2 Triangulated P -ﬁbred categories in algebraic geometry 43 Here again, we can appl y the preceding proposition and conclude. 2.2.c Further properties W e will be par ticularl y interested in the follo wing properties of the tr iangulated P -ﬁbred categor y T . Deﬁnition 2.2.13 Let f : Y / / X be a morphism in S . W e introduced the f ollowing properties f or T , assuming in the third case that T is monoidal: (A dj f ) The functor f ∗ admits a r ight adjoint. U nder this assumption, we denote b y f ! the right adjoint of f ∗ . (BC f ) Any car tesian square of S of the f or m Y 0 f 0 / / g 0   ∆ X 0 g   Y f / / X , is T -transversal (Def. 1.1.17 ) – i.e. the ex chang e transf ormation E x ( ∆ ∗ ∗ ) : g ∗ f ∗ / / f 0 ∗ g 0∗ associated with ∆ is an isomor phism. (PF f ) For any object premotive M ov er Y , and N o v er X , the ex chang e transf or mation (see paragraph 1.1.31 ) E x ( f ∗ ∗ , ⊗ X ) : ( f ∗ M ) ⊗ X N / / f ∗ ( M ⊗ Y f ∗ N ) is an isomor phism. W e denote by (Adj) (resp. (BC), (PF)) the proper ty (Adj f ) (resp. (BC f ), (PF f )) f or an y proper mor phism f in S and call it the adjoint property (resp. proper base c hang e property , projection f ormula ). W e can summar ize the construction and proper ties introduced in this section as f ollow s: Theorem 2.2.14 Assume T satisﬁes the properties (Supp) and (Adj). Then f or any separated mor phism of ﬁnite type f : Y / / X in S , there exists an essentially unique pair of adjoint functors f ! : T ( Y ) / / o o T ( X ) : f ! called the e x ceptional functors , such that: 1. Ther e exists a structur e of a cov ariant (resp. contrav ariant) 2 -functor on f  / / f ! (r esp. f  / / f ! ). 44 Fibred categor ies and the six functors formalism 2. Ther e exists a natural transf or mation α f : f ! / / f ∗ compatible with composition in f whic h is an isomor phism when f is proper . 3. F or any open immersion j , j ! = j ] and j ! = j ∗ . 4. F or any car tesian squar e Y 0 f 0 / / g 0   ∆ X 0 g   Y f / / X , in whic h f is separat ed and of ﬁnite type, ther e exists natural transf or mations E x ( ∆ ∗ ! ) : g ∗ f ! / / f 0 ! g 0 ∗ , E x ( ∆ ! ∗ ) : g 0 ∗ f 0 ! / / f ! g ∗ whic h are isomor phisms in the f ollowing thr ee cases: • f is an open immersion. • g is a P -mor phism. • T satisﬁes the proper base chang e property (BC). Assume that T is in addition monoidal. Then the follo wing property holds: (5) F or any separated mor phism of ﬁnite type f : Y / / X in S , ther e exists natural transf ormations E x ( f ∗ ! , ⊗ ) : ( f ! K ) ⊗ X L / / f ! ( K ⊗ Y f ∗ L ) , Hom X ( f ! ( L ) , K ) / / f ∗ Hom Y ( L , f ! ( K )) , f ! Hom X ( L , M ) / / Hom Y ( f ∗ ( L ) , f ! ( M )) . whic h are isomor phisms in the f ollowing cases: • f is an open immersion. • T satisﬁes the projection formula (PF). Indeed the e xistence of f ! f ollow s from Proposition 2.2.7 while that of f ! f ollow s directly from assumption (Adj). Asser tions (1) and (3) follo ws from the construction, (2) is Proposition 2.2.10 , (4) (resp. (5)) f ollow s from Corollar y 2.2.12 and the deﬁnition of (BC) (resp. (PF)). Note also that the second and third isomor phisms in (5) are obtained by transposition from E x ( f ! , ⊗ ) . 2.2.15 While the proper ties (BC f ) and (PF f ) are only reasonable in practice f or proper mor phisms, this is not the case f or the proper ty (A dj f ). Recall that an ex act functor between well generated tr iangulated categor ies admits a r ight adjoint if and only if it commutes with small sums: this is an immediate consequence of the Brown r epresentability theor em prov ed by Neeman ( cf. [ Nee01 , 8.4.4]). Proposition 2.2.16 Assume that T is a compactly τ -g enerat ed triangulated premo- tivic categor y ov er S . Then, for any mor phism of schemes f : T / / S , the functor f ∗ : T ( T ) / / T ( S ) admits a right adjoint. 2 Triangulated P -ﬁbred categories in algebraic geometry 45 Proof This f ollo ws directly from Proposition 1.3.19 .  2.3 The localization property 2.3.a Deﬁnition 2.3.1 Consider a closed immersion i : Z / / S in S . Let U = S − Z be the comple- ment open subscheme of S and j : U / / S the canonical immersion. W e will use the f ollo wing consequence of the tr iangulated P -ﬁbred structure on T : (a) The unit 1 / / j ∗ j ] is an isomor phism. (b) The counit j ∗ j ∗ / / 1 is an isomor phism. (c) i ∗ j ] = 0 . (d) j ∗ i ∗ = 0 . (e) The composite map j ] j ∗ a d 0 ( j ] , j ∗ ) / / 1 a d ( i ∗ , i ∗ ) / / i ∗ i ∗ is zero. In fact, the ﬁrst f our relations all f ollow from the base chang e proper ty ( P -BC). Relation (e) is a consequence of (d) once we ha v e noticed that the follo wing square is commutativ e j ] j ∗ / /   1   j ] j ∗ i ∗ i ∗ / / i ∗ i ∗ . For the closed immersion i and the triangulated categor y T , w e introduce the property (Loc i ) made of the follo wing assumptions: (a) The pair of functors ( j ∗ , i ∗ ) is conser v ativ e. (b) The counit i ∗ i ∗ a d 0 ( i ∗ , i ∗ ) / / 1 is an isomor phism. Deﬁnition 2.3.2 W e say that T satisﬁes the localization property , denoted by (Loc), if: 1. T ( ∅ ) = 0 . 2. For any closed immersion i in S , (Loc i ) is satisﬁed. The main consequence of the localization axiom is that it leads to the situation of the six gluing functor ( cf. [ BBD82 , prop. 1.4.5]): Proposition 2.3.3 Let i : Z / / S be a closed immersion with complementar y open immersion j : U / / S such that (Loc i ) is satisﬁed. 1. The functor i ∗ admits a right adjoint i ! . 2. F or any K in T ( S ) , ther e exists a unique map ∂ i , K : i ∗ i ∗ K / / j ] j ∗ K [ 1 ] suc h that the triangle j ] j ∗ K a d 0 ( j ] , j ∗ ) / / K a d ( i ∗ , i ∗ ) / / i ∗ i ∗ K ∂ i , K / / j ] j ∗ K [ 1 ] 46 Fibred categories and the six functors formalism is distinguished. The map ∂ i , K is functorial in K . 3. F or any K in T ( S ) , ther e exists a unique map ∂ 0 i , K : j ∗ j ∗ K / / i ∗ i ! K [ 1 ] suc h that the triangle i ∗ i ! K a d 0 ( i ∗ , i ! ) / / K a d ( j ∗ , j ∗ ) / / j ∗ j ∗ K ∂ 0 i , K / / i ∗ i ! K [ 1 ] is distinguished. The map ∂ 0 i , K is functorial in K . U nder the proper ty (Loc i ), the canonical tr iangles appear ing in (2) and (3) abo ve are called the localization triang les associated with i . Proof W e ﬁrst consider point (2). For the exis tence, w e consider a distinguished triangle j ] j ∗ K a d 0 ( j ] , j ∗ ) / / K π / / C + 1 / / Applying 2.3.1 (e), we obtain a factorization K a d ( i ∗ , i ∗ ) / / π & & i ∗ i ∗ K C w 6 6 W e prov e w is an isomor phism. According to the abo v e tr iangle, j ∗ C = 0 . From 2.3.1 (d), j ∗ i ∗ i ∗ K = 0 so that j ∗ w is an isomorphism. Appl ying i ∗ to the abo ve distinguished triangle, we obtain from 2.3.1 (c) that i ∗ π is an isomor phism. Thus, applying i ∗ to the abov e commutative diagram together with (Loc i ) (b), we obtain that i ∗ w is an isomor phism which concludes. Consider a map K u / / L in T ( S ) and suppose we hav e chosen maps a and b in the diagram: j ] j ∗ K a d 0 ( j ] , j ∗ ) / / u   K a d ( i ∗ , i ∗ ) / / u   i ∗ i ∗ K a / / j ] j ∗ K [ 1 ] u   j ] j ∗ L a d 0 ( j ] , j ∗ ) / / L a d ( i ∗ , i ∗ ) / / i ∗ i ∗ L b / / j ] j ∗ L [ 1 ] such that the hor izontal lines are distinguished tr iangles. W e can ﬁnd a map h : i ∗ i ∗ K / / i ∗ i ∗ L completing the previous diagram into a mor phism of triangles. Then the map w = h − i ∗ i ∗ ( u ) satisfy the relation w ◦ a d ( i ∗ , i ∗ ) = 0 . Thus it can be lifted to a map in Hom ( j ] j ∗ K [ 1 ] , i ∗ i ∗ L ) . But this is zero by adjunction and the relation 2.3.1 (d). This pro v es both the naturality of ∂ i , K and its uniqueness. For point (1) and (3), f or an y object K of T ( S ) , we consider a distinguished triangle D / / K a d ( j ∗ , j ∗ ) / / j ∗ j ∗ K + 1 / / A ccording to 2.3.1 (b), j ∗ D = 0 . Thus according to the tr iangle of point (2) applied to D , w e obtain D = i ∗ i ∗ D . Arguing as f or point (2), we thus obtain that D is unique and depends functorially on K so that, if we put i ! K = i ∗ D , point (1) and (3) follo ws.  2 Triangulated P -ﬁbred categories in algebraic geometry 47 Remar k 2.3.4 Consider the hypothesis and notations of the previous proposition. 1. By transposition from 2.3.1 (d), we deduce that i ! j ∗ = 0 . 2. Assume that i is a P -mor phism. Then the P -base change f or mula implies that i ∗ j ∗ = 0 . Dually , w e g et that i ! j ] = 0 . By adjunction, w e thus obtain ∂ i , K = 0 and ∂ 0 i , K = 0 f or an y object K so that both localization tr iangles are split. In that case, w e get that T ( S ) = T ( Z ) × T ( U ) . 46 The preceding proposition admits the follo wing reciprocal statement: Lemma 2.3.5 Consider a closed immersion i : Z / / S in S with complementar y open immersion j : U / / S . Then the f ollowing pr operties are equiv alent: (i) T satisﬁes (Loc i ). (ii) (a) The functor i ∗ is conser v ative. (b) F or any object K of T ( S ) , ther e exists a map i ∗ i ∗ ( K ) / / j ] j ∗ ( K )[ 1 ] which ﬁts into a distinguished triang le j ] j ∗ ( K ) a d 0 ( j ] , j ∗ ) / / K a d ( i ∗ , i ∗ ) / / i ∗ i ∗ ( K ) / / j ] j ∗ ( K )[ 1 ] Proof The fact (i) implies (ii) f ollow s from Proposition 2.3.3 . Conv ersely , (ii)(b) implies that the pair ( i ∗ , j ∗ ) is conser vativ e and it remains to prov e (Loc i ) (b). Let K be an object of T ( S ) . Consider the distinguished triangle given b y (ii)(b): j ] j ∗ ( K ) a d 0 ( j ] , j ∗ ) / / K a d ( i ∗ , i ∗ ) / / i ∗ i ∗ ( K ) / / j ] j ∗ ( K )[ 1 ] . If w e apply i ∗ on the left to this tr iangle, we get using 2.3.1 (d) that the mor phism i ∗ ( K ) a d ( i ∗ , i ∗ ) . i ∗ / / i ∗ i ∗ i ∗ ( K ) is an isomor phism. Hence, by the zig-zag equation, the mor phism i ∗ i ∗ i ∗ ( K ) i ∗ . a d 0 ( i ∗ , i ∗ ) / / i ∗ ( K ) is an isomor phism. Proper ty (ii)(a) thus implies that i ∗ i ∗ ( K ) ' K .  2.3.b First consequences of localization The f ollo wing statement is straightf or w ard. Proposition 2.3.6 Assume T satisﬁes the localization property and consider a sc heme S in S . 46 This remark explains why the localization property is too strong for generalized premotivic categories. 48 F ibred categor ies and the six functors formalism 1. Let S r e d be the reduced sc heme associated with S . The canonical immersion S r e d ν / / S induces an equivalence of categories: ν ∗ : T ( S ) / / T ( S r e d ) . 2. F or any any par tition 3. partition ( S i ν i / / S ) i ∈ I of S by locally closed subse ts, the f amily of functors ( ν ∗ i ) i ∈ I is conser vativ e ( S i is considered with its canonical structure of a r educed subsc heme of S ). Lemma 2.3.7 If T satisﬁes the localization property (Loc) then it is additive. Proof N ote that, by assumption, T ( ∅ ) = 0 . Then the asser tion f ollow s directly from Lemma 2.2.2 .  Proposition 2.3.8 If T satisﬁes the localization property then it satisﬁes the c dh - separation property . Proof Consider a car tesian square of schemes B / /   Q Y p   A e / / X . A ccording to Lemma 2.1.12 , we ha v e only to check that the pair of functors ( e ∗ , p ∗ ) is conser vativ e when Q is a Nisnevich (or respectivel y a proper cdh) distinguished square. Let ν : A 0 / / X be the complementar y closed (resp. open) immersion to e , where A 0 has the induced reduced subscheme (resp. induced subscheme) structure. Consider the car tesian square Y p   B 0 q   o o X A 0 ν o o By assumption on Q , q is an isomor phism. A ccording to (Loc) (ii), ( e ∗ , ν ∗ ) is conservativ e. This concludes.  The f ollowing proposition can be f ound in a slightly less precise and general f orm in [ A y o07a , 2.1.162]. 47 Proposition 2.3.9 Assume T satisﬁes the localization property . Then the f ollowing conditions ar e equiv alent: (i) T is separat ed. (ii) F or a morphism f : T / / S in S , f ∗ : T ( S ) / / T ( T ) is conser vativ e when- ev er f is: 47 A warning: the proof in loc. cit. seems to require that the schemes are ex cellent. 2 Triangulated P -ﬁbred categories in algebraic geometry 49 (a) a ﬁnite étale cov er ; (b) ﬁnite, faithfully ﬂat and radicial. Proof Onl y ( ii ) ⇒ ( i ) requires a proof. Consider a surjective morphism of ﬁnite type f : T / / S in S . According to [ GD67 , 17.16.4], there e xists a par tition ( S i ) i ∈ I of S by (aﬃne) subschemes and a famil y of maps of the form S 00 i g i / / S 0 i h i / / S i such that g i (resp. h i ) satisﬁes assumption (a) (resp. (b)) abo v e and such that f or an y i ∈ I , f × S S 00 i admits a section. Thus, Proposition 2.3.6 concludes.  2.3.c Localization and ex change properties 2.3.10 Consider a mor phism of complete tr iangulated P -ﬁbred categor ies ov er S : ϕ ∗ : T / / T 0 . Recall that for any mor phism f : Y / / X , there is an ex chang e transf ormation ( 1.2.5.1 ): E x ( ϕ ∗ , f ∗ ) : ϕ ∗ X f ∗ / / f ∗ ϕ ∗ Y . If T and T 0 satisﬁes the support axiom and f is separated of ﬁnite type, we hav e constructed (Proposition 2.2.11 ) another ex chang e transf or mation: E x ( ϕ ∗ , f ! ) : ϕ ∗ X f ! / / f ! ϕ ∗ Y . Proposition 2.3.11 Consider a morphism ϕ ∗ : T / / T 0 as abov e. 1. Let i : Z / / X be a closed immersion suc h that T and T 0 satisfy property (Loc i ). Then the exc hang e E x ( ϕ ∗ , i ∗ ) : ϕ ∗ X i ∗ / / i ∗ ϕ ∗ Z is an isomorphism. 2. Assume T and T 0 satisfy property (Loc). Then the f ollowing conditions ar e equiv alent: (i) F or any integ er n > 0 and any scheme X in S , the exchang e E x ( ϕ ∗ , p n ∗ ) is an isomorphism where p n : P n X / / X is the canonical projection. (ii) F or any proper mor phism f : Y / / X , the exc hang e E x ( ϕ ∗ , f ∗ ) is an iso- morphism. 3. Assume T and T 0 satisfy properties (Loc) and (Supp). Then conditions (i) and (ii) abov e are equivalent to the follo wing one: (iii) F or any separated morphism f : Y / / X of ﬁnite type, the exchang e E x ( ϕ ∗ , f ! ) is an isomor phism. Remar k 2.3.12 W e will simply say that ϕ ∗ commutes with f ! when asser tion (iii) is fulﬁlled. For an important case where this happens, see Proposition 2.4.53 . 50 Fibred categories and the six functors formalism Proof Assertion (1) f ollow s easily from the conser vativity of ( i ∗ , j ∗ ) where j is the complementary open immersion and the relations of paragraph 2.3.1 . Asser tion (3) is an easy consequence of the deﬁnition of f ! and the e x chang e E x ( ϕ ∗ , f ! ) . Concerning asser tion (2), we ha v e to prov e that (i) implies (ii). W e ﬁx a mor - phism f : Y / / X and prov e that the e x chang e E x ( ϕ ∗ , f ∗ ) : ϕ ∗ Y f ∗ / / f ∗ ϕ ∗ X is an isomorphism. W e ﬁrst treat the case where f is projectiv e. According to Proposition 2.3.8 , T 0 satisﬁes the Zariski separation proper ty . Using the ( P -BC) proper ty , we see that the problem is local in X so that we can assume X is aﬃne. Then X admits an ample line bundle and there e xists an integ er n > 0 such that f can be factored ([ GD61 , (5.5.4)(ii)]) into a closed immersion i : Y / / P n X and the projection p n : P n X / / X . Thus, asser tion (1) and assumption (i) allow us to conclude. T o treat the general case, we argue b y noether ian induction on Y , assuming that f or any proper closed subscheme T of Y , the result is known f or the restriction of f to T . In fact, the case T = ∅ is obvious because T ( ∅ ) = 0 . A ccording to Chow’ s lemma [ GD61 , 5.6.2], there exis ts a mor phism p : Y 0 / / Y such that: (a) p and f ◦ p are projective mor phisms. (b) There e xists a dense open subscheme V 0 of Y o v er which p is an isomor phism. Let T be the complement of V in Y equipped with its reduced subscheme str ucture. Let j and i be the respective immersion of T and V in Y . A ccording to point (3) of Proposition 2.3.3 , it is suﬃcient to prov e that the f ollowing natural transformations are isomor phisms: ϕ ∗ Y f ∗ i ∗ / / f ∗ ϕ ∗ X i ∗ . (2.3.12.1) ϕ ∗ Y f ∗ j ∗ / / f ∗ ϕ ∗ X j ∗ . (2.3.12.2) Concerning the ﬁrst one, w e consider the follo wing commutative diag ram: ϕ ∗ Y f ∗ i ∗ E x ( ϕ ∗ , f ∗ ) / / f ∗ ϕ ∗ X i ∗ E x ( ϕ ∗ , i ∗ ) / / f ∗ i ∗ ϕ ∗ X ϕ ∗ Y ( f i ) ∗ E x ( ϕ ∗ , ( f i ) ∗ ) / / ( f i ) ∗ ϕ ∗ X . Thus the result f ollo ws from asser tion (1) and the induction hypothesis. Concerning the natural transf ormation ( 2.3.12.2 ), w e consider the pullback square V 0 l / / q   Y 0 p   V j / / Y . Assumption (b) abo v e say s that q is an isomor phism which implies the relation: j ∗ = p ∗ l ∗ q ∗ . In par ticular , it is suﬃcient to prov e that the natural transf or mation 2 Triangulated P -ﬁbred categories in algebraic geometry 51 ϕ ∗ Y f ∗ p ∗ / / f ∗ ϕ ∗ X p ∗ is an isomor phism. This follo ws from the commutativity of the f ollowing diag ram ϕ ∗ Y f ∗ p ∗ E x ( ϕ ∗ , f ∗ ) / / f ∗ ϕ ∗ X p ∗ E x ( ϕ ∗ , p ∗ ) / / f ∗ p ∗ ϕ ∗ X ϕ ∗ Y ( f p ) ∗ E x ( ϕ ∗ , ( f p ) ∗ ) / / ( f p ) ∗ ϕ ∗ X , according to the projective case treated abo v e and assumption (b). The proof is complete.  Corollary 2.3.13 In the next statements, we assume T is monoidal when it is needed. 1. Let i : Z / / X be a closed immersion suc h that T satisﬁes property (Loc i ). Then T satisﬁes property (Supp i ) (r esp. (BC i ), (PF i )). 2. Assume T satisﬁes the localization property . Then the following properties of T ar e equivalent : (i) F or any int eg er n > 0 and any sc heme X in S , p n : P n X / / X being the canonical projection, T satisﬁes (Supp p n ) (r esp. (BC p n ), (PF p n )). (ii) T satisﬁes (Supp) (resp. (BC), (PF)). 3. Assume T is well gener ated and satisﬁes the localization property . Then the f ollowing pr operties of T ar e equiv alent: (i’) F or any int eg er n > 0 and any sc heme X in S , p n : P n X / / X being the canonical projection, T satisﬁes (Adj p n ). (ii’) T satisﬁes (Adj). Proof As in the proof of Corollar y 2.2.12 , each respective case of asser tions (1) and (2) f ollo ws from the previous proposition applied to a par ticular type of mor phisms ϕ ∗ : T 0 / / T 00 of complete P -ﬁbred tr iangulated categories o ver a subcategor y S 0 of S . For proper ty (Supp), we proceed as f ollo ws. W e ﬁx an open immersion j : U / / X and let S 0 = S / X . For any Y / X , we let j Y = Y × X U / / Y be the pullback of j . W e put T 0 ( Y ) = T ( Y × X U ) and T 00 ( Y ) = T ( Y ) and let ϕ ∗ Y be the functor: j Y ] : T ( Y × X U ) / / T ( Y ) . For the proper ty (BC) (resp. (PF)), we refer the reader to the proof of asser tion (1) (resp. (2)) in Corollar y 2.2.12 . Finall y w e consider asser tion (3). It is suﬃcient to pro v e that ( i’ ) implies ( ii’ ). A ccording to the Brown representability theorem [ Nee01 , 8.4.4], the proper ty (Adj f ) f or a proper mor phism f is equiv alent to ask that f ∗ preserves small sum. Consider an arbitrar y set I . F or an y scheme S , we put T I ( S ) = T ( S ) I , that is the category of families of object of T ( S ) inde xed by I . Then T I is ob viously a complete triangulated P -ﬁbred categor y ov er S (limits and colimits are computed termwise). F or any scheme S , we consider the functor: 52 Fibred categor ies and the six functors formalism ϕ ∗ S : T I ( S ) / / T ( S ) , ( M i ) i ∈ I  / / Ê i ∈ I M i . Then ϕ ∗ : T I / / T is ob viously a mor phism of complete P -ﬁbred categor ies. Thus, giv en condition ( i’ ), the preceding proposition applied to ϕ ∗ sho ws that f or an y proper mor phism f , f ∗ commutes with sums inde xed by I . As this is true f or any I , we obtain ( ii’ ).  2.3.d Localization and monoidal structure 2.3.14 Assume T is monoidal and let M denote its geometric sections. Fix a closed immersion i : Z / / S in S with complementary open immersion j : U / / S . W e ﬁx an object M S ( S / S − Z ) of T ( S ) and a distinguished tr iangle (2.3.14.1) M S ( S − Z ) j ∗ / / 1 S p i / / M S ( S / S − Z ) d i / / M S ( S − Z )[ 1 ] . Remark that according to 2.3.1 (c), the map i ∗ ( p i ) : 1 Z / / i ∗ M S ( S / S − Z ) is an isomorphism. Giv en any object K in T ( S ) , w e thus obtain an isomor phism i ∗ ( M S ( S / S − Z ) ⊗ S K ) = i ∗ ( M S ( S / S − Z )) ⊗ Z i ∗ ( K ) ( i ∗ p i ) − 1 / / 1 Z ⊗ Z i ∗ ( K ) = i ∗ ( K ) which is natural in K . It induces by adjunction a map (2.3.14.2) ψ i , K : M S ( S / S − Z ) ⊗ S K / / i ∗ i ∗ ( K ) which is natural in K . For any P -scheme X / S , we put M S ( X / X − X Z ) = M S ( S / S − Z ) ⊗ S M S ( X ) so that w e get from ( 2.3.14.1 ) a canonical distinguished triangle: M S ( X − X Z ) j X ∗ / / M S ( X ) / / M S ( X / X − X Z ) / / M S ( X − X Z )[ 1 ] . The map ( 2.3.14.2 ) f or K = M S ( X ) giv es a canonical map (2.3.14.3) ψ i , X : M S ( X / X − X Z ) / / i ∗ ( M Z ( X Z )) . Proposition 2.3.15 Consider the pr evious hypothesis and notations. Then t he f ol- lowing conditions ar e equivalent : (i) T satisﬁes the property (Loc i ). (ii) (a) The functor i ∗ is conser v ative. (b) The morphism ψ i , S : M S ( S / S − Z ) / / i ∗ ( 1 Z ) is an isomor phism. (c) F or any object K of T ( S ) , the exc hang e transf ormation E x ( i ∗ ∗ , ⊗ ) : ( i ∗ 1 Z ) ⊗ S K / / i ∗ i ∗ K 2 Triangulated P -ﬁbred categories in algebraic geometry 53 is an isomorphism. (iii) (a) The functor i ∗ is conser v ative. (b) The morphism ψ i , S : M S ( S / S − Z ) / / i ∗ ( 1 Z ) is an isomor phism. (c) F or any objects K and L of T ( S ) , the exc hang e transf ormation E x ( i ∗ ∗ , ⊗ ) : ( i ∗ K ) ⊗ S L / / i ∗ ( K ⊗ Z i ∗ L ) is an isomorphism. Assume in addition that T is w ell g enerated and τ -g enerated as a triangulated P -ﬁbred categor y . Then the abov e conditions are equivalent to the follo wing one: (iv) (a) The functor i ∗ is conser v ativ e, commutes with direct sums and with τ -twists. (b) The morphism ψ i , X : M S ( X / X − X Z ) / / i ∗ ( M Z ( X Z )) is an isomorphism f or any P -sc heme X / S . In par ticular , (Loc i ) implies that f or any object K of T ( S ) , the localization tr iangle of 2.3.3 j ] j ∗ ( K ) / / K / / i ∗ i ∗ ( K ) ∂ K / / j ] j ∗ ( K )[ 1 ] is canonically isomor phic (through e x chang e transformations) to the tr iangle ( 2.3.14.1 ) tensored with K . Proof ( i ) ⇒ ( iii ) : According to (Loc i ) (a), w e need only to check that the maps in (iii)(b) and (iii)(c) are isomor phisms after applying i ∗ and j ∗ . This f ollo ws easil y from (Loc i ) (b). ( iii ) ⇒ ( ii ) : Obvious ( ii ) ⇒ ( i ) : According to (ii)(b), the distinguished triangle ( 2.3.14.1 ) is isomor phic to a tr iangle of the f orm j ] j ∗ ( 1 S ) a d 0 ( j ] , j ∗ ) / / 1 S a d ( i ∗ , i ∗ ) / / i ∗ i ∗ ( 1 S ) / / j ] j ∗ ( 1 S ) . A ccording to (ii)(c), this latter triangle tensored with K is isomor phic through e x chang e transf ormations to a tr iangle of the f orm j ] j ∗ ( K ) a d 0 ( j ] , j ∗ ) / / K a d ( i ∗ , i ∗ ) / / i ∗ i ∗ ( K ) / / j ] j ∗ ( K ) . Thus Lemma 2.3.5 allow s us to conclude. T o end the proof, we remark by using the equations for the adjunction ( i ∗ , i ∗ ) that f or an y object M of T ( S ) , the f ollo wing diagram is commutativ e: i ∗ i ∗ ( 1 S ) ⊗ K i ∗ ( 1 Z ) ⊗ K E x ( i ∗ ∗ , ⊗ )   M S ( S / S − Z ) ⊗ K ψ i ⊗ 1 K 3 3 ψ i , K + + i ∗ i ∗ ( K ) i ∗ ( 1 Z ⊗ i ∗ i ∗ ( K )) . 54 Fibred categories and the six functors formalism Note that (i) implies that i ∗ is conservativ e and commutes with direct sums (see 2.3.3 ) and (ii)(c) implies it commutes with twists. According to the abov e diagram, (ii)(b) implies (iv)(b). W e prov e that reciprocall y that (iv) implies (ii). Because (ii)(b) (resp. (ii)(a)) is a particular case of (iv)(b) (resp. (iv)(a)), we ha v e only to prov e (ii)(b). In view of the previous diagram, w e are reduced to pro v e that f or an y object K of T ( S ) , the map ψ i , K is an isomor phism. Consider the full subcategor y U of T ( S ) made of the objects K such that ψ i , K is an isomor phism. Then U is tr iangulated. Using (iv)(a), U is stable by small sums and τ -twists. By assumption, it contains the objects of the f or m M S ( X ) f or a P -scheme X / S . Thus, because T is well g enerated by assumption, Lemma 1.3.17 concludes.  Lemma 2.3.16 Consider a closed immersion i : Z / / S . W e assume the follo wing conditions ar e satisﬁed in addition to that of 2.0.1 : • T is w ell g enerated, τ -gener ated, and satisﬁes the Zariski separation property. • F or any P -sc heme X 0 / Z and any point x 0 of X 0 , ther e exists an open neigh- borhood U 0 of x 0 in X 0 and a P -scheme U / S suc h that U 0 = U × S Z . 48 Then the functor i ∗ is conser v ative. Proof Consider an object K of T ( Z ) such that i ∗ ( K ) = 0 . W e pro ve that K = 0 . Because T is τ -generated, it is suﬃcient to prov e that for a P -mor phism p 0 : X 0 / / Z and a twist ( n , m ) ∈ Z × τ , Hom T ( Z ) ( M Z ( X 0 ){ m } [ n ] , K ) = 0 . Because M Z ( X 0 ) = p 0 ] ( 1 X 0 ) , this equiv alent to prov e that Hom T ( X 0 ) ( 1 X 0 { m }[ n ] , p ∗ 0 ( K )) = 0 . Using the Zariski separation property on T , this latter assumption is local in X 0 . Thus, according to the assumption on the class P , we can assume there exis ts a P -scheme X / S such that X 0 = X × S Z . Thus M Z ( X 0 ){ m } [ n ] = i ∗ ( M S ( X ){ m } [ n ]) and the initial assumption on K allo ws us to conclude.  Note for future applications the f ollowing interesting corollar ies: Corollary 2.3.17 Assume T is a pr emotivic triangulated categor y whic h is com- pactly τ -g enerat ed for a gr oup of twists τ ( i.e. any twists in τ admits a tensor inver se) and whic h satisﬁes the Zariski separation pr operty. Then, f or any closed immersion i , the functor i ∗ is conser v ativ e, commutes with sums and with twists. This is a consequence of lemmas 2.3.16 and 2.2.16 . In fact, under these conditions, i ∗ commutes with arbitrar y τ -twists because it is tr ue f or its (left) adjoint i ∗ . 48 This proper ty is trivial when P is the class of open immersions or the class of mor phisms of ﬁnite type in S . It is also true when P is the class of étale mor phism or P = Sm ( cf. [ GD67 , 18.1.1]). 2 Triangulated P -ﬁbred categories in algebraic geometry 55 Corollary 2.3.18 Assume T satisﬁes the assump tions of the preceding corollar y . Then the f ollowing conditions on a closed immersion i ar e equivalent : (i) T satisﬁes the property (Loc i ). (ii) F or any scheme S in S and any smooth S -sc heme X , the map ( 2.3.14.3 ) ψ i , X : M S ( X / X − X Z ) / / i ∗ M Z ( X Z ) is an isomorphism. W e ﬁnish this section with the follo wing useful result: Proposition 2.3.19 Assume T is τ -g enerated and consider a τ 0 -g enerated triangu- lated P -ﬁbr ed categor y T 0 and a morphism ϕ ∗ : ( T , τ ) / / o o ( T 0 , τ 0 ) : ϕ ∗ . W e assume the f ollowing pr operties: (a) the morphism ϕ ∗ is strictly compatible with twists; (b) T 0 is w ell g enerat ed. W e consider a closed immersion i : Z / / S and further assume the follo wing properties: (c) T satisﬁes the property (Loc i ). (d) The exc hang e transf or mation E x ( ϕ ∗ , i ∗ ) : ϕ ∗ i ∗ / / i ∗ ϕ ∗ is an isomorphism. (e) The functor i ∗ : T 0 ( Z ) / / T 0 ( S ) commutes with τ 0 -twists. 49 Then T 0 satisﬁes the pr operty (Loc i ). Proof N ote that, under the abo v e assumptions, ϕ ∗ is conser vativ e (in f act, for an y P -scheme X / S and an y twists i ∈ τ 0 , the premotiv e M S ( X ){ i } is in the essential image of ϕ ∗ ). Thus, if i ∗ : T ( Z ) / / T ( S ) is conser vativ e (resp. commute with sums), then i ∗ : T 0 ( S ) / / T 0 ( S ) is conservativ e (resp. commute with sums) using the isomor phism ϕ ∗ i ∗ ' i ∗ ϕ ∗ . Let M (resp. M 0 ) be the geometric sections of T (resp. T 0 ). As in 2.3.14 , we ﬁx a distinguished tr iangle M S ( S − Z ) j ∗ / / 1 S p i / / M S ( S / S − Z ) d i / / M S ( S − Z )[ 1 ] . and we put M 0 S ( S / S − Z ) = ϕ ∗ M S ( S / S − Z ) . According to loc. cit. , we thus g et f or an y P -scheme X / S canonical maps ψ i , X : M S ( X / X − X Z ) / / i ∗ M Z ( X Z ) , ψ 0 i , X : M 0 S ( X / X − X Z ) / / i ∗ M 0 Z ( X Z ) . 49 This will be satisﬁed if an y τ 0 -twists is inv ertible because the left adjoint of i ∗ commutes with τ 0 -twists. 56 Fibred categor ies and the six functors formalism By construction, the follo wing diagram is commutativ e: ϕ ∗ M S ( X / X − X Z ) ϕ ∗ ψ i , X / / ϕ ∗ i ∗ M Z ( X Z ) E x ( ϕ ∗ , i ∗ ) / / i ∗ ϕ ∗ M Z ( X Z ) M 0 S ( X / X − X Z ) ψ 0 i , X / / M 0 Z ( X Z ) Thus, Proposition 2.3.15 allow s us to conclude.  2.4 Purity and the theorem of V oe v odsky-R öndigs-A y oub Recall we assume P = Sm in this section. 2.4.a The stability property The f ollowing section is directly inspired b y the w ork of A y oub in [ A yo07a , §1.5]. 50 W e claim no or iginality e xcept for a closer look on the needed axioms. Deﬁnition 2.4.1 A pointed smooth S -sc heme will be a couple ( f , s ) of mor phisms of S such that f : X / / S is a smooth separated mor phism of ﬁnite type and s : S / / X is a section of f . W e associate with a pointed smooth scheme ( f , s ) the f ollo wing endofunctor of T ( S ) T h ( f , s ) : = f ] s ∗ called the associated Thom transf or mation . If T satisﬁes (Adj s ) (recall: s ∗ admits a r ight adjoint denoted b y s ! ), w e put T h 0 ( f , s ) : = s ! f ∗ and call it the associated adjoint Thom transf or mation . Remar k 2.4.2 Note that because f is separated, s is a closed immersion. Example 2.4.3 1. Let p : E / / X be a v ector bundle and s 0 be its zero section. Follo wing [ A yo07a ], w e put T h ( E ) : = T h ( p , s 0 ) and simply call it the Thom transf or mation associated with E / X . 2. Consider a pointed smooth S -scheme ( f , s ) such that f is étale. Then s is an open and closed immersion. Thus, if T is additive, s ∗ = s ] according to Lemma 2.2.2 . In par ticular , T h ( f , s ) = I d S . Deﬁnition 2.4.4 W e will say that T satisﬁes the stability property , denoted b y (Stab), if for any point smooth scheme ( f , s ) , the Thom transf or mation T h ( f , s ) is an equiv alence of categor ies. 50 See also [ Del01 , §5]. 2 Triangulated P -ﬁbred categories in algebraic geometry 57 2.4.5 Consider a commutativ e diagram in S of the f orm S t # # t 0   Y 0 s 0 / / p 0   ∆ Y g " " p   S s / / X f / / S (2.4.5.1) such that ∆ is a car tesian square, ( f , s ) , ( g, t ) are smooth pointed schemes and g is a smooth separated mor phism of ﬁnite type. Then we get a canonical e x chang e morphism: (2.4.5.2) T h ( g , t ) = f ] p ] s 0 ∗ t 0 ∗ E x ( ∆ ] ∗ ) / / f ] s ∗ p 0 ] t 0 ∗ = T h ( f , s ) T h ( p 0 , t 0 ) . This is an isomor phism as soon as E x ( ∆ ] ∗ ) is an isomor phism. The f ollo wing lemma giv es a suﬃcient condition for this to happen. Lemma 2.4.6 Consider the abov e notations. If T satisﬁes (Loc s ) then the natural transf ormations E x ( ∆ ] ∗ ) is an isomor phism f or any squar e ∆ as abov e. This lemma follo ws easily from the deﬁnition of (Loc s ), the relations of paragraph 2.3.1 and the P -base chang e f or mula ( P -BC). It motivates the next deﬁnition: Deﬁnition 2.4.7 W e say that T satisﬁes the w eak localization property (wLoc) if it satisﬁes (Loc s ) f or an y closed immersion s which admits a smooth retraction. Proposition 2.4.8 Assume that T satisﬁes the Nisnevich separation property . Then the f ollowing conditions ar e equivalent : (i) T satisﬁes (wLoc). (ii) F or any scheme S and any closed immersion i : Z / / X between smooth S - sc hemes, T satisﬁes (Loc i ). Proof Of course, (ii) implies (i). W e prov e the reciprocal statement. The Nisnevic h separation proper ty say s that f or any Nisnevich cov er f : X 0 / / X , the functor f ∗ is conservativ e. W e deduce from that point the proper ties (Loc i ) (a) and (Loc i ) (b) are local in X with respect to the Nisnevic h topology – f or (b), one also uses the smooth projection formula. Thus, we can conclude as locally for the Nisnevich topology , i admits a smooth retraction (see for ex ample [ Dég07 , 4.5.11]).  Applying the second point of Example 2.4.3 , we easil y deduce from that con- struction the f ollo wing kind of ex cision proper ty: Lemma 2.4.9 Assume that T satisﬁes (wLoc). Then, given any diagr am ( 2.4.5.1 ) satisfying the assumption as abo v e and suc h that p is étale, the natural transf or mation ( 2.4.5.2 ) gives an isomor phism: T h ( g , t ) ∼ / / T h ( f , s ) . 58 F ibred categor ies and the six functors formalism 2.4.10 T o an y shor t e xact sequence of v ector bundles o v er a scheme S ( σ ) 0 / / E 0 ν / / E π / / E 00 / / 0 , w e can associate a commutative diag ram S ! !   E 0 ν / /   ∆ E π   S / / E 00 / / S where the non labelled map are either the canonical projections or the zero sections of the rele vant vector bundles, and ∆ is cartesian. Using the notation of Example 2.4.3 , the e x chang e transformation ( 2.4.5.2 ) associated with this diag ram has the f ollowing form: T h ( σ ) : T h ( E ) / / T h ( E 00 ) ◦ T h ( E 0 ) . Recall from the abo v e that this natural transf or mation is an isomor phism as soon as T satisﬁes (wLoc). Proposition 2.4.11 Assume T satisﬁes (wLoc) and ( Zar -sep). Then the f ollowing conditions ar e equivalent : (i) The complete triangulated Sm -ﬁbred category T satisﬁes the stability property. (ii) F or any scheme S , the Thom tr ansformation T h ( A 1 S ) is an equivalence of cate- gories. Proof W e ha v e to pro v e that (ii) implies (i). Note that according to the abo ve paragraph, w e already now that for an y scheme S and any integ er n ≥ 0 , T h ( A n S ) ' T h ( A 1 S ) ◦ , n is an equiv alence. W e consider a smooth pointed scheme ( f : X / / S , s ) and w e pro v e that T h ( f , s ) is an equiv alence. Recall that (Loc s ) implies (Adj) s (ﬁrst point of Proposition 2.3.3 ). In par ticular , T h ( f , s ) admits a right adjoint T h 0 ( f , s ) and we ha ve to prov e that the adjunction morphisms are isomor phisms. Consider an open immersion j : U / / S and let ( f 0 , s 0 ) be the restr iction of the smooth S -point ( f , s ) ov er U . Proper ty (Loc s ) implies (BC s ) (Corollar y 2.3.13 ). Thus, using also proper ty ( P -BC), we obtain a canonical isomor phism: j ∗ T h ( f , s ) ∼ / / T h ( f 0 , s 0 ) j ∗ . Recall also that (Loc s ) implies (Supp s ) (again Corollar y 2.3.13 ). Thus we get a canonical isomor phism: j ] T h ( f 0 , s 0 ) ∼ / / T h ( f , s ) j ] 2 Triangulated P -ﬁbred categories in algebraic geometry 59 which gives by adjunction an isomor phism: T h 0 ( f 0 , s 0 ) j ∗ ∼ / / j ∗ T h 0 ( f , s ) . Thus, ( Zar -sep) show s that the proper ty f or T h ( f , s ) to be an equivalence is Zar iski local in S . Consider a point a ∈ S , x = s ( a ) . As X is smooth o ver S , there e xists an open subscheme U ⊂ X , an integer n ≥ 0 and an étale S -morphism π : U / / A n S which ﬁts into the f ollo wing car tesian square: S 0 / /   U π   S ν / / A n S where ν is the zero section ( cf. [ GD67 , 17.12.2]). Note that the scheme S 0 = s − 1 ( U ) is an open neighborhood of a in S . Let us put X 0 = f − 1 ( S 0 ) and U 0 = U ∩ X 0 . Then w e get the f ollowing commutativ e diag ram: X 0 f 0 ' ' S 0 s 0 7 7 ν 0 & & s 0 0 / / U 0 ?  O O π 0   f 0 0 / / S 0 A n S 0 8 8 where π 0 is the restriction of π abo ve S 0 and ν 0 is again the zero section. A ccording to Lemma 2.4.9 , we get isomorphisms T h ( f 0 , s 0 ) ' T h ( f 0 0 , s 0 0 ) ' T h ( A n S ) . Thus, according to the beginning of the proof, T h ( f 0 , s 0 ) is an equiv alence. This concludes because S 0 is an open neighborhood of a in S .  Deﬁnition 2.4.12 Assume that T is monoidal. 1. For any smooth pointed scheme ( f : X / / S , s ) , we put M S  X / X − s ( S )  : = f ] s ∗ ( 1 S ) . 2. For any vector bundle E / S with projection f and zero section s , we deﬁne the Thom pr emotiv e associated with E o v er S as M T h S ( E ) = f ] s ∗ ( 1 S ) . 2.4.13 W e assume T is monoidal and satisﬁes proper ties (wLoc) and ( Zar -sep). In each case of the previous deﬁnition, if we apply f ] to the distinguished tr iangle obtained from point (2) of Proposition 2.3.3 applied to s , we get the f ollo wing canonical distinguished triangles: 60 Fibred categor ies and the six functors formalism M S  X − s ( S )  / / M S ( X ) / / M S  X / X − s ( S )  + 1 / / M S ( E × ) / / M S ( E ) / / M T h S ( E ) + 1 / / where the ﬁrst map is induced by the obvious open immersion. Moreo v er , proper ty (Loc s ) implies (PF s ) (see Corollar y 2.3.13 ). Thus for any premotiv e K o v er S , the f ollo wing composite map is an isomor phism: T h ( f , s ) . K = f ] s ∗ ( K ) = f ] s ∗ ( 1 S ⊗ S s ∗ f ∗ ( K )) E x ( s ∗ ∗ , ⊗ ) − 1 / / f ] ( s ∗ ( 1 S ) ⊗ X f ∗ ( K )) E x ( f ∗ ] , ⊗ ) / / ( f ] s ∗ ( 1 S )) ⊗ S K = M S ( X / X − s ( S )) ⊗ S K (2.4.13.1) Similarl y , in the case of a vector bundle E / S , w e get a canonical isomor phism: T h ( E ) . K ∼ / / M T h S ( E ) ⊗ S K . From these isomor phisms, we deduce easily the f ollo wing corollar y of the previous proposition: Corollary 2.4.14 Consider the abov e notations and assumptions. Then the follo wing properties are equiv alent: (i) T satisﬁes the stability pr operty. (ii) F or any smooth pointed scheme ( X / / S , s ) , the premo tive M S ( X / X − s ( S )) is ⊗ -inv ertible. (iii) F or any vector bundle E / S the Thom pr emotiv e M T h S ( E ) is ⊗ -invertible. (iv) F or any scheme S , the premo tive M T h S ( A 1 S ) is ⊗ -invertible. Remar k 2.4.15 Assume that T satisﬁes the assumptions and the equiv alent condi- tions of the previous corollar y . Then, under the notations of P aragraph 2.4.10 , we associate with the ex act sequence ( σ ) a canonical isomor phism (2.4.15.1) T h S ( σ ) : M T h S ( E ) / / M T h S ( E 00 ) ⊗ S M T h S ( E 0 ) . Recall that Deligne introduced in [ Del87 , 4.12] the Picard category K ( S ) of virtual v ector bundle o v er a scheme S . Then, it follo ws from the abov e isomorphism and the univ ersal proper ties of K ( S ) (see [ Del87 , 4.3]) that the functor M T h S can be extended uniquel y to a symmetr ic monoidal functor: M T h S : K ( S ) / / T ( S ) . The reader is referred to [ A y o07a , th. 1.5.18] for a detailed argument. 2.4.16 Assume T is monoidal. For an y scheme S , the canonical projection p : P 1 S / / S is a split epimor phism. A splitting is given b y the inclusion of the inﬁnite point ν : S / / P 1 S . The induced map p ∗ : M S ( P 1 S ) / / 1 S is a split epimorphism. Thus it admits a kernel K in the triangulated category T ( S ) . 2 Triangulated P -ﬁbred categories in algebraic geometry 61 Deﬁnition 2.4.17 U nder the abov e assumption and notations, we deﬁne the T ate pr emotiv e o v er S as the object 1 S ( 1 ) = K [− 2 ] of T ( S ) . The monoid generated b y the car tesian section ( 1 S ) S deﬁnes a canonical N -twist on T called the T ate twist . The n -th T ate twist of an object K is denoted b y K ( n ) . 2.4.18 Consider again the assumption of Paragraph 2.4.13 . A ccording to Lemma 2.4.9 , we g et a canonical isomor phism M T h S ( A 1 S ) = M S ( A 1 S / A 1 S − { 0 } ) / / M S ( P 1 S / P 1 S − { 0 } ) . On the other hand, 1 S ( 1 )[ 2 ] is by deﬁnition the cokernel of the monomor phism ν ∗ : 1 S / / M S ( P 1 S ) . Thus we get a canonical mor phism: (2.4.18.1) 1 S ( 1 )[ 2 ] / / M S ( P 1 S / P 1 S − { 0 } ) ∼ / / M T h S ( A 1 S ) . From this deﬁnition and Corollar y 2.4.14 the f ollo wing result is obvious: Corollary 2.4.19 Consider the abov e assumption and notations. Then the follo wing conditions ar e equivalent : (i) T satisﬁes the homotopy property . (ii) F or any scheme S , the arro w ( 2.4.18.1 ) is an isomor phism. When these equivalent asser tions ar e satisﬁed, the f ollowing conditions are equiv a- lent : (iii) T satisﬁes the stability pr operty. (iv) F or any scheme S , the T ate pr emotiv e 1 S ( 1 ) is ⊗ -invertible. 2.4.b The purity property 2.4.20 Let f : X / / S be a smooth proper morphism in S . W e consider the f ollowing car tesian square: X × S X f 0 0 / / f 0   ∆ X f   X f / / S (2.4.20.1) where f 0 (resp. f 00 ) is the projection on the ﬁrst (resp. second) f actor . Let δ : X / / X × S X be the diagonal embedding. Note that ( f 0 , δ ) is a smooth pointed scheme which depends only on f . W e put: Σ f : = T h ( f 0 , δ ) = f 0 ] δ ∗ . W e then deﬁne a canonical mor phism: 62 Fibred categor ies and the six functors f or malism p f : f ] = f ] f 00 ∗ δ ∗ E x ( ∆ ] ∗ ) / / f ∗ f 0 ] δ ∗ = f ∗ ◦ Σ f using the e x chang e transf or mation introduced in paragraph 1.1.15 . Deﬁnition 2.4.21 W e sa y that f is T -pure , or simply pure when T is clear, when the f ollo wing conditions are satisﬁed: 1. The natural transf ormation Σ f is an equiv alence. 2. The mor phism p f : f ] / / f ∗ ◦ Σ f is an isomor phism. Then p f is called the purity isomorphism associated with f . W e say also that f is univ ersally T -pure if f is pure after an y base change along a mor phism of S . W e introduce the f ollo wing proper ties on T : • T satisﬁes the purity property (Pur) if any proper smooth mor phism is pure. • T satisﬁes the w eak purity property (wPur) if f or an y scheme S and any integer n > 0 , the canonical projection p n : P n S / / S is pure. Remar k 2.4.22 Consider the abov e notations and assume f is pure. Then f ∗ admits a right adjoint f ! and w e deduce b y transposition from p f a canonical isomor phism: p 0 f : f ∗ / / Σ − 1 f ◦ f ! . Recall also that, when δ ∗ admits a right adjoint δ ! , Σ f admits as a r ight adjoint the transf or mation Ω f : = δ ! f ∗ . In par ticular , Ω f = Σ − 1 f . The f ollo wing lemma sho ws the impor tance of the purity proper ty . Lemma 2.4.23 Assume that T satisﬁes (wLoc). Let f : Y / / X be a proper smooth morphism. If f is univer sally pure then the follo wing conditions hold: 1. T satisﬁes (Supp f ) and (BC f ). 2. F or any car tesian squar e Z ˜ f / / h   ∆ Y g   X f / / S suc h that g is smooth, the exchang e transf ormation: E x ( ∆ ] ∗ ) : g ] ˜ f ∗ / / f ∗ h ] is an isomorphism. 3. If mor eov er T is monoidal then T satisﬁes (PF f ). Proof W e ﬁrst pro v e condition (2). By assumption, the natural transformation Σ ˜ f is an equiv alence. for f and ˜ f : by assumption the natural transf ormations Σ f = f 0 ] δ ∗ and Σ ˜ f = ˜ f 0 ˜ δ ∗ ) are equivalences. Thus, it is suﬃcient to pro ve that the natural transf or mation 2 Triangulated P -ﬁbred categories in algebraic geometry 63 g ] ˜ f ∗ Σ ˜ f E x ( ∆ ] ∗ ) / / f ∗ h ] Σ ˜ f is an isomor phism. For matter of notations, let us also introduce the follo wing car tesian squares: Z ˜ δ / / h   Γ Z × Y Z ˜ f 0 / / k   Θ Z h   X δ / / X × S X f 0 / / X using the notations of 2.4.20 . Thus, by deﬁnition: Σ f = f 0 ] δ ∗ , Σ ˜ f = ˜ f 0 ˜ δ ∗ . Then w e consider the f ollo wing diagram of e x chang e transf ormations: g ] ˜ f ] p ˜ f / / g ] ˜ f ∗ ˜ f 0 ] ˜ δ ∗ E x ( ∆ ] ∗ )   f ] h ] p f / / f ∗ f 0 ] δ ∗ h ] f ∗ f 0 ] k ] ˜ δ ∗ E x ( Γ ] ∗ ) o o f ∗ h ] ˜ f 0 ] ˜ δ ∗ Note that it only inv olv es e x chang e transf ormations of type E x ( ? ] ∗ ) : it is commutativ e b y compatibility of these e x chang e transf ormations with composition. By assump- tion, the transf ormations p f and p ˜ f are isomor phisms. Moreov er the proper ty (Loc δ ) is satisﬁed and it implies (Supp δ ) according to Corollary 2.3.13 . Thus E x ( Γ ] ∗ ) is an isomorphism and this concludes the proof of (2). For condition (1), w e note that (2) already implies (Supp f ). Thus we hav e only to prov e (BC f ). W e consider a square of shape ∆ as in the statement of the lemma without assuming that g is smooth. W e hav e to prov e that E x ( ∆ ∗ ∗ ) : g ∗ f ∗ / / ˜ f ∗ h ∗ is an isomor phism. W e proceed as f or condition (2). It is suﬃcient to prov e that E x ( ∆ ∗ ∗ ) is an isomor phism after composition on the r ight with Σ f . Then we consider the f ollo wing commutativ e diagram of e x chang e transf ormations: g ∗ f ] E x ( ∆ ∗ ] )   p f / / g ∗ f ∗ f 0 ] δ ∗ E x ( ∆ ∗ ∗ )   ˜ f ] h ∗ p ˜ f / / ˜ f ∗ ˜ f 0 ] ˜ δ ∗ h ∗ ˜ f ∗ ˜ f 0 ] k ∗ δ ∗ E x ( Γ ∗ ∗ ) o o ˜ f ∗ h ∗ f 0 ] δ ∗ E x ( Θ ∗ ] ) o o A ccording to ( P -BC), E x ( ∆ ∗ ] ) and E x ( Θ ∗ ] ) are isomorphisms. By assumption, p f and p ˜ f are isomor phisms. Moreo ver , proper ty (Loc δ ) is satisﬁed and this implies E x ( Γ ∗ ∗ ) is an isomor phism according to Corollar y 2.3.13 . Condition (1) is prov ed. 64 Fibred categories and the six functors formalism It remains to prov e (3). W e consider again the notations of the car tesian diag ram ( 2.4.20.1 ). F or an y premotiv es K o ver X and L ov er S , w e consider the f ollowing commutativ e diagram of ex change transf or mations (see Remark 1.1.32 ): f ]  K ⊗ f ∗ ( L )  E x ( f ∗ ] , ⊗ )   p f / / f ∗ f 0 ] δ ∗  K ⊗ δ ∗ f 0∗ f ∗ ( L )  f ∗ f 0 ]  δ ∗ ( K ) ⊗ f 0∗ f ∗ ( L )  E x ( f 0∗ ] , ⊗ )   E x ( δ ∗ ∗ , ⊗ ) O O f ∗  f 0 ] δ ∗ ( K ) ⊗ f ∗ ( L )  f ] ( K ) ⊗ L p f / / f ∗ f 0 ] δ ∗ ( K ) ⊗ L . E x ( f ∗ ∗ , ⊗ ) O O By deﬁnition, the ex chang es E x ( f ∗ ] , ⊗ ) and E x ( f 0∗ ] , ⊗ ) are isomor phisms. By as- sumption, the ar ro w s labeled p f are isomor phisms. Moreo v er , the proper ty (Loc δ ) is satisﬁed: Corollary 2.3.13 implies that E x ( δ ∗ ∗ , ⊗ ) is an isomor phism. W e deduce from this that the ar ro w E x ( f ∗ ∗ , ⊗ ) is an isomor phism. This concludes the proof of (3) as the functor Σ f = f 0 ] δ ∗ is an equiv alence according to the h ypothesis on f .  2.4.24 Assume that T satisﬁes the suppor t property (Supp). Then w e can extend Deﬁnition 2.4.21 to the case of a smooth separated mor phism of ﬁnite type f : X / / S . W e still consider the car tesian square ( 2.4.20.1 ) and the diagonal embedding δ : X / / X × S X . Again, ( f 0 , δ ) is a smooth pointed scheme so that we can put Σ f : = T h ( f 0 , δ ) = f 0 ] δ ∗ and w e deﬁne a canonical mor phism: (2.4.24.1) p f : f ] = f ] f 00 ! δ ! E x ( ∆ ] ! ) / / f ! f 0 ] δ ! = f ! ◦ Σ f . using the e x chang e transf or mation of point (2) in Corollar y 2.2.12 . Deﬁnition 2.4.25 Using the notations abov e, we say that f is T -pure , or simpl y pur e when T is clear, when the f ollowing conditions are satisﬁed: 1. The natural transf ormation Σ f is an equiv alence. 2. The mor phism p f : f ] / / f ! ◦ Σ f is an isomor phism. W e can easil y deduce from the construction of the ex chang e transf or mation E x ( ∆ ] ! ) that, when T satisﬁes proper ties (Stab) and (Pur), any smooth separated mor phism of ﬁnite type f is pure. The f ollo wing theorem is a consequence of the f ormalism dev eloped previousl y . Theorem 2.4.26 Assume that T satisﬁes the localization and weak purity proper - ties. Then the f ollowing conditions hold: 2 Triangulated P -ﬁbred categories in algebraic geometry 65 1. T satisﬁes the stability pr operty. 2. T satisﬁes the support and base chang e properties. If mor eov er T is monoidal, it satisﬁes the projection f ormula. 3. Any smooth separated mor phism of ﬁnite type is pure. 4. F or any projectiv e morphism f , the property (Adj f ) holds. If mor eov er T is well g enerat ed, then the adjoint pr operty holds in g eneral. Proof W e start by pro ving condition (1). As (Loc) implies ( Zar -sep), we can apply Proposition 2.4.11 and we ha v e only to pro ve that for any scheme S , T h ( A 1 S ) is an equiv alence. Let s : S / / A 1 S be the zero section and j : A 1 S / / P 1 S be the canonical open immersion. Put t = j ◦ s . A ccording to Lemma 2.4.9 , j induces an isomorphism T h ( A 1 S ) ' T h ( p 1 , s ) . Consider now the follo wing car tesian squares: S s / / s   P 1 S p 1 / / s 0   ∆ S s   P 1 S δ / / P 1 S × S P 1 S p 0 1 / / P 1 S where p 0 1 (resp. δ ) is the projection on the ﬁrst factor (resp. diagonal embedding). The property (Loc s ) implies that s ∗ s ∗ = 1 and that the e x chang e transformation E x ( ∆ ] ∗ ) is an isomor phism according to Corollary 2.3.13 . Thus we get an isomorphism of functors: T h ( p 1 , s ) = p 1 ] s ∗ = s ∗ s ∗ p 1 ] s ∗ E x ( ∆ ] ∗ ) − 1 / / s ∗ p 0 1 ] s 0 ∗ s ∗ = s ∗ p 0 1 ] δ ∗ s ∗ = s ∗ Σ p 1 s ∗ and this pro v es (1) because p 1 is pure. Condition (2) f ollow s simply from Corollar y 2.3.13 . In fact, f or any scheme S , the w eak purity assumption on T implies that p n : P n S / / S is univ ersally pure. Thus, Lemma 2.4.23 implies proper ties (Supp p n ) and (BC p n ) so that we can apply Corollary 2.3.13 to g et (Supp) and (BC). The same argument applies to the proper ty (PF) in the monoidal case. For condition (3), w e consider a smooth separated morphism of ﬁnite type g : Y / / S and w e pro ve it is pure. According to (1), Σ g is an equiv alence. Thus, b y deﬁnition of p g , it is suﬃcient to prov e that for any car tesian square: Z ˜ f / / h   ∆ Y g   X f / / S with f separated of ﬁnite type, the ex chang e transf ormation E x ( ∆ ] ! ) : g ] ˜ f ! / / f ! h ] 66 Fibred categories and the six functors formalism is an isomor phism. T o do this, w e appl y Proposition 2.3.11 , as in the case of Corollar y 2.3.13 . W e consider the obvious complete Sm -ﬁbred tr iangulated categor ies T 0 and T 00 o v er S / S which to an S -scheme Y associates: • T 0 ( Y ) = T ( Y × S X ) . • T 00 ( Y ) = T ( Y ) . W e consider the mor phism ϕ ∗ : T 0 / / T 00 such that f or an y S -scheme Y , ϕ ∗ Y = ( Y × S p ) ] . As f or an y sc heme S , p n : P n S / / S is univ ersally pure, Lemma 2.4.23 show s that ϕ ∗ satisﬁes condition (i) of Proposition 2.3.11 . According to that Proposition, (i) is equivalent to condition (iii), and (iii) is precisely what w e want. It remains only to prov e condition (4). According to proper ty (Pur), any smooth proper mor phism f satisﬁes (Adj f ). A ccording to (Loc) and Proposition 2.3.3 any closed immersion i satisﬁes (Adj i ). It follo ws easily that any projectiv e mor phism f satisﬁes (Adj f ). When T is w ell generated, we simpl y apply point (4) of Corollary 2.3.13 .  Remar k 2.4.27 In par ticular , in the assumption of the previous theorem, if T satisﬁes properties (Loc), (wPur) and (Adj) 51 , w e can appl y Theorem 2.2.14 to T so that w e get a complete f or malism of operations ( f ∗ , f ∗ , f ! , f ! ) satisfying all the desired f or mulas. Thus the preceding theorem giv es another look at the main result of [ A yo07a , 1.4.2]. In fact, the proof giv en here is simpler as the assumptions of our theorem are strong er . How ev er , we do not use the homotopy proper ty in our theorem. W e end up this section with a theorem due to A y oub [ A y o07a , 1.4.2]. The par ticular case T ( X ) = SH ( X ) was also established b y Röndigs in [ Rön05 ], after V oe vodsky , with a proof which e xtends immediately to A y oub’ s axiomatic setting. It ma y be stated in a simpler f or m, according to theorem 2.4.26 abo ve: Theorem 2.4.28 (V oev odsky-R öndigs-A y oub) Assume T satisﬁes the localization, homotopy and stability properties. Then T is weakly pure. In fact, this theorem is stated e xplicitly in [ A y o07a , Theorem 1.7.9]. Remar k 2.4.29 Recall that A youb prov es more than just this theorem: indeed he constructs the whole f or malism of the six functors for quasi-projectiv e mor phisms f or his monoidal homotopy stable functors — see again [ A y o07a ]. Similarl y , the f act that one can deduce the proper base chang e f or mula from relativ e purity was also observed b y Röndigs [ Rön05 ]. The work w e hav e done here is to isolate the cr ucial properties of pur ity and w eak pur ity . Also, using the construction of Deligne, we see ho w to av oid the assumption of quasi-projectiv eness made b y A y oub. Finally , the interest of Theorem 2.4.26 is to giv e a possible approac h to the six functor s f ormalism without requiring the homotopy property . 51 Note that under the assumptions of the previous theorem, w e know that for any proper smooth morphism f , f ∗ admits a r ight adjoint. The same is tr ue f or a proper mor phism whic h can be factorized as a closed immersion follo wed b y a smooth proper mor phism according to (Loc). 2 Triangulated P -ﬁbred categories in algebraic geometry 67 2.4.c Duality , purity and orient ation 2.4.30 This section is concerned with the relation between purity and duality . W e will assume that T is premotivic. Recall that an object M of a monoidal categor y M is called strong ly dualizable if there e xists an object M 0 such that ( M 0 ⊗ −) is both r ight and left adjoint to ( M ⊗ −) . Then, M 0 is called the strong dual of M . In case M is closed monoidal, we will sa y that a mor phism of the form µ : M ⊗ M 0 / / 1 is a per f ect pairing if the natural transformation ( M ⊗ −) / / Hom ( M 0 , −) obtained from µ b y adjunction is an isomor phism. Then M is s trongly dualizable with dual M 0 . Proposition 2.4.31 Let f : X / / S be a smooth proper morphism. If f is pur e then the pr emotiv e M S ( X ) is str ong ly dualizable in T ( S ) with dual: f ∗ ( 1 X ) ' f ]  Ω f ( 1 X )  wher e Ω f denot es the inv erse of Σ f . Proof By assumption, Σ f is an automor phism of the categor y T ( X ) . Moreo ver , the identiﬁcation ( 2.4.13.1 ) can be rewritten as Σ f ( M ) = Σ f ( 1 X ) ⊗ X M for any premotive M ov er X . The fact Σ f is an equiv alence means that Σ f ( 1 X ) is a ⊗ -in v er tible object, whose in verse is T : = Ω f ( 1 S ) . In par ticular , we g et: Ω f ( M ) = T ⊗ M . A ccording to the Sm -projection f or mula, the functor M S ( X ) ⊗ . is isomor phic to f ] f ∗ . Thus, its right adjoint is f ∗ f ∗ . As f is pure b y assumption, this last functor is isomorphic to f ] Ω f f ∗ . Using the obser v ation at the beginning of the proof and the Sm -projection f or mula again, we obtain: f ] Ω f f ∗ ( N ) = f ] ( T ⊗ f ∗ ( N )) = f ] ( T ) ⊗ N . Moreo v er , the r ight adjoint of f ] Ω f f ∗ is f ∗ Σ f f ∗ . Using again the pur ity isomor phism f or f , this last functor can be identiﬁed with f ] f ∗ and this concludes.  2.4.32 Assume again that the premotivic tr iangulated categor y T satisﬁes proper ties (wLoc) and ( Nis -sep). Let S be a scheme. A smooth closed S -pair will be pair ( X , Z ) of smooth S -schemes such that Z is closed subscheme of X . W e consider the canonical projection p : X / / S and the immersion i : Z / / X associated with ( X , Z ) . Note that according to Proposition 2.4.8 , T satisﬁes proper ty (Loc i ). Then we deﬁne the premotive of ( X , Z ) as f ollo ws: (2.4.32.1) M S ( X / X − Z ) : = p ] i ∗ ( 1 Z ) . 68 Fibred categories and the six functors formalism A ccording to proper ty (Loc i ), w e thus get a canonical distinguished triangle: (2.4.32.2) M S ( X − Z ) j ∗ / / M S ( X ) / / M S ( X / X − Z ) + 1 / / Note that given any smooth mor phism p : S / / S 0 , w e get ob viously : (2.4.32.3) p ] M S ( X / X − Z ) = M S 0 ( X / X − Z ) . Moreo v er , giv en any mor phism f : T / / S , w e get an e x chang e isomor phism: (2.4.32.4) f ∗ M S ( X / X − Z ) ∼ / / M T ( X T / X T − Z T ) . A mor phism of smooth closed S -pairs ( Y , T ) / / ( X , Z ) will be a couple ( f , g ) which ﬁts into a commutative diagram T k / / g   ∆ Y f   Z i / / X , with i , k the canonical immersions, and such that T = f − 1 ( Z ) as a set. W e can associate with ( f , g ) a mor phism of premotiv es: M S ( Y / Y − T ) = q ] k ∗ g ∗ ( 1 Z ) E x ( ∆ ∗ ∗ ) − 1 / / q ] f ∗ i ∗ ( 1 Z ) E x ∗ ] / / p ] i ∗ ( 1 Z ) = M S ( X / X − Z ) . Indeed, the e x chang e map E x ( ∆ ∗ ∗ ) is an isomor phism according to (Loc i ) and Corol- lary 2.3.13 . It is easy to c heck that the tr iangle ( 2.4.32.2 ) is functorial with respect to mor- phisms of closed S -pairs. Bef ore pro ving the ne xt theorem, w e state the f ollowing lemma. Lemma 2.4.33 Consider the assumptions and notations abo ve. Let ( f , g ) : ( Y , T ) / / ( X , Z ) be a mor phism of smooth closed S -pairs such that f is étale and g is an isomorphism. Then the induced map M S ( Y / Y − T ) / / M S ( X / X − Z ) is an isomorphism. Proof A ccording to the identiﬁcation 2.4.32.3 , it is suﬃcient to treat the case where X = Z . Let U = X − Z and j : U / / X be the obvious immersion. Then ( f , j ) is a Nisnevic h cov er of X . According to ( Nis -sep), it is suﬃcient to prov e that the pullback of M X ( Y / Y − T ) / / M X ( X / X − Z ) along f and j is an isomor phism. This is obvious using 2.4.32.4 .  2.4.34 W e consider ag ain the assumption of the paragraph preceding the abo v e lemma. Fix a smooth closed S -pair ( X , Z ) . Let B Z X (resp. B Z ( A 1 X ) be the blow -up of X (resp. A 1 X ) with center in Z (resp. { 0 } × Z ). W e deﬁne the deformation space 2 Triangulated P -ﬁbred categories in algebraic geometry 69 associated with ( X , Z ) as the S -scheme D Z X = B Z ( A 1 X )− B Z X . Note also D Z Z = A 1 Z is a closed subscheme of D Z X ; the couple ( D Z X , A 1 Z ) is a smooth closed S -pair . Let N Z X be the normal bundle of Z in X . The scheme D Z X is ﬁbred o ver A 1 . Moreo v er , the 0 -ﬁber of ( D Z X , A 1 ) is the closed pair ( N Z X , Z ) cor responding to the zero section and the 1 -ﬁber is the closed pair ( X , Z ) . In par ticular , w e g et the f ollowing mor phisms of closed pairs: (2.4.34.1) ( X , Z ) d 1 / / ( D Z X , A 1 Z ) d 0 o o ( N Z X , Z ) W e are no w ready to state the pur ity theorem f or smooth closed pairs in our abstract f or malism. Though our assumptions are more general, this theorem f ollow s exactl y from the method of Morel and V oev odsky used to pro v e this result in the homotopy category H (see [ MV99 , §3, 2.24]): Theorem 2.4.35 Consider the abov e assumptions and no tations and suppose that T satisﬁes the homotopy property . Then the mor phisms M S ( X / X − Z ) d 1 ∗ / / M S ( D Z X / D Z X − A 1 Z ) d 0 ∗ o o M S ( N Z X / N × Z X ) = : M T h S ( N Z X ) . ar e isomorphisms. Proof By noether ian induction and the preceding lemma, the statement is local in X f or the Nisnevich topology . Thus, because ( X , Z ) is a smooth closed S -pair , we can assume that there e xists an étale map π : X / / A n + c S such that π − 1 ( A c S ) = Z – cf. [ GD67 , 17.12.2]. Consider the pullback square X 0 p / / q   X π   A n × Z 1 × π | Z / / A n × A c S . There is an obvious closed immersion Z / / X 0 and its image is contained in q − 1 ( Z ) . As q is étale, Z is a direct factor of q − 1 ( Z ) . Put W = q − 1 ( Z ) − Z and Ω = X 0 − W . Thus Ω is an open subscheme of X 0 , and the reader can chec k that p and q induces morphisms of smooth closed S -pairs ( X , Z ) o o ( Ω , Z ) / / ( A n Z , Z ) . Applying again the preceding lemma, these mor phisms induces isomor phisms on the associated premotives. Thus we are reduced to the case of the closed S -pair ( A n Z , Z ) . A direct computation show s that D Z ( A n Z ) ' A 1 × A n Z . Under this isomor phism d 0 (resp. d 1 ) cor responds to the 0 -section (resp. 1 -section) of A 1 × A n Z corresponding to the ﬁrst f actor . Thus, w e conclude using the homotop y proper ty .  2.4.36 The interest of the pre vious theorem is to simplify the purity isomor phism. Let us restate the assumptions on the tr iangulated premotivic categor y T : • T satisﬁes proper ties ( Nis -sep), (wLoc) and (Htp). 70 Fibred categor ies and the six functors formalism Then applying the abov e theorem, w e get f or an y smooth closed S -pair ( X , Z ) a canonical isomor phism (2.4.36.1) p X , Z : M S ( X / X − Z ) / / M T h S ( N Z X ) Corollary 2.4.37 Consider the assumptions and notations abov e. 1. F or any smooth pointed S -scheme ( f , s ) and any premo tive K ov er S , we g et a canonical isomorphism T h ( f , s ) . K ' M S ( X / X − s ( S )) ⊗ S K p X , S / / M T h S ( N s ) ⊗ S K . wher e the ﬁrst isomor phism is given by the map ( 2.4.13.1 ) and N s is the normal bundle of s . 2. F or any smoot h separated mor phism of ﬁnite type f : X / / S with tang ent bundle 52 T f , and any premo tiv e K o v er X , w e g et a canonical isomorphism: p X X , X : Σ f ( K ) ∼ / / M T h X ( T f ) ⊗ X K — here, ( X X , X ) stands for the closed pair corr esponding to the diagonal em- bedding of X / S . In the assumption of point (2), we thus get a canonical map: (2.4.37.1) f ] ( K ) p f / / f ! ( Σ f K ) ∼ / / f !  M T h X ( T f ) ⊗ X K  that w e will still denote by p f and call the purity isomor phism associated with f . Deﬁnition 2.4.38 Assume the tr iangulated premotivic categor y T satisﬁes (wLoc). As usual, M ( 1 ) denotes the T ate twist of a premotive M . An orientation t of T will be the data f or each smooth scheme X and each v ector bundle E / X of rank n of an isomor phism t E : M T h X ( E ) / / 1 X ( n )[ 2 n ] , called the Thom isomor phism , satisfying the f ollo wing coherence proper ties: (a) Giv en a scheme X and an isomor phism of vector bundles ϕ : E / / F of ranks n o v er X , the f ollo wing diagram is commutative: M T h X ( E ) t E ) ) ϕ ∗ / / M T h X ( F ) . t F u u 1 X ( n )[ 2 n ] 52 W e deﬁne T f as the normal bundle of the diagonal immersion δ : X / / X × S X . 2 Triangulated P -ﬁbred categories in algebraic geometry 71 (b) For any mor phism f : Y / / X of schemes, and any vector bundle E / X of rank n with pullback F o ver Y , the follo wing diagram commutes: f ∗ ( M T h X ( E )) ∼   f ∗ t E / / f ∗ ( 1 X ( n )[ 2 n ]) ∼   M T h Y ( F ) t F / / 1 Y ( n )[ 2 n ] where the v er tical maps are the canonical isomor phisms. (c) For any scheme X and an y ex act sequence ( σ ) of v ector bundles o ver X 0 / / E 0 ν / / E π / / E 00 / / 0 , if n (resp. m ) denotes the rank of the v ector bundle E 0 (resp. E 00 ), the f ollowing diagram commutes: M T h X ( E ) t E   T h X ( σ ) / / M T h X ( E 0 ) ⊗ M T h X ( E 00 ) t E 0 ⊗ t E 0 0   1 X ( n + m )[ 2 n + 2 m ] / / 1 X ( n )[ 2 n ] ⊗ 1 X ( m )[ 2 m ] where the map T h X ( σ ) is the isomor phism ( 2.4.15.1 ) associated with ( σ ) and the bottom v er tical one is the obvious identiﬁcation. W e will also say that T is oriented when the choice of one par ticular or ientation is not essential. Note that the Thom isomor phism can be view ed as a cohomology class in H 2 n , n T ( T h X ( E )) : = Hom T ( X )  M T h X ( E ) , 1 S ( n )[ 2 n ]  which in classical homotopy theor y is called the Thom class . 2.4.39 Suppose the tr iangulated premotivic categor y T satisﬁes the follo wing prop- erties: • T satisﬁes proper ties ( Nis -sep), (wLoc), (Htp). • T admits an or ientation t . Consider a smooth closed S -pair ( X , Z ) of codimension n . Let p (resp. q ) be the structural mor phism of X / S (resp. Z / S ) and i : Z / / X the associated immersion. Then w e associate with ( X , Z ) the f ollo wing f orm of the pur ity isomorphism: (2.4.39.1) p t X , Z : M S ( X / X − Z ) p X , Z / / M T h S ( N Z X ) q ] ( t N Z X ) / / M S ( Z )( n )[ 2 n ] where p X , Z is the isomorphism ( 2.4.36.1 ). For future reference, note that we deduce from this the so-called Gysin morphism: 72 Fibred categor ies and the six functors f ormalism (2.4.39.2) i ∗ : M S ( X ) π / / M S ( X / X − Z ) p t X , Z / / M S ( Z )( n )[ 2 n ] where π is the f ollowing map: M S ( X ) = p ] ( 1 X ) a d ( i ∗ , i ∗ ) / / p ] i ∗ i ∗ ( 1 X ) = M S ( X / X − Z ) . As a particular case, w e get using the notation of Corollar y 2.4.37 , point (2), an isomorphism: p t X X , X : Σ f ( K ) p X X , X / / M T h X ( T f ) ⊗ K t T f / / K ( d )[ 2 d ] In par ticular , when T satisﬁes proper ty (Supp), the pur ity compar ison map associ- ated with f can be rewritten as: (2.4.39.3) p t f : f ] p f / / f ! ◦ Σ f p t X X , X / / f ! ( d )[ 2 d ] Example 2.4.40 Assume as in the abo v e deﬁnition that T is premotivic and satisﬁes properties (wLoc) and ( Nis -sep). W e suppose the f ollo wing tw o additional conditions are fulﬁlled: (a ’) There e xists a mor phism of triangulated premotivic categories: ϕ ∗ : SH / / o o T : ϕ ∗ where SH is the stable homotop y category of Morel and V oev odsky — see Example 1.4.3 . (b’) For an y scheme X , let Pic ( X ) be the Picard g roup of X . W e assume there exis ts an application c 1 : Pic ( X ) / / H 2 , 1 T ( X ) : = Hom T ( X ) ( M ( X ) , 1 X ( 1 )[ 2 ]) which is natural with respect to contrav ariant functor iality — we do not require c 1 is a mor phism of abelian groups. Then one can apply the results of [ Dég08 ] to T ( X ) f or an y scheme X . All the ref erences which f ollow s will be within loc. cit. : according to section 2.3.2, the tr i- angulated categor y T ( X ) satisﬁes the axioms of Paragraph 2.1. 53 Then the exis tence of the Thom isomor phism f ollo ws from Proposition 4.3 and, more e xplicitly , from Paragraph 4.4. Proper ty (a) and (b) of the abo v e deﬁnition are easy — explicitl y , this is a consequence of 4.10 — and Proper ty (c) follo ws from Lemma 4.30. 53 Note in par ticular that f or an y smooth closed S -pair, we obtain a canonical isomor phism in T ( S ) of the f orm: ϕ ∗ ( Σ ∞ X / X − Z ) ' M S ( X / X − Z ) where one the left-hand side X / X − Z stands f or the homotopy coﬁber of the open immersion ( X − Z ) / / X while the left-hand side is deﬁned b y Equality ( 2.4.32.1 ). 2 Triangulated P -ﬁbred categories in algebraic geometry 73 T o sum up, the assumptions (a ’) and (b’) guarantees the e xistence of a canon- ical orientation of T in the sense of the abo ve deﬁnition. Moreov er, the pur ity isomorphism ( 2.4.39.1 ) as w ell as the Gysin mor phism ( 2.4.39.2 ) associated in the preceding paragraph f or this par ticular orientation coincide with the one deﬁned in [ Dég08 ] (see in par ticular the uniqueness statement of [ Dég08 , Prop. 4.3]). Note moreo v er that assuming T satisﬁes all the proper ties abov e e x cept (b’), the data of an or ientation of T is equivalent to the data of a map c 1 as in (b’). Indeed, if t is an orientation of T , given any line bundle L / X with zero section s , we put c 1 ( L ) = ρ ( t L ) where ρ is the follo wing composite map: H 2 , 1 T ( T h X ( L )) / / H 2 , 1 T ( L ) s ∗ / / H 2 , 1 T ( X ) where the ﬁrst map is induced by the canonical projection M X ( L ) / / M T h X ( L ) . Then c 1 depends onl y on the isomor phism classes of L / X — property (a) of the abo v e deﬁnition — and it is compatible with pullbacks — proper ty (c) of the abov e deﬁnition. 2.4.41 W e no w assume the f ollowing conditions on the triangulated premotivic category T : • T satisﬁes proper ties ( Nis -sep), (wLoc), (Htp) and (Stab). • T admits an or ientation t . Let f : X / / S be a smooth proper mor phism of dimension d . Note we do not need that T satisﬁes proper ty (Supp) to rewrite the pur ity comparison map as f ollo ws: (2.4.41.1) p t f : f ] / / f ∗ ( d )[ 2 d ] (see Paragraph 2.4.39 ). Note also that using the Gysin mor phism ( 2.4.39.2 ) associated with the diagonal immersion δ : X / / X × S X , we get the f ollo wing mor phism: (2.4.41.2) µ t f : M S ( X ) ⊗ M S ( X )(− d )[− 2 d ] = M S ( X × S X )(− d )[− 2 d ] δ ∗ / / M S ( X ) f ∗ / / 1 S . Theorem 2.4.42 Consider the assumptions and notations abov e. Then the f ollowing conditions ar e equivalent : (i) f is pur e: p f is an isomorphism. (i’) The natural transf or mation p f . f ∗ is an isomorphism. (ii) The pr emotiv e M S ( X ) is str ong ly dualizable and µ t f is a per f ect pairing. Proof In this proof, w e put τ ( K ) = K ( d )[ 2 d ] . As T satisﬁes proper ty (Stab), f ∗ commutes with T ate twist (def. 1.1.44 ). This means the f ollo wing e xc hange transf or - mation is an isomor phism: (2.4.42.1) E x τ : τ f ∗ / / f ∗ τ . 74 Fibred categor ies and the six functors formalism W e ﬁrst pro ve that (i) is equivalent to (i’). One implication is obvious so that we ha v e only to pro v e that (i’) implies (i). Guided b y a method of A y oub (see [ A y o07a , 1.7.14, 1.7.15]), we will construct a r ight in v erse φ 1 and a left inv erse φ 2 to the morphism p t f as the f ollo wing composite maps: φ 1 : f ∗ τ a d ( f ∗ , f ∗ ) / / f ∗ f ∗ f ∗ τ E x − 1 τ / / f ∗ f ∗ τ f ∗ = f ∗ τ f ∗ f ∗ ( p t f . f ∗ f ∗ ) − 1 / / f ] f ∗ f ∗ a d 0 ( f ∗ , f ∗ ) / / f ] φ 2 : f ∗ τ β f / / f ∗ τ f ∗ f ] ( p t f . f ∗ f ] ) − 1 / / f ] f ∗ f ] a d 0 ( f ] , f ∗ ) / / f ] . Let us check that p t f ◦ φ 1 = 1 . T o pro v e this relation, we pro ve that the follo wing diagram is commutative: f ∗ τ a d ( f ∗ , f ∗ ) / / f ∗ f ∗ f ∗ τ E x − 1 τ / / f ∗ τ f ∗ f ∗ ( p t f f ∗ f ∗ ) − 1 / / f ] f ∗ f ∗ a d 0 ( f ∗ , f ∗ ) / / ( 1 ) f ] p t f / / f ∗ τ f ∗ τ f ∗ f ∗ ( p t f f ∗ f ∗ ) − 1 / / ( 2 ) f ] f ∗ f ∗ p t f f ∗ f ∗ / / f ∗ τ f f ∗ a d 0 ( f ∗ , f ∗ ) / / f ∗ τ f ∗ f ∗ f ∗ τ E x − 1 τ / / ( 3 ) f ∗ τ f ∗ f ∗ a d 0 ( f ∗ , f ∗ ) / / f ∗ τ f ∗ τ a d ( f ∗ , f ∗ ) / / f ∗ f ∗ f ∗ τ a d 0 ( f ∗ , f ∗ ) / / f ∗ τ . The commutativity of (1) and (2) is obvious and the commutativity of (3) f ollow s from Formula ( 2.4.42.1 ) deﬁning E x τ . Then the result follo ws from the usual f or mula betw een the unit and counit of an adjunction. The relation φ 2 ◦ p t f = 1 is pro ved using the same kind of computations. It remains to prov e that (i) and (i’) are equiv alent to (ii). W e already know from Proposition 2.4.31 that (i) implies the premotive M S ( X ) is strongl y dualizable. Saying that µ t f is a perfect pair ing amounts to prov e that the natural transf or mation obtained b y adjunction d t f : ( M S ( X ) ⊗ −) / / Hom ( M S ( X ) , −( d )[ 2 d ]) is an isomor phism. On the other hand, as we ha v e already seen previousl y , the smooth projection f ormula implies an identiﬁcation of functors: f ] f ∗ ' ( M S ( X ) ⊗ −) , f ∗ f ∗ ' Hom ( M S ( X ) , −) . (2.4.42.2) Thus, to ﬁnish the proof, it will be enough to show that the map f ] f ∗ p t f f ∗ / / f ∗ τ f ∗ = f ∗ f ∗ τ . 2 Triangulated P -ﬁbred categories in algebraic geometry 75 is equal to d t f through the identiﬁcations ( 2.4.42.2 ). Let us consider the follo wing car tesian square X × S X f 0 0 / / f 0   ∆ X f   X f / / S and put g = f ◦ f 00 . According to the deﬁnition of µ t f , and notably Formula ( 2.4.39.2 ) f or the Gysin map δ ∗ , the natural transformation of functors ( µ t f ⊗ −) can be descr ibed as the f ollo wing composition: f ] f ∗ f ] f ∗ E x ( ∆ ∗ ] ) / / f ] f 0 ] f 00∗ f ∗ = g ] g ∗ a d ( δ ∗ , δ ∗ ) / / g ] δ ∗ δ ∗ g ∗ = f ] f 0 ] δ ∗ f ∗ p t X X , X / / f ] τ f ∗ = f ] f ∗ τ a d 0 ( f ] , f ∗ ) / / τ . Note in par ticular that the base change map E x ( ∆ ∗ ] ) corresponds to the ﬁrst identiﬁ- cation in Formula ( 2.4.41.2 ). Thus we hav e to prov e the preceding composite map is equal to the f ollo wing one, obtained by adjunction from p t f : f ] f ∗ f ] f ∗ = f ] f ∗ f ] f 00 ∗ δ ∗ f ∗ E x ( ∆ ] ∗ ) / / f ] f ∗ f ∗ f 0 ] δ ∗ f ∗ p t X X , X / / f ] f ∗ f ∗ τ f ∗ = f ] f ∗ f ∗ f ∗ τ a d 0 ( f ∗ , f ∗ ) / / f ] f ∗ τ a d 0 ( f ] , f ∗ ) / / τ This amounts to prov e, after some easy cancellation, the commutativity of the fol- lo wing diagram: f ∗ f ] E x ( ∆ ∗ ] )   f ∗ f ] f 00 ∗ δ ∗ E x ( ∆ ] ∗ ) / / f ∗ f ∗ f 00 ] δ ∗ a d 0 ( f ∗ , f ∗ )   f 0 ] f 00∗ a d ( δ ∗ , δ ∗ ) / / f 0 ] δ ∗ δ ∗ f 00∗ f 0 ] δ ∗ . A ccording to the deﬁnition of the e xc hange transf ormation E x ( ∆ ] ∗ ) (cf Paragraph 1.1.14 ), w e can divide this diagram into the follo wing pieces: 76 Fibred categories and the six functors formalism f ∗ f ] f 00 ∗ δ ∗ a d ( f ∗ , f ∗ ) / / E x ( ∆ ∗ ] )   f ∗ f ∗ f ∗ f ] f 00 ∗ δ ∗ E x ( ∆ ∗ ] ) / / f ∗ f ∗ f 0 ] f 00∗ f 00 ∗ δ ∗ a d 0 ( f 0 0∗ , f 0 0 ∗ ) / / a d 0 ( f ∗ , f ∗ )   f ∗ f ∗ f 00 ] δ ∗ a d 0 ( f ∗ , f ∗ )   f 0 ] f 00∗ f 00 ∗ δ ∗ a d ( f ∗ , f ∗ ) 4 4 f 0 ] f 00∗ f 00 ∗ δ ∗ a d 0 ( f 0 0∗ , f 0 0 ∗ ) / / f 0 ] δ ∗ f 0 ] f 00∗ a d ( δ ∗ , δ ∗ ) / / (∗) f 0 ] δ ∗ . Ev ery par t of this diagram is obviousl y commutative e x cept f or par t (∗) . As f 00 δ = 1 , the axioms of a 2-functors (for f ∗ and f ∗ sa y) implies that the unit map α : f 0 ] f 00∗ / / f 0 ] f 00∗ ( f 00 δ ) ∗ ( f 00 δ ) ∗ is the canonical identiﬁcation that we g et using 1 ∗ = 1 and 1 ∗ = 1 . W e can consider the f ollo wing diagram: f 0 ] f 00∗ α f 0 ] f 00∗ ( f 00 δ ) ∗ ( f 00 δ ) ∗ f 0 ] f 00∗ f 00 ∗ δ ∗ a d 0 ( f 0 0∗ , f 0 0 ∗ )   f 0 ] f 00∗ a d ( f 0 0∗ , f 0 0 ∗ ) / / f 0 ] f 00∗ f 00 ] f 00∗ a d ( δ ∗ , δ ∗ ) / / a d 0 ( f 0 0∗ , f 0 0 ∗ )   f 0 ] f 00∗ ( f 00 δ ) ∗ ( f 00 δ ) ∗ a d 0 ( f 0 0∗ , f 0 0 ∗ )   f 0 ] f 00∗ f 0 ] f 00∗ a d ( δ ∗ , δ ∗ ) / / f 0 ] δ ∗ δ ∗ f 00∗ f 0 ] δ ∗ f or which each par t is obviousl y commutativ e. This concludes.  As a corollar y , together with the results of [ Dég08 ], we get the f ollowing theorem: Corollary 2.4.43 Let us assume the follo wing conditions on the triangulated pre- motivic categor y T : (a) T satisﬁes properties ( Nis -sep), (wLoc), (Htp) and (Stab). (b) T admits an orientation t . (c) Ther e exists a morphism of triangulated pr emotivic categories: ϕ ∗ : SH / / o o T : ϕ ∗ . Then any smooth projectiv e morphism is T -pure. In particular , T is weakly pure. 2 Triangulated P -ﬁbred categories in algebraic geometry 77 Proof A ccording to Example 2.4.40 , one can appl y the results of [ Dég08 ] to the triangulated category T ( X ) . Then it f ollo ws from [ Dég08 , 5.23] that condition (ii) of the abo v e theorem is satisﬁed.  Remar k 2.4.44 This theorem is to be compared with the result of A youb recalled in Theorem 2.4.28 . On the one hand, if T satisﬁes the localization property , w e get another proof of this result under the additional assumption that T is oriented. On the other hand, the abov e theorem does not require the assumption that T satisﬁes (Loc); this is impor tant as we can only pro v e (wLoc) for the categor y DM Λ introduced in Deﬁnition 11.1.1 . 2.4.d Motivic categories This section summar izes the main constructions of this par t and draw s a conclusiv e theorem. Deﬁnition 2.4.45 A motivic triangulated category ov er S is a premotivic trian- gulated categor y ov er S which satisﬁes the homotopy , stability , localization and adjoint proper ty . Remar k 2.4.46 Without the adjoint property , this deﬁnition cor responds to what A y oub called a monoidal stable homotopy 2 -functor (cf [ A yo07a , def. 2.3.1]). W e think our shorter terminology ﬁts w ell in the spir it of the cur rent theor y of mix ed motiv es. Remar k 2.4.47 Assume T is a premotivic triangulated category such that: 1. T is well generated. 2. T satisﬁes the homotopy and stability proper ties. 3. T satisﬁes the localization proper ty . Then T is a motivic tr iangulated categor y in the abo v e sense. Indeed, proper ty (A dj) is pro ved under the abov e assumptions in point (4) of Theorem 2.4.26 . N ote also that if T is compactly τ -generated, we simply obtain proper ty (Adj) from Lemma 2.2.16 . 54 Example 2.4.48 A ccording to the previous remark, the premotivic categor y SH of Example 1.4.3 is a motivic categor y . In fact, proper ty (1) is pro ved in [ A yo07a , 4.5.67], property (2) f ollo ws by deﬁnition and proper ty (3) is prov ed in [ A y o07a , 4.5.44]. 2.4.49 In the ne xt theorem, we summarize what is no w called the Gr othendiec k six functor s formalism . In fact, this is a consequence of the axioms in the abo v e deﬁnition, as a result of the work done in previous sections. More precisely : 54 In our ex amples, (1) will alwa ys be satisﬁed, (2) will be obtained b y construction and (3) will be the hard point. 78 Fibred categories and the six functors formalism • W e apply Theorem 2.4.26 using the theorem of A y oub recalled in 2.4.28 , and use the g eneralized theorem of Morel and V oev odsky , Theorem 2.4.35 , to get the f orm ( 2.4.37.1 ) of the pur ity isomorphism. • In the case where T is or iented, w e use the f orm ( 2.4.41.1 ) of the pur ity isomorphism. Recall that, when T satisﬁes assumption (c) of Corollar y 2.4.43 , then we hav e given a diﬀerent proof of the Theorem of A y oub and the theorem belo w f ollo ws from 2.4.26 and 2.4.43 . Theorem 2.4.50 Let T be a motivic triangulated category. Then, for any separat ed mor phism of ﬁnite type f : Y / / X in S , ther e exists a pair of adjoint functors, the ex ceptional functors , f ! : T ( Y ) / / o o T ( X ) : f ! suc h that : 1. Ther e exists a structur e of a cov ariant (resp. contrav ariant) 2 -functor on f  / / f ! (r esp. f  / / f ! ). 2. Ther e exists a natur al transf ormation α f : f ! / / f ∗ whic h is an isomorphism when f is pr oper . Mor eov er , α is a mor phism of 2 -functor s. 3. F or any smooth separated mor phism of ﬁnite type f : X / / S in S with tangent bundle T f , ther e ar e canonical natural isomorphisms p f : f ] / / f !  M T h X ( T f ) ⊗ X .  p 0 f : f ∗ / / M T h X (− T f ) ⊗ X f ! whic h are dual to each other – the Thom premo tive M T h X ( T f ) is ⊗ -invertible with inv erse M T h X (− T f ) . If T admits an orientation t and f has dimension d then there are canonical natural isomor phisms p t f : f ] / / f ! ( d )[ 2 d ] p 0 t f : f ∗ / / f ! (− d )[− 2 d ] whic h are dual to each other . 4. F or any Car tesian squar e: Y 0 f 0 / / g 0   ∆ X 0 g   Y f / / X , suc h that f is separ ated of ﬁnite type, ther e exist natur al isomor phisms E x ( ∆ ∗ ! ) : g ∗ f ! ∼ / / f 0 ! g 0 ∗ , E x ( ∆ ! ∗ ) : g 0 ∗ f 0 ! ∼ / / f ! g ∗ . 2 Triangulated P -ﬁbred categories in algebraic geometry 79 5. F or any separated morphism of ﬁnite type f : Y / / X in S , ther e exist natural isomorphisms E x ( f ∗ ! , ⊗ ) : ( f ! K ) ⊗ X L ∼ / / f ! ( K ⊗ Y f ∗ L ) , Hom X ( f ! ( L ) , K ) ∼ / / f ∗ Hom Y ( L , f ! ( K )) , f ! Hom X ( L , M ) ∼ / / Hom Y ( f ∗ ( L ) , f ! ( M )) . Remar k 2.4.51 It is impor tant to precise that in the case where the mor phisms in S are assumed to be quasi-projectiv e, this theorem is pro v ed by A youb in [ A yo07a ] if w e ex cept the case where T is oriented in point (3). 55 Reg arding this theorem, our contr ibution is to e xtend the result of A y oub to the non quasi-projectiv e case and to consider the or iented case — which is cr ucial in the theory of motives. Recall also we ha v e giv en another proof of this result in the case where the motivic categor y T satisﬁes in addition the assumptions of Corollary 2.4.43 — which will alwa ys be the case for the diﬀerent categor ies of motiv es introduced here. Remar k 2.4.52 The pur ity isomorphism is compatible with composition. Giv en smooth separated mor phisms of ﬁnite type Y g / / X f / / S w e obtain ( cf. [ GD67 , 17.2.3]) an e xact sequence of v ector bundles o v er Y ( σ ) 0 / / g − 1 T f / / T f g / / T g / / 0 . which according to Remark 2.4.15 induces an isomor phism:  σ : M T h Y ( T f g ) M T h Y ( σ ) / / M T h Y ( T g ) ⊗ Y M T h Y ( g − 1 T f ) ∼ / / g ∗ M T h X ( T f ) ⊗ Y M T h Y ( T g ) . One can chec k the follo wing diagram is commutativ e: 55 This theorem was ﬁrst announced by V oev odsky but only notes co v er ing the basic setting were to be f ound by the time A youb wrote the proof. 80 Fibred categor ies and the six functors formalism ( f g ) ] ( K ) p f g   f ] g ] ( K ) p f ◦ p g   f !  M T h X ( T f ) ⊗ X g !  M T h Y ( T g ) ⊗ Y K   E x ( g ∗ ! , ⊗ ) − 1   f ! g !  g ∗ M T h Y ( T f ) ⊗ Y M T h Y ( T g ) ⊗ Y K   − 1 σ   ( f g ) ! ( M T h ( T f g ) ⊗ K ) f ! g ! ( M T h ( T f g ) ⊗ K ) . This is not an easy chec k. 56 In fact, this is one of the ke y technical point in the proof of the main Theorem of A y oub ([ A y o07a , 1.4.2]). W e ref er the reader to [ A y o07a , 1.5] f or details. Note also that given the commutativity of the abo v e diag ram, if T admits an orientation t , it readily follo ws from axiom (c) of Deﬁnition 2.4.38 that the follo wing diagram is commutative: ( f g ) ] ( K ) p t f g   f ] g ] ( K ) p t f ◦ p t g   ( f g ) ! ( K )( n + m )[ 2 n + 2 m ] f ! g ! ( K )( n + m )[ 2 n + 2 m ] where n (resp. m ) is the relativ e dimension of f (resp. g ). Morphisms of triangulated motivic categories are compatible with Grothendieck 6 operations in the follo wing sense: Proposition 2.4.53 Let T and T 0 be motivic triangulated categories and ϕ ∗ : T / / o o T 0 : ϕ ∗ be an adjunction of premo tivic categories. Then ϕ ∗ (r esp. ϕ ∗ ) commutes with the operations f ∗ (r esp. f ∗ ), for any morphism of schemes f , as well as with the operation p ! (r esp. p ! ), for any separat ed morphism of ﬁnite type p . Mor eov er , ϕ ∗ is monoidal and for any pr emotiv e M ∈ T ( S ) , N ∈ T 0 ( S ) , the canonical map Hom ( M , ϕ ∗ ( N )) / / ϕ ∗ Hom ( ϕ ∗ ( M ) , N ) is an isomorphism. 56 The main point is to check that the isomor phism of Theorem 2.4.35 is compatible with compo- sition (of closed immersions). On that particular point, see [ Dég08 , Th. 4.32, Cor . 4.33]. 3 Descent in P -ﬁbred model categor ies 81 Proof The only thing to pro v e is that ϕ ∗ commutes with p ! as the other statements f ollow s either from the deﬁnitions or by adjunction. This f ollo ws from Proposition 2.3.11 , the purity proper ty in T and T 0 (property (3) in the abo v e theorem) and the fact ϕ ∗ commutes with p ] when p is smooth by assumption.  Remar k 2.4.54 With additional assumptions on T and T 0 , and ov er a ﬁeld, we will see that ϕ ∗ commutes with all the six operations (see Theorem 4.4.25 ). 3 Descent in P -ﬁbred model categories 3.0.1 In this section, S is an abstract categor y and P an admissible class of morphisms in S . In section 3.3 ho we ver , we will consider as in 2.0.1 a noether ian base scheme S and w e will assume that S is an adequate categor y of S -schemes satisfying the f ollowing condition on S : (a) An y scheme in S is ﬁnite dimensional. Moreo v er , in sections 3.3.c and 3.3.d , we will ev en assume: (a 0 ) An y scheme in S is quasi-e xcellent and ﬁnite dimensional. W e ﬁx an admissible class P of morphisms in S which contains the class of étale mor phisms in S and a stable combinator ial P -ﬁbred model categor y M o v er S (See Paragraph 1.3.21 ). In section 3.3.d , we will assume fur thermore that: (b) The stable model P -ﬁbred categor y M is Q -linear (see 3.2.14 ). 3.1 Extension of P -ﬁbred categories to diagrams 3.1.a The general case 3.1.1 Assume giv en a P -ﬁbered category M o v er S . Then M can be extended to S -diagrams (i.e. functors from a small categor y to S ) as follo ws. Let I be a small category , and X a functor from I to S . For an object i of I , we will denote by X i the ﬁber of X at i (i.e. the ev aluation of X at i ), and, f or a map u : i / / j in I , w e will still denote by u : X i / / X j the morphism induced by u . W e deﬁne the category M ( X , I ) as f ollo ws. An object of M ( X , I ) is a couple ( M , a ) , where M is the data of an object M i in M ( X i ) f or any object i of I , and a is the data of a morphism a u : u ∗ ( M j ) / / M i f or any mor phism u : i / / j in I , such that, f or an y object i of I , the map a 1 i is the identity of M i (w e will alw a ys assume that 1 ∗ i is the identity functor), and, f or 82 Fibred categor ies and the six functors f or malism an y composable mor phisms u : i / / j and v : j / / k in I , the f ollowing diagram commutes. u ∗ v ∗ ( M k ) u ∗ ( a v )   ' / / ( v u ) ∗ ( M k ) a v u   u ∗ ( M j ) a u / / M i A mor phism p : ( M , a ) / / ( N , b ) is a collection of mor phisms p i : M i / / N i in M ( X i ) , f or each object i in I , such that, f or any mor phism u : i / / j in I , the f ollowing diag ram commutes. u ∗ ( M j ) u ∗ ( p j ) / / a u   u ∗ ( N j ) b u   M i p i / / N i In the case where M is a monoidal P -ﬁbred categor y , the category M ( X , I ) is naturally endo w ed with a symmetr ic monoidal structure. Given two objects ( M , a ) and ( N , b ) of M ( X , I ) , their tensor product ( M , a ) ⊗ ( N , b ) = ( M ⊗ N , a ⊗ b ) is deﬁned as f ollo ws. For any object i of I , ( M ⊗ N ) i = M i ⊗ N i , and f or any map u : i / / j in I , the map ( a ⊗ b ) u is the composition of the isomorphism u ∗ ( M j ⊗ N j ) ' u ∗ ( M j ) ⊗ u ∗ ( N j ) with the mor phism a u ⊗ b u : u ∗ ( M j ) ⊗ u ∗ ( N j ) / / M i ⊗ N i . Note ﬁnally that if M is a complete monoidal P -ﬁbred categor y , then M ( X , I ) admits an inter nal Hom. 3.1.2 Evaluation functors . Assume no w that f or any S , M ( S ) admits small sums. For each object i of I , we hav e a functor (3.1.2.1) i ∗ : M ( X , I ) / / M ( X i ) ( M , a )  / / M i called the evaluation functor associated with i . This functor i ∗ has a left adjoint (3.1.2.2) i ] : M ( X i ) / / M ( X , I ) 3 Descent in P -ﬁbred model categor ies 83 deﬁned as f ollow s. If M is an object of M ( X i ) , then i ] ( M ) is the data ( M 0 , a 0 ) such that f or an y object j of I , (3.1.2.3)  i ] ( M )  j = M 0 j = Þ u ∈ Hom I ( j , i ) u ∗ ( M ) , and, f or any mor phism v : k / / j in I , the map a 0 v is the canonical map induced by the collection of maps (3.1.2.4) v ∗ u ∗ ( M ) ' ( u v ) ∗ ( M ) / / Þ w ∈ Hom I ( k , i ) w ∗ ( M ) f or u ∈ Hom I ( j , i ) . If we assume that M is a complete P -ﬁbred category and that M ( S ) admits small products f or any S , then i ∗ has a r ight adjoint (3.1.2.5) i ∗ : M ( X i ) / / M ( X , I ) giv en, f or any object M of M ( X i ) b y the formula (3.1.2.6) ( i ∗ ( M ) ) j = Ö u ∈ Hom I ( i , j ) u ∗ ( M ) , with transition map given by the dual f ormula of 3.1.2.4 . 3.1.3 F unctoriality . Assume that M if a P -ﬁbred categor y such that f or an y object S of S , M ( S ) has small colimits. Remember that, if X and Y are S -diag rams, inde x ed respectivel y b y small categories I and J , a mor phism of S -diagrams ϕ : ( X , I ) / / ( Y , J ) is a couple ϕ = ( α , f ) , where f : I / / J is a functor, and α : X / / f ∗ ( Y ) is a natural transf or mation (where f ∗ ( Y ) = Y ◦ f ). In par ticular , f or an y object i of I , we ha v e a mor phism α i : X i / / Y f ( i ) in S . This tur ns S -diagrams into a s tr ict 2 -category: the identity of ( X , I ) is the couple ( 1 X , 1 I ) , and, if ϕ = ( α , f ) : ( X , I ) / / ( Y , J ) and ψ = ( β , g ) : ( Y , J ) / / ( Z , K ) are tw o composable mor phisms, the composed mor phism ψ ◦ ϕ : ( X , I ) / / ( Z , K ) is the couple ( g f , γ ) , where f or each object i of I , the map γ i : X i / / Z g ( f ( i )) is the composition X i α i / / Y f ( i ) β f ( i ) / / Z g ( f ( i )) . There is also a notion of natural transf or mation betw een morphisms of S -diagrams: if ϕ = ( α, f ) and ϕ 0 = ( α 0 , f 0 ) are two mor phisms from ( X , I ) to ( Y , J ) , a natural transf or mation t from ϕ to ϕ 0 is a natural transf ormation t : f / / f 0 such that the f ollowing diag ram of functors commutes. 84 F ibred categor ies and the six functors formalism X α { { α 0 # # Y ◦ f t / / Y ◦ f 0 This makes the category of S -diagrams a (strict) 2 -categor y . T o a mor phism of diag rams ϕ = ( α, f ) : ( X , I ) / / ( Y , J ) , w e associate a functor ϕ ∗ : M ( Y , J ) / / M ( X , I ) as f ollo ws. For an object ( M , a ) of M ( Y ) , ϕ ∗ ( M , a ) = ( ϕ ∗ ( M ) , ϕ ∗ ( a )) is the object of M ( X ) deﬁned by ϕ ∗ ( M ) i = α ∗ i ( M f ( i ) ) f or i in I , and by the formula ϕ ∗ ( a ) u = α ∗ i ( a f ( u ) ) : α ∗ i f ( u ) ∗ ( M f ( j ) ) = u ∗ α ∗ j ( M f ( j ) ) / / α ∗ i ( M f ( i ) ) f or u : i / / j in I . W e will say that a mor phism ϕ : ( X , I ) / / ( Y , J ) is a P -mor phism if, for any object i in I , the mor phism α i : X i / / Y f ( i ) is a P -mor phism. For such a mor phism ϕ , the functor ϕ ∗ has a left adjoint which we denote by ϕ ] : M ( X , I ) / / M ( Y , J ) . For instance, giv en a S -diagram X inde xed by a small categor y I , each object i of I deﬁnes a P -mor phism of diagrams i : X i / / ( X , I ) (where X i is inde x ed by the terminal category), so that the corresponding the functor i ] corresponds precisely to ( 3.1.2.2 ). Assume that M is a complete P -ﬁbred category such that M ( S ) has small limits f or an y object S of S . Then the functor ϕ ∗ has a r ight adjoint which we denote b y ϕ ∗ : M ( X , I ) / / M ( Y , J ) . In the case where ϕ is the mor phism i : X i / / ( X , I ) deﬁned b y an object i of I , i ∗ corresponds precisely to ( 3.1.2.5 ). Remar k 3.1.4 This construction can be applied in par ticular to any Grothendieck abelian (monoidal) P -ﬁbred category ( cf. deﬁnition 1.3.8 ). The triangulated case cannot be treated in g eneral without assuming a thorough str ucture – this is the purpose of the next section. 3.1.b The model category case 3.1.5 Let M be a P -ﬁbred model category ov er S ( cf. 1.3.21 ). Giv en a S -diagram X inde x ed b y a small categor y I , we will sa y that a morphism of M ( X , I ) is a termwise weak equivalence (resp. a termwise ﬁbr ation , resp. a termwise coﬁbration ) if, f or an y object i of I , its image by the functor i ∗ is a weak equiv alence (resp. a ﬁbration, resp. a coﬁbration) in M ( X i ) . 3 Descent in P -ﬁbred model categor ies 85 Proposition 3.1.6 If M is a coﬁbrantly g enerat ed P -ﬁbred model category ov er S , then, for any S -diagram X indexed by a small category I , the categor y M ( X , I ) is a coﬁbrantly gener ated model categor y whose weak equiv alences (resp. ﬁbrations) ar e the termwise w eak equiv alences (r esp. the termwise ﬁbr ations). This model category structure on M ( X , I ) will be called the projectiv e model structure. Mor eov er , any coﬁbration of M ( X , I ) is a termwise coﬁbration, and the family of functors i ∗ : Ho ( M )( X , I ) / / Ho ( M )( X i ) , i ∈ Ob ( I ) , is conser v ative. If M is lef t pr oper (r esp. right pr oper , resp. combinatorial, r esp. stable), then so is the pr ojective model categor y structur e on M ( X ) . Proof Let X δ be the S -diag ram index ed by the set of objects of I (seen as a discrete category), whose ﬁber at i is X i . Let ϕ : ( X δ , Ob I ) / / ( X , I ) be the inclusion (i.e. the map which is the identity on objects and which is the identity on each ﬁber). As ϕ is clearl y a P -mor phism, we hav e an adjunction ϕ ] : M ( X δ , Ob I ) ' Ö i M ( X i ) / / o o M ( X , I ) : ϕ ∗ . The functor ϕ ] can be made explicit: it sends a famil y of objects ( M i ) i (with M i in M ( X i ) ) to the sum of the i ] ( M i ) ’ s index ed by the set of objects of I . Note also that this proposition is tr ivially veriﬁed whenev er X δ = X . Using the explicit formula f or i ] giv en in 3.1.2 , it is then straightf or w ard to check that the adjunction ( ϕ ] , ϕ ∗ ) satisﬁes the assumptions of [ Cra95 , Theorem 3.3], which prov es the e xistence of the projectiv e model structure on M ( X , I ) . Further more, the g enerating coﬁbrations (resp. tr ivial coﬁbrations of M ( X , I ) ) can be described as f ollo ws. For each object i of I , let A i (resp. B i ) be a generating set of coﬁbrations (resp. of tr ivial coﬁbrations in M ( X i ) . The class of ter mwise trivial ﬁbrations (resp. of ter m wise ﬁbrations) of M ( X , I ) is the class of maps which hav e the right lifting proper ty with respect to the set A = ∪ i ∈ I i ] ( A i ) (resp. to the set B = ∪ i ∈ I i ] ( B i ) ). Hence, the set A (resp. B ) generates the class of coﬁbrations (resp. of trivial coﬁbrations). In particular, as an y element of A is a ter m wise coﬁbration (which f ollo ws immediatel y from the e xplicit f or mula f or i ] giv en in 3.1.2 ), and as ter m wise coﬁbrations are s table by pushouts, transﬁnite compositions and retracts, any coﬁbration is a ter m wise coﬁbration (b y the small object argument). As any ﬁbration (resp. coﬁbration) of M ( X , I ) is a term wise ﬁbration (resp. a termwise coﬁbration), it is clear that, whenev er the model categor ies M ( X i ) are right (resp. left) proper , the model categor y M ( X , I ) has the same proper ty . The functor ϕ ∗ preserves ﬁbrations and coﬁbrations, while it also preser v es and detects w eak equivalences (by deﬁnition). This implies that the induced functor ϕ ∗ : Ho ( M )( X , I ) / / Ho ( M )( X δ , Ob I ) ' Ö i Ho ( M )( X i ) 86 Fibred categories and the six functors formalism is conser v ativ e (using the facts that the set of maps from a coﬁbrant object to a ﬁbrant object in the homotopy categor y of a model categor y is the set of homotopy classes of maps, and that a mor phism of a model categor y is a weak equiv alence if and only if it induces an isomor phism in the homotopy categor y). As ϕ ∗ commutes to limits and colimits, this implies that it commutes to homotopy limits and to homotop y colimits (up to w eak equivalences). Using the conser vativity proper ty , this implies that a commutativ e square of M ( X , I ) is a homotopy pushout (resp. a homotopy pullback) if and only if it is so in M ( X δ , Ob I ) . Remember that stable model categories are characterized as those in which a commutative square is a homotop y pullback square if and only if it is a homotopy pushout sq uare. As a consequence, if all the model categor ies M ( X i ) are stable, as M ( X δ , Ob I ) is then ob viously stable as w ell, the model categor y M ( X , I ) has the same property . It remains to pro ve that, if M ( X , I ) is a combinator ial model categor y f or an y object X of S , then M ( X , I ) is combinator ial as w ell. For each object i in I , let G i be a set of accessible generators of M ( X i ) . Note that, for any object i of I , the functor i ] has a left adjoint i ∗ which commutes to colimits (having itself a r ight adjoint i ∗ ). It is then easy to chec k that the set of objects of shape i ] ( M ) , for M in G i and i in I , is a small set of accessible g enerators of M ( X , I ) . This implies that M ( X , I ) is accessible and ends the proof.  Proposition 3.1.7 Let M be a combinatorial P -ﬁbred model categor y ov er S . Then, for any S -diagr am X indexed by a small categor y I , the category M ( X , I ) is a combinatorial model categor y whose weak equivalences (r esp. coﬁbrations) ar e the termwise weak equivalences (r esp. the termwise coﬁbrations). This model category structur e on M ( X , I ) will be called the injectiv e model structure 57 . Moreo ver , any ﬁbration of the injective model structure on M ( X , I ) is a termwise ﬁbration. If M is lef t proper (r esp. right proper , resp. stable), then so is the injective model category structure on M ( X , I ) . Proof See [ Bar10 , Theorem 2.28] f or the e xistence of such a model structure (if, f or an y object X in S , all the coﬁbrations of M ( X ) are monomorphisms, this can also be done f ollowing mutatis mutandis the proof of [ A y o07a , Proposition 4.5.9]). Any tr ivial coﬁbration of the projective model str ucture being a ter m wise trivial coﬁbration, any ﬁbration of the injective model str ucture is a ﬁbration of the projectiv e model structure, hence a termwise ﬁbration. The asser tions about proper ness f ollo w from their analogs for the projectiv e model structure and from [ Cis06 , Corollary 1.5.21] (or can be prov ed directly ; see [ Bar10 , Proposition 2.31]). Similarl y , the asser tion on stability f ollo ws from their analogs f or the projectiv e model structure.  3.1.8 From no w on, w e assume that a combinatorial P -ﬁbred model category M o v er S is giv en. Then, for an y S -diagram ( X , I ) , we ha ve tw o model categor y structures on M ( X , I ) , and the identity deﬁnes a left Quillen equiv alence from the projectiv e model structure to the injectiv e model structure. This fact will be used 57 Quite unf or tunatel y , this cor responds to the ‘semi-projective ’ model structure introduced in [ A yo07a , Def. 4.5.8]. 3 Descent in P -ﬁbred model categor ies 87 f or the understanding of the functor ialities coming from mor phisms of diagrams of S -schemes. 3.1.9 The categor y of S -diag rams admits small sums. If {( Y j , I j )} j ∈ J is a small famil y of S -diagrams, then their sum is the S -diag ram ( X , I ) , where I = Þ j ∈ J I j , and X is the functor from I to S deﬁned b y X i = Y j whenev er i ∈ I j . Proposition 3.1.10 F or any small family of S -diagr ams { ( Y j , I j )} j ∈ J , the canonical functor Ho ( M )  Þ j ∈ J Y j  / / Ö j ∈ J Ho ( M )( Y j ) is an equiv alence of categories. Proof The functor M  Þ j ∈ J Y j  / / Ö j ∈ J M ( Y j ) is an equivalence of categor ies. It thus remains an equiv alence after localization. T o conclude, it is suﬃcient to see that the homotopy categor y of a product of model categories is the product of their homotopy categor ies, which f ollo ws rather easily from the e xplicit description of the homotop y categor y of a model categor y ; see e.g. [ Ho v99 , Theorem 1.2.10].  Proposition 3.1.11 Let ϕ = ( α , f ) : ( X , I ) / / ( Y , J ) be a morphism of S - diagr ams. (i) The adjunction ϕ ∗ : M ( Y , J ) / / o o M ( X , I ) : ϕ ∗ is a Quillen adjunction with r espect to the injective model structur es. In par ticular , it induces a derived adjunction L ϕ ∗ : Ho ( M )( Y , J ) / / o o Ho ( M )( X , I ) : R ϕ ∗ . (ii) If ϕ is a P -mor phism, then the adjunction ϕ ] : M ( X , I ) / / o o M ( Y , J ) : ϕ ∗ is a Quillen adjunction with r espect to the projectiv e model structur es, and the functor ϕ ∗ pr eser v es weak equivalences. In par ticular , we g et a deriv ed adjunction L ϕ ] : Ho ( M )( X , I ) / / o o Ho ( M )( Y , J ) : L ϕ ∗ = ϕ ∗ = R ϕ ∗ . Proof The functor ϕ ∗ obviousl y preserves termwise coﬁbrations and ter m wise tr ivial coﬁbrations (we reduce to the case of a mor phism of S using the e xplicit descr iption of ϕ ∗ giv en in 3.1.3 ), which pro v es the ﬁrst asser tion. Similarl y , the second asser tion f ollow s from the f act that, under the assumption that ϕ is a P -mor phism, the functor 88 Fibred categor ies and the six functors f or malism ϕ ∗ preserves ter m wise weak equiv alences (see Remark 1.3.22 ), as well as ter mwise ﬁbrations.  3.1.12 The computation of the (der iv ed) functors R ϕ ∗ (and L ϕ ] whenev er it makes sense) giv en b y Proposition 3.1.11 has to do with homotopy limits (and homotop y colimits). It is easier to ﬁrst understand this in the non der iv ed v ersion as f ollo ws. Consider ﬁrst the tr ivial case of a constant S -diagram: let X be an object of S , and I a small categor y . Then, seeing X as the constant functor I / / S with value X , w e hav e a projection map p I : ( X , I ) / / X . From the v ery deﬁnition, the categor y M ( X , I ) is simply the categor y of on I with v alues in M ( X ) , so that the in verse image functor (3.1.12.1) p ∗ I : M ( X ) / / M ( X , I ) = M ( X ) I op is the ‘constant diagram functor’ , while its r ight adjoint (3.1.12.2) lim o o I op = p I , ∗ : M ( X , I ) / / M ( X ) is the limit functor , and its left adjoint, (3.1.12.3) lim / / I op = p I , ] : M ( X , I ) / / M ( X ) is the colimit functor . Let S be an object of S . A S -diagr am ov er S is the data of a S -diagram ( X , I ) , together with a morphism of S -diag rams p : ( X , I ) / / S (i.e. its a S / S -diag ram). Such a map p factors as (3.1.12.4) ( X , I ) π / / ( S , I ) p I / / S , where π = ( p , 1 I ) . Then one checks easily that, f or an y object M of M ( X , I ) , and f or an y object i of I , one has (3.1.12.5) π ∗ ( M ) i ' p i , ∗ ( M i ) , where p i : X i / / S is the str uctural map, from which we deduce the f ormula (3.1.12.6) p ∗ ( M ) ' lim o o i ∈ I op π ∗ ( M ) i ' lim o o i ∈ I op p i , ∗ ( M i ) , Remark that, if I is a small categor y with a ter minal object ω , then any S -diag ram X inde xed by I is a S -diag ram o v er X ω , and we deduce from the computations abo v e that, if p : ( X , I ) / / X ω denotes the canonical map, then, for any object M of M ( X , I ) , (3.1.12.7) p ∗ ( M ) ' M ω . 3 Descent in P -ﬁbred model categor ies 89 Consider now a morphism of S -diag rams ϕ = ( α , f ) : ( X , I ) / / ( Y , J ) . F or each object j , we can f or m the f ollo wing pullback square of categories. I / j u j / / f / j   I f   J / j v j / / J (3.1.12.8) in which J / j is the categor y of objects of J o ver j (which has a terminal object, namely ( j , 1 j ) , and v j is the canonical projection; the categor y I / j is thus the category of pairs ( i , a ) , where i is an object of I , and a : f ( i ) / / j a mor phism in J . From this, w e can form the f ollo wing pullback of S -diagrams ( X / j , I / j ) µ j / / ϕ / j   ( X , I ) ϕ   ( Y / j , J / j ) ν j / / ( Y , J ) (3.1.12.9) in which X / j = X ◦ u j , Y / j = Y ◦ v j , and the maps µ j and ν j are the one induced by u j and v j respectiv ely . For an object M of M ( X , I ) (resp. an object N of M ( Y , J ) ), w e deﬁne M / j (resp. N / j ) as the object of M ( X / j , I / j ) (resp. of M ( Y / j , J / j ) ) obtained as M / j = µ ∗ j ( M ) (resp. N / j = ν ∗ j ( N ) ). W ith these con v entions, f or any object M of M ( X , I ) and any object j of the inde xing categor y J , one gets the formula (3.1.12.10) ϕ ∗ ( M ) j ' ( ϕ / j ) ∗ ( M / j ) ( j , 1 j ) ' lim o o ( i , a ) ∈ I / j op α i , ∗ ( M i ) . This implies that the natural map (3.1.12.11) ϕ ∗ ( M )/ j = ν ∗ j ϕ ∗ ( M ) / / ( ϕ / j ) ∗ µ ∗ j ( M ) = ( ϕ / j ) ∗ ( M / j ) is an isomor phism: to pro ve this, it is suﬃcient to obtain an isomorphism from ( 3.1.12.11 ) after e valuating by any object ( j 0 , a : j 0 / / j ) of J / j , which f ollow s readily from ( 3.1.12.10 ) and from the obvious f act that ( I / j )/( j 0 , a ) is canonically isomorphic to I / j 0 . In order to deduce from the computations abov e their der iv ed versions, we need tw o lemmata. Lemma 3.1.13 Let X be a S -diagr am indexed by a small categor y I , and i an object of I . Then the ev aluation functor i ∗ : M ( X , I ) / / M ( X i ) 90 Fibred categor ies and the six functors formalism is a right Quillen functor with respect to the injective model structure, and it preserves w eak equivalences. Proof Pro ving that the functor i ∗ is a right Quillen functor is equiv alent to proving that its left adjoint ( 3.1.2.2 ) is a left Quillen functor with respect to the injective model structure, which f ollo ws immediatel y from its computation ( 3.1.2.3 ), as, in an y model category , coﬁbrations and tr ivial coﬁbrations are s table by small sums. The last assertion is obvious from the v ery deﬁnition of the weak equiv alences in M ( X , I ) .  Lemma 3.1.14 F or any pullback squar e of S -diagr ams of shape ( 3.1.12.9 ) , the functors µ ∗ j : M ( X , I ) / / M ( X / j , I / j ) , M  / / M / j ν ∗ j : M ( Y , I ) / / M ( Y / j , J / j ) , N  / / N / j ar e right Quillen functors with r espect to the injective model structure, and they pr eser v e weak equivalences. Proof It is suﬃcient to pro v e this for the functor µ ∗ j (as ν ∗ j is simply the special case where I = J and f is the identity). The fact that µ ∗ j preserves weak equiv alences is obvious, so that it remains to pro ve that it is a r ight Quillen functor . W e thus ha v e to pro v e that left adjoint of µ ∗ j , µ j , ] : M ( X / j , I / j ) / / M ( X , I ) , is a left Quillen functor . In other words, w e hav e to pro v e that, f or any object i of I , the functor i ∗ µ j , ] : M ( X , I ) / / M ( X ) is a left Quillen functor . For any object M of M ( X , I ) , we ha v e a natural isomor - phism i ∗ µ j , ] ( M ) ' Þ a ∈ Hom J ( f ( i ) , j ) ( i , a ) ] ( M i ) . But we know that the functors ( i , a ) ] are left Quillen functors, so that the stability of coﬁbrations and tr ivial coﬁbrations b y small sums and this descr iption of the functor i ∗ µ j , ] achie ves the proof.  Proposition 3.1.15 Let S be an object of S , and p : ( X , I ) / / S a S -diagr am ov er S , and consider the canonical f actorization ( 3.1.12.4 ) . F or any object M of Ho ( M )( X , I ) , ther e ar e canonical isomor phisms and Ho ( M )( S ) : R π ∗ ( M ) i ' R p i , ∗ ( M i ) and R p ∗ ( M ) ' R lim o o i ∈ I op R p i , ∗ ( M i ) . In particular , if further mor e the category I has a terminal object ω , then R p ∗ ( M ) ' R p ω, ∗ ( M ω ) . 3 Descent in P -ﬁbred model categor ies 91 Proof This f ollo ws immediately from F ormulas ( 3.1.12.5 ), ( 3.1.12.6 ) and from the fact that der iving (right) Quillen functors is compatible with composition.  Proposition 3.1.16 W e consider the pullbac k squar e of S -diagr ams ( 3.1.12.9 ) (as w ell as the notations ther eof). F or any object M of Ho ( M )( X , I ) , and any object j of J , we have natural isomorphisms R ϕ ∗ ( M ) j ' R lim o o ( i , a ) ∈ I / j op R α i , ∗ ( M i ) and R ϕ ∗ ( M )/ j ' R ( ϕ / j ) ∗ ( M / j ) in Ho ( M )( Y j ) and in Ho ( M )( Y / j , J / j ) respectiv ely. Proof Using again the fact that der iving r ight Quillen functors is compatible with composition, by vir tue of Lemma 3.1.13 and Lemma 3.1.14 , this is a direct translation of ( 3.1.12.10 ) and ( 3.1.12.11 ).  Proposition 3.1.17 Let u : T / / S be a P -mor phism of S , and p : ( X , I ) / / S a S -diagr am ov er S . Consider the pullback square of S -diagr ams ( Y , I ) ϕ / / q   ( X , I ) p   T u / / S (i.e. Y i = T × S X i f or any object i of I ). Then, for any object M of Ho ( M )( X , I ) , the canonical map L u ∗ R p ∗ ( M ) / / R q ∗ L v ∗ ( M ) is an isomorphism in Ho ( M )( T ) . Proof By Remark 1.3.22 , the functor ν ∗ is both a left and a right Quillen functor which preser v es weak equiv alences, so that the functor L ν ∗ = ν ∗ = R ν ∗ preserves homotop y limits. Hence, by Proposition 3.1.15 , one reduces to the case where I is the terminal categor y , i.e. to the transposition of the isomor phism given by the P -base chang e formula ( P -BC) f or the homotop y P -ﬁbred categor y Ho ( M ) (see 1.1.19 ).  3.1.18 A mor phism of S -diagrams ν = ( α, f ) : ( Y 0 , J 0 ) / / ( Y , J ) , is cartesian if, f or an y ar ro w i / / j in J 0 , the induced commutative square Y 0 i / / α i   Y 0 j α j   Y f ( i ) / / Y f ( j ) is car tesian. A morphism of S -diagrams ν = ( α , f ) : ( Y 0 , J 0 ) / / ( Y , J ) is reduced if J = J 0 and f = 1 J . 92 Fibred categor ies and the six functors f ormalism Proposition 3.1.19 Let ν : ( Y 0 , J ) / / ( Y , J ) be a reduced car t esian P -mor phism of S -diagr ams, and ϕ = ( α, f ) : ( X , I ) / / ( Y , J ) a morphism of S -diagr ams. Consider the pullbac k squar e of S -diagrams ( X 0 , I ) µ / / ψ   ( X , I ) ϕ   ( Y 0 , J ) ν / / ( Y , J ) (i.e. X 0 i = Y 0 f ( i ) × Y f ( i ) X i f or any object i of I ). Then, f or any object M of Ho ( M )( X , I ) , the canonical map L ν ∗ R ϕ ∗ ( M ) / / R ψ ∗ L µ ∗ ( M ) is an isomorphism in Ho ( M )( Y 0 , J ) . Proof By vir tue of Proposition 3.1.6 , it is suﬃcient to pro v e that the map j ∗ L ν ∗ R ϕ ∗ ( M ) / / j ∗ R ψ ∗ L µ ∗ ( M ) is an isomorphism f or an y object j of J . Let p : ( X / j , I / j ) / / Y j and q : ( X 0 / j , J , j ) / / Y 0 j be the canonical maps. As ν is cartesian, w e ha ve a pullback square of S -diagrams ( X 0 / j , I / j ) µ / j / / q   ( X / j , I / j ) p   Y 0 j ν j / / Y j But ν j being a P -mor phism, b y vir tue of Proposition 3.1.17 , we thus hav e an isomorphism L ν ∗ j R p ∗ ( M / j ) ' R q ∗ L ( µ / j ) ∗ ( M / j ) = R q ∗ ( L µ ∗ ( M )/ j ) . Applying Proposition 3.1.16 and the last asser tion of Proposition 3.1.15 twice, we also ha ve canonical isomor phisms j ∗ R ϕ ∗ ( M ) ' R p ∗ ( M / j ) and j ∗ R ψ ∗ L µ ∗ ( M ) ' R q ∗ ( L µ ∗ ( M )/ j ) . The obvious identity j ∗ L ν ∗ = L ν ∗ j j ∗ achie ves the proof.  Corollary 3.1.20 Under the assumptions of Proposition 3.1.19 , for any object N of the category Ho ( M )( Y 0 , j ) , the canonical map L µ ] L ψ ∗ ( N ) / / L ϕ ∗ L ν ] ( N ) 3 Descent in P -ﬁbred model categor ies 93 is an isomorphism in Ho ( M )( X , I ) . Remar k 3.1.21 The class of cartesian P -mor phisms f or m an admissible class of morphisms in the categor y of S -diag rams, which we denote b y P c art . Proposition 3.1.11 and the preceding corollar y thus asser ts that Ho ( M ) is a P c art -ﬁbred categor y o v er the categor y of S -diagrams. 3.1.22 W e shall sometimes deal with diagrams of S -diagrams. Let I be a small category , and F a functor from I to the category of S -diagrams. For each object i of I , we hav e a S -diagram ( F ( i ) , J i ) , and, for each map u : i / / i 0 , w e hav e a functor f u : J i / / J i 0 as well as a natural transf or mation α u : F ( i ) / / F ( i 0 ) ◦ f u , subject to coherence identities. In particular, the correspondence i  / / J i deﬁnes a functor from I to the categor y of small categor ies. Let I F be the coﬁbred categor y o v er I associated to it; see [ Gro03 , Exp. VI]. Explicitly , I F is descr ibed as f ollo ws. The objects are the couples ( i , x ) , where i is an object of I , and x is an object of J i . A mor phism ( i , x ) / / ( i 0 , x 0 ) is a couple ( u , v ) , where u : i / / i 0 is a mor phism of I , and v : f u ( x ) / / x 0 is a morphism of J i 0 . The identity of ( i , x ) is the couple ( 1 i , 1 x ) , and, for tw o mor phisms ( u , v ) : ( i , x ) / / ( i 0 , x 0 ) and ( u 0 , v 0 ) : ( i 0 , x 0 ) / / ( i 00 , x 00 ) , their composition ( u 00 , v 00 ) : ( i , x ) / / ( i 00 , x 00 ) is deﬁned by u 00 = u 0 ◦ u , while v 00 is the composition of the map f u 0 0 ( x ) = f u 0 ( f u ( x )) f u 0 ( v ) / / f u 0 ( x 0 ) v 0 / / x 00 . The functor p : I F / / I is simply the projection ( i , x )  / / i . For each object i of I , w e get a canonical pullback square of categor ies J i q   ` i / / I F p   e i / / I (3.1.22.1) in which i is the functor from the ter minal category e which cor responds to the object i , and ` i is the functor deﬁned by ` i ( x ) = ( i , x ) . The functor F deﬁnes a S -diag ram ( ∫ F , I F ) : f or an object ( i , x ) of I F , ( ∫ F ) ( i , x ) = F ( i ) x , and f or a mor phism ( u , v ) : ( i , x ) / / ( i 0 , x 0 ) , the map ( u , v ) : ( ∫ F ) ( i , x ) = F ( i ) x / / ( ∫ F ) ( i 0 , x 0 ) = F ( i 0 ) x 0 is simply the mor phism induced by α u and v . For each object i of I , there is a natural morphism of S -diag rams (3.1.22.2) λ i : ( F ( i ) , J i ) / / ( ∫ F , I F ) , giv en by λ i = ( 1 F ( i ) , ` i ) 94 Fibred categor ies and the six functors formalism Proposition 3.1.23 Let X be an object of S , and f : F / / X a morphism of functors (where X is consider ed as the constant functor fr om I to S -diagrams with value the functor from e to S deﬁned by X ). Then, for eac h object i of I , we have a canonical pullbac k squar e of S -diagr ams ( F ( i ) , J i ) λ i / / ϕ i   ( ∫ F , I F ) ϕ   X i / / ( X , I ) in whic h ϕ and ϕ i ar e the obvious morphisms induced by f (wher e, this time, ( X , I ) is seen as the constant functor fr om I to S with v alue X ). Mor eov er , for any object M of Ho ( M )( ∫ F , I F ) , the natur al map i ∗ R ϕ ∗ ( M ) = R ϕ ∗ ( M ) i / / R ϕ i , ∗ λ ∗ i ( M ) is an isomorphism. In particular , if we also write by abuse of notation f for the induced map of S -diagrams from ( ∫ F , I F ) to X , we have a natur al isomor phism R f ∗ ( M ) ' R lim o o i ∈ I op R ϕ i , ∗ λ ∗ i ( M ) . Proof This pullback square is the one induced by ( 3.1.22.1 ). W e shall pro v e ﬁrst that the map i ∗ R ϕ ∗ ( M ) = R ϕ ∗ ( M ) i / / R ϕ i , ∗ λ ∗ i ( M ) is an isomor phism in the par ticular case where I has a ter minal object ω and i = ω . By vir tue of Propositions 3.1.15 and 3.1.16 , we hav e isomor phisms (3.1.23.1) ω ∗ R ϕ ∗ ( M ) ' R lim o o i ∈ I op R ϕ ∗ ( M ) i ' R lim o o ( i , x ) ∈ I op F R ϕ i , x , ∗ ( M ( i , x ) ) , where ϕ i , x : F ( i ) x / / X denotes the map induced b y f . W e are thus reduced to pro v e that the canonical map (3.1.23.2) R lim o o ( i , x ) ∈ I op F R ϕ i , x , ∗ ( M ( i , x ) ) / / R lim o o x ∈ J op ω R ϕ ω, x , ∗ ( M ( ω, x ) ) ' R ϕ ω, ∗ λ ∗ ω ( M ) is an isomor phism. As I F is coﬁbred ov er I , and as ω is a ter minal object of I , the inclusion functor ` ω : J ω / / I F has a left adjoint, whence is coaspher ical in an y w eak basic localizer (i.e. is homotopy coﬁnal); see [ Mal05 , 1.1.9, 1.1.16 and 1.1.25]. As any model categor y deﬁnes a Grothendieck der iv ator ([ Cis03 , Thm. 6.11]), it f ollow s from [ Cis03 , Cor . 1.15] that the map ( 3.1.23.2 ) is an isomor phism. T o prov e the general case, we proceed as f ollo ws. Let F / i be the functor obtained b y composing F with the canonical functor v i : I / i / / I . Then, keeping track of the conv entions adopted in 3.1.12 , w e c heck easil y that ( I / i ) F / i = ( I F )/ i and that 3 Descent in P -ﬁbred model categor ies 95 ∫ ( F / i ) = ( ∫ F )/ i . Moreo v er , the pullback square ( 3.1.22.1 ) is the composition of the f ollo wing pullback squares of categor ies. J i a i / / q   I F / i u i / / p / i   I F p   e ( i , 1 i ) / / I / i v i / / I The pullback square of the proposition is thus the composition of the f ollo wing pullback squares. ( F ( i ) , J i ) α i / / ϕ i   ( ∫ F / i , I F / i ) µ i / / ϕ / i   ( ∫ F , I F ) ϕ   X ( i , 1 i ) / / ( X , I / i ) v i / / ( X , I ) The natural transf ormations ( i , 1 i ) ∗ R ( ϕ / i ) ∗ / / R ϕ i , ∗ α ∗ i and v ∗ i R ϕ ∗ / / R ( ϕ / i ) ∗ µ ∗ i are both isomor phisms: the ﬁrst one comes from the fact that ( i , 1 i ) is a ter minal object of I / i , and the second one from Proposition 3.1.16 . W e thus get: i ∗ R ϕ ∗ ( M ) ' ( i , 1 i ) ∗ v ∗ i R ϕ ∗ ( M ) ' ( i , 1 i ) ∗ R ( ϕ / i ) ∗ µ ∗ i ( M ) ' R ϕ i , ∗ α ∗ i µ ∗ i ( M ) ' R ϕ i , ∗ λ ∗ i ( M ) . The last asser tion of the proposition is then a straightf or ward application of Propo- sition 3.1.15 .  Proposition 3.1.24 If M is a monoidal P -ﬁbred combinatorial model category ov er S , then, f or any S -diagr am X indexed by a small categor y I , the injec- tiv e model structure tur ns M ( X , I ) into a symmetric monoidal model categor y . In particular , the categories Ho ( M )( X , I ) ar e canonically endow ed with a closed symmetric monoidal structur e, in such a way that, f or any mor phism of S -diagr ams ϕ : ( X , I ) / / ( Y , J ) , the functor L ϕ ∗ : Ho ( M )( Y , J ) / / Ho ( M )( X , I ) is sym- metric monoidal. Proof This is obvious from the deﬁnition of a symmetric monoidal model category , as the tensor product of M ( X , I ) is deﬁned termwise, as well as the coﬁbrations and the tr ivial coﬁbrations.  Proposition 3.1.25 Assume that M is a monoidal P -ﬁbred combinatorial model category ov er S , and consider a reduced cartesian P -mor phism ϕ = ( α, f ) : 96 Fibred categories and the six functors formalism ( X , I ) / / ( Y , I ) . Then, for any object M in Ho ( M )( X , I ) and any object N in Ho ( M )( Y , I ) , the canonical map L ϕ ] ( M ⊗ L ϕ ∗ ( N )) / / L ϕ ] ( M ) ⊗ L N is an isomorphism. Proof Let i be an object of I . It is suﬃcient to prov e that the map i ∗ L ϕ ] ( M ⊗ L ϕ ∗ ( N )) / / i ∗ L ϕ ] ( M ) ⊗ L N is an isomorphism in Ho ( M )( X i ) . Using Corollary 3.1.20 , we see that this map can be identiﬁed with the map L ϕ i , ] ( M i ⊗ L ϕ ∗ i ( N i )) / / L ϕ i , ] ( M i ) ⊗ L N i , which is an isomor phism according to the P -projection f ormula f or the homotopy P -ﬁbred categor y Ho ( M ) .  3.1.26 Let ( X , I ) be a S -diagram. An object M of M ( X , I ) is homot opy carte- sian if, f or any map u : i / / j in I , the s tr uctural map u ∗ ( M j ) / / M i induces an isomorphism L u ∗ ( M i ) ' M j in Ho ( M )( X , I ) (i.e. if there e xists a weak equivalence M 0 j / / M j with M 0 j coﬁbrant in M ( X j ) such that the map u ∗ ( M 0 j ) / / M i is a weak equivalence in M ( X i ) ). W e denote by Ho ( M )( X , I ) hc art the full subcategor y of Ho ( M )( X , I ) spanned b y homotopy car tesian sections. Deﬁnition 3.1.27 A coﬁbrantly g enerated model category V is tractable if there e xist sets I and J of coﬁbrations between coﬁbrant objects which g enerate the class of coﬁbrations and the class of tr ivial coﬁbrations respectivel y . Remar k 3.1.28 If M is a combinatorial and tractable P -ﬁbred model category ov er S , then so are the projective and the injective model s tr uctures on M ( X , I ) ; see [ Bar10 , Thm. 2.28 and 2.30]. Proposition 3.1.29 If M is tractable, then the inclusion functor Ho ( M )( X , I ) hc art / / Ho ( M )( X , I ) admits a right adjoint. Proof This follo ws from the fact that the coﬁbrant homotopy cartesian sections are the coﬁbrant objects of a r ight Bousﬁeld localization of the injective model structure on M ( X , I ) ; see [ Bar10 , Theorem 5.25].  Deﬁnition 3.1.30 Let M and M 0 tw o P -ﬁbred model categor ies ov er S . A Quillen morphism γ from M to M 0 is a mor phism of P -ﬁbred categor ies γ : M / / M 0 such that γ ∗ : M ( X ) / / M 0 ( X ) is a left Quillen functor for any object X of S . 3 Descent in P -ﬁbred model categor ies 97 Remar k 3.1.31 If γ : M / / M 0 is a Quillen morphism betw een P -ﬁbred combina- torial model categor ies, then, f or any S -diagram ( X , I ) , we get a Quillen adjunction γ ∗ : M ( X , I ) / / o o M 0 ( X , I ) : γ ∗ (with the injective model structures as well as with the projectiv e model structures). Proposition 3.1.32 F or any Quillen mor phism γ : M / / M 0 , the deriv ed adjunc- tion L γ ∗ : Ho ( M )( X ) / / o o Ho ( M 0 )( X ) : R γ ∗ deﬁnes a mor phism of P -ﬁbr ed categories Ho ( M ) / / Ho ( M 0 ) ov er S . If mor e- ov er M and M 0 ar e combinatorial, then the morphism Ho ( M ) / / Ho ( M 0 ) extends to a morphism of P c art -ﬁbr ed categories ov er the category of S -diagr ams. Proof This f ollo ws immediately from [ Ho v99 , Theorem 1.4.3].  3.2 Hyperco v ers, descent, and deriv ed global sections 3.2.1 Let S be an essentially small category , and P an admissible class of mor - phisms in S . W e assume that a Grothendieck topology t on S is given. W e shall write S q f or the full subcategor y of the category of S -diagrams whose objects are the small families X = { X i } i ∈ I of objects of S (seen as functors from a discrete category to S ). The category S q is equivalent to the full subcategor y of the cat- egory of preshea v es of sets on S spanned by sums of representable preshea v es. In particular, small sums are representable in S q (but note that the functor from S to S q does not preser v e sums). Finall y , w e remark that the topology t extends naturally to a Grothendieck topology on S q such that the topology t on S is the topology induced from the inclusion S ⊂ S q . The cov er ing maps for this topology on S q will be called t -cov ers (note that the inclusion S ⊂ S q is continuous and induces an equivalence between the topos of t -shea v es on S and the topos of t -sheav es on S q ). Let ∆ be the categor y of non-empty ﬁnite ordinals. Remember that a simplicial object of S q is a presheaf on ∆ with values in S q . For a simplicial set K and an object X of S q , w e denote by K × X the simplicial object of S q deﬁned b y ( K × X ) n = Þ x ∈ K n X , n ≥ 0 . W e write ∆ n f or the standard combinatorial simple x of dimension n , and i n : ∂ ∆ n / / ∆ n f or its boundar y inclusion. A mor phism p : X / / Y between simplicial objects of S q is a t -hypercov er if, locally for the t -topology , it has the right lifting proper ty with respect to boundar y inclusions of standard simplices, which, in a more precise w ay , means that, f or any integer n ≥ 0 , an y object U of S q , and any commutative square 98 Fibred categories and the six functors formalism ∂ ∆ n × U x / / i n × 1   X p   ∆ n × U y / / Y , there e xists a t -co v er ing q : V / / U , and a morphism of simplicial objects z : ∆ n × V / / X , such that the diag ram bello w commutes. ∂ ∆ n × V x ( 1 × q ) / / i n × 1   X p   ∆ n × V y ( 1 × q ) / / z : : Y A t -hyperco ver of an object X of S q is a a t -h yperco ver p : X / / X (where X is considered as a constant simplicial object). Remar k 3.2.2 This deﬁnition of t -h yperco v er is equiv alent to the one giv en in [ A GV73 , Exp. V , 7.3.1.4]. 3.2.3 Let X be a simplicial object of S q . It is in par ticular a functor from the category ∆ op to the category of S -diagrams, so that the constructions and consid- erations of 3.1.22 apply to X . In par ticular , there is a S -diagram ˜ X associated to X , namely ˜ X = ( ∫ X , ( ∆ op ) X ) . More e xplicitly , f or each integer n ≥ 0 , there is a famil y { X n , x } x ∈ K n of objects of S , such that (3.2.3.1) X n = Þ x ∈ K n X n , x . In f act, the sets K n f or m a simplicial set K , and the categor y ( ∆ op ) X can be identiﬁed o v er ∆ op to the categor y ( ∆ / K ) op , where ∆ / K is the ﬁbred categor y o v er ∆ whose ﬁber o v er n is the set K n (seen as a discrete categor y), i.e. the categor y of simplices of K . W e shall call K the underlying simplicial set of X , while the decomposition ( 3.2.3.1 ) will be called the local pr esentation of X . The construction X  / / ˜ X is functorial. If p : X / / Y is a morphism of simplicial objects of S q , we shall still denote b y p : ˜ X / / ˜ Y the induced mor phism of S -diagrams. In par ticular , f or a morphism of p : X / / X , where X is an object of S q , p : ˜ X / / X denotes the corresponding mor phism of S -diagrams. Let M be a P -ﬁbred combinator ial model category o v er S . Giv en a simplicial object X of S q , w e deﬁne the categor y Ho ( M )( X ) by the formula: (3.2.3.2) Ho ( M )( X ) = Ho ( M )( ∫ X , ( ∆ op ) X ) . Giv en an object X of S q and a morphism p : X / / X , we hav e a derived adjunction (3.2.3.3) L p ∗ : Ho ( M )( X ) / / o o Ho ( M )( X ) : R p ∗ . 3 Descent in P -ﬁbred model categor ies 99 Proposition 3.2.4 Consider an object X of S , a simplicial object X of S q , as w ell as a mor phism p : X / / X . Denot e by K the underlying simplicial set of X , and for each integ er n ≥ 0 and each simplex x ∈ K n , write p n , x : X n , x / / X f or the morphism of S q induced by the local pr esentation of X ( 3.2.3.1 ) . Then, for any object M of Ho ( M )( X ) , ther e ar e canonical isomor phisms R p ∗ R p ∗ ( M ) ' R lim o o n ∈ ∆ R p n , ∗ L p ∗ n ( M ) ' R lim o o n ∈ ∆  Ö x ∈ K n R p n , x , ∗ L p ∗ n , x ( M )  . Proof The ﬁrst isomor phism is a direct application of the last asser tion of Proposition 3.1.23 for F = X , while the second one follo ws from the ﬁrst one by Proposition 3.1.10 .  Deﬁnition 3.2.5 Giv en an object Y of S q , an object M of Ho ( M )( Y ) will be said to satisfy t -descent if it has the f ollo wing property: for any mor phism f : X / / Y and an y t -h yperco v er p : X / / X , the map R f ∗ L f ∗ ( M ) / / R f ∗ R p ∗ L p ∗ L f ∗ ( M ) is an isomor phism in Ho ( M )( Y ) . W e shall say that M (or by abuse, that Ho ( M ) ) satisﬁes t -descent if, for an y object Y of S q , an y object of Ho ( M )( Y ) satisﬁes t -descent. Proposition 3.2.6 If Y = { Y i } i ∈ I is a small family of objects of S (seen as an object of S q ), t hen an object M of Ho ( M )( Y ) satisﬁes t -descent if and only if, for any i ∈ I , any mor phism f : X / / Y i of S , and any t -hyper cov er p : X / / X , the map R f ∗ L f ∗ ( M i ) / / R f ∗ R p ∗ L p ∗ L f ∗ ( M i ) is an isomorphism in Ho ( M )( Y i ) . Proof This f ollo ws from the deﬁnition and from Proposition 3.1.10 .  Corollary 3.2.7 The P -ﬁbred model category M satisﬁes t -descent if and only if, f or any object X of S , and any t -hypercov er p : X / / X , the functor L p ∗ : Ho ( M )( X ) / / Ho ( M )( X ) is fully fait hful. Proposition 3.2.8 If M satisﬁes t -descent, then, for any t -cov er f : Y / / X , the functor L f ∗ : Ho ( M )( X ) / / Ho ( M )( Y ) is conser v ative. Proof Let f : Y / / X be a t -cov er , and u : M / / M 0 a morphism of Ho ( M )( X ) whose image b y L f ∗ is an isomor phism. W e can consider the Čech t -hyperco v er associated to f , that is the simplicial object Y ov er X deﬁned by 100 Fibred categor ies and the six functors f or malism Y n = Y × X Y × X · · · × X Y | {z } n + 1 times . Let p : Y / / X be the canonical map. For each n ≥ 0 , the map p n : Y n / / X factor through f , from which w e deduce that the functor L p ∗ n : Ho ( M )( X ) / / Ho ( M )( Y n ) sends u to an isomor phism. This implies that the functor L p ∗ : Ho ( M )( X ) / / Ho ( M )( Y ) sends u to an isomor phism as well. But, as Y is a t -hyperco v er of X , the functor L p ∗ is full y faithful, from which we deduce that u is an isomor phism by the Y oneda Lemma.  3.2.9 Let V be a complete and cocomplete categor y . For an object X of S , deﬁne PSh ( S / X , V ) as the category of presheav es on S / X with values in V . Then PSh ( C /− , V ) is a P -ﬁbred category (where, by abuse of notations, S denotes also the class of all maps in S ): this is a special case of the constructions explained in 3.1.2 applied to V , seen as a ﬁbred categor y ov er the ter minal categor y . T o be more e xplicit, f or each object X of S q , w e hav e a V -enr iched Y oneda embedding (3.2.9.1) S q / X × V / / PSh ( S / X , V ) , ( U , M }  / / U ⊗ M , where, if U = { U i } i ∈ I is a small famil y of objects of S / X , U ⊗ M is the presheaf (3.2.9.2) V  / / Þ i ∈ I Þ a ∈ Hom S / S ( V , U i ) M . For a mor phism f : X / / Y in S , the functor f ∗ : PSh ( S / Y , V ) / / PSh ( S / X , V ) is the functor deﬁned by composition with the cor responding functor S / X / / S / Y . The functor f ∗ has alwa ys a left adjoint f ] : PSh ( S / X , V ) / / PSh ( S / Y , V ) , which is the unique colimit preserving functor deﬁned by f ] ( U ⊗ M ) = U ⊗ M , where, on the left hand side U is considered as an object o v er X , while, on the right hand side, U is considered as an object o ver Y by composition with f . Similarl y , if all the pullbac ks b y f are representable in S (e.g. if f is a P -mor phism), the functor f ∗ can be descr ibed as the colimit preser ving functor deﬁned by the f or mula 3 Descent in P -ﬁbred model categor ies 101 f ∗ ( U ⊗ M ) = ( X × Y U ) ⊗ M . If V is a coﬁbrantly generated model categor y , then, f or each object X of S , the category PSh ( S / X , V ) is naturall y endo w ed with the pr ojective model categor y structur e , i.e. with the coﬁbrantly generated model categor y structure whose w eak equiv alences and ﬁbrations are deﬁned ter m wise (this is Proposition 3.1.6 applied to V , seen as a ﬁbred category o ver the ter minal categor y). The coﬁbrations of the projectiv e model categor y structure on PSh ( S / X , V ) will be called the projectiv e coﬁbrations. If moreo v er V is combinatorial (resp. left proper , resp. right proper , resp. stable), so is PSh ( S / X , V ) . In par ticular , if V is a combinatorial model category , then PSh ( S /− , V ) is a P -ﬁbred combinator ial model category ov er S . A ccording to Deﬁnition 3.2.5 , it thus makes sense to speak of t -descent in PSh ( S /− , V ) . If U = { U i } i ∈ I is a small famil y of objects of S ov er X , and if F is a presheaf o v er S / X , we deﬁne (3.2.9.3) F ( U ) = Ö i ∈ I F ( U i ) . the functor F  / / F ( U ) is a r ight adjoint to the functor E  / / U ⊗ E . W e remark that a ter m wise ﬁbrant presheaf F on S / X satisﬁes t -descent if and only if, for any object Y of S q , and any t -h yperco v er Y / / Y ov er X , the map F ( Y ) / / R lim o o n ∈ ∆ F ( Y n ) is an isomor phism in Ho ( V ) . Proposition 3.2.10 If V is combinatorial and lef t proper , then t he category of pr esheaves PSh ( S / X , V ) admits a combinatorial model categor y structure whose coﬁbrations ar e the projectiv e coﬁbrations, and whose ﬁbrant objects ar e the termwise ﬁbrant objects which satisfy t -descent. This model category structure will be called the t -local model categor y s tr ucture , and the corresponding homotopy category will be denot ed by Ho t ( PSh ( S / X , V ) ) . Mor eov er , any termwise w eak equiv alence is a w eak equiv alence for the t -local model structur e, and the induced functor a ∗ : Ho ( PSh ( S / X , V ) ) / / Ho t ( PSh ( S / X , V ) ) admits a fully faithful right adjoint a ∗ : Ho t ( PSh ( S / X , V ) ) / / Ho ( PSh ( S / X , V ) ) whose essential imag e consists precisely of the full subcategor y of Ho ( PSh ( S / X , V ) ) spanned by the pr esheaves which satisfy t -descent. Proof Let H be the class of maps of shape 102 Fibred categor ies and the six functors f ormalism (3.2.10.1) ho colim n ∈ ∆ op Y n ⊗ E / / Y ⊗ E , where Y is an object of S q o v er X , Y / / Y is a t -hyperco v er , and E is a coﬁ- brant replacement of an object whic h is either a source or a target of a generating coﬁbration of V . Deﬁne the t -local model category structure as the left Bousﬁeld localization of Pr ( S / X , V ) by H ; see [ Bar10 , Theorem 4.7]. W e shall call t -local w eak equiv alences the w eak equiv alences of the t -local model category structure. For each object Y o ver X , the functor Y ⊗ (−) is a left Quillen functor from V to Pr ( S / X , V ) . W e thus get a total left derived functor Y ⊗ L (−) : Ho ( V ) / / Ho t ( PSh ( S / X , V ) ) whose right adjoint is the ev aluation at Y . F or an y object E of V and an y t -local ﬁbrant presheaf F on S / X with values in V , we thus hav e natural bijections (3.2.10.2) Hom ( E , F ( Y )) ' Hom ( Y ⊗ L E , F ) , and, f or an y simplicial object Y of S / X , identiﬁcations (3.2.10.3) Hom ( E , R lim o o n ∈ ∆ F ( Y n )) ' Hom ( L lim / / n ∈ ∆ Y n ⊗ L E , F ) , One sees easily that, f or any t -h yperco v er Y / / Y and any coﬁbrant object E of V , the map (3.2.10.4) L lim / / n ∈ ∆ Y n ⊗ L E / / Y ⊗ L E is an isomor phism in the t -local homotopy category Ho t ( PSh ( S / X , V ) ) : b y the small object argument, the smallest full subcategory of Ho ( PSh ( S / X , V ) ) which is stable by homotopy colimits and which contains the source and the targ ets of the generating coﬁbrations is Ho t ( PSh ( S / X , V ) ) itself, and the class of objects E of V such that the map ( 3.2.10.4 ) is an isomor phism in Ho ( V ) is sable by homotopy colimits. Similarly , w e see that, for an y object E , the functor (−) ⊗ L E preser v es sums. As a consequence, we get from ( 3.2.10.2 ) and ( 3.2.10.3 ) that the ﬁbrant objects of the t -local model categor y str ucture are precisel y the ter mwise ﬁbrant objects F of the projectiv e model structure which satisfy t -descent. The last par t of the proposition f ollow s from the general yog a of left Bousﬁeld localizations.  3.2.11 Let M be a P -ﬁbred combinatorial model category ov er S , and S an object of S . Denote b y S : S / S / / S the canonical f org etful functor . Then there is a canonical mor phism of S -diag rams (3.2.11.1) σ : ( S , S / S ) / / ( S , S / S ) (where ( S , S / S ) stands f or the constant diag ram with value S ). This deﬁnes a functor 3 Descent in P -ﬁbred model categor ies 103 (3.2.11.2) R σ ∗ : Ho ( M )( S , S / S ) / / Ho ( M )( S , S / S ) = Ho ( PSh ( S / S , M ( S ) ) ) . For an object M of Ho ( M )( S ) , one deﬁnes the presheaf of geome tric derived g lobal sections of M o v er S by the f ormula (3.2.11.3) R Γ ge om (− , M ) = R σ ∗ L σ ∗ ( M ) . This is a presheaf on S / S with values in M ( S ) whose ev aluation on a mor phism f : X / / S is, by vir tue of Propositions 3.1.15 and 3.1.16 , (3.2.11.4) R Γ ge om ( X , M ) ' R f ∗ L f ∗ ( M ) . Proposition 3.2.12 F or an object M of Ho ( M )( S ) , the f ollowing conditions ar e equiv alent. (a) The object M satisﬁes t -descent. (b) The pr esheaf R Γ ge om (− , M ) satisﬁes t -descent. Proof F or an y mor phism f : X / / S and any t -hyperco ver p : X / / X o v er S , we ha v e, by Proposition 3.2.4 and f ormula ( 3.2.11.4 ), an isomor phism R f ∗ R p ∗ L p ∗ L f ∗ ( M ) ' R lim o o n ∈ ∆ R Γ ge om ( X n , M ) . From there, we see easily that conditions (a) and (b) are equivalent.  3.2.13 The preceding proposition allo ws us to reduce descent problems in a ﬁbred model category to descent problems in a categor y of preshea v es with values in a model category . On can e v en go further and reduce the problem to categor y of preshea v es with values in an ‘elementary model categor y’ as f ollow s. Consider a model category V . Then one can associate to V its correspond- ing prederiv ator Ho ( V ) , that is the strict 2 -functor from the 2 -categor y of small categories to the 2 -categor y of categories, deﬁned b y (3.2.13.1) Ho ( V )( I ) = Ho ( V I op ) = Ho ( PSh ( I , V ) ) f or any small category I . More e xplicitly : f or any functor u : I / / J , one gets a functor u ∗ : Ho ( V )( J ) / / Ho ( V )( I ) (induced b y the composition with u ), and f or any mor phism of functors I u ' ' v 7 7   α J , one has a mor phism of functors 104 Fibred categor ies and the six functors formalism Ho ( V )( I ) Ho ( V )( J ) v ∗ l l u ∗ r r K S α ∗ . Moreo v er , the prederivator Ho ( V ) is then a Grothendieck der ivator; see [ Cis03 , Thm. 6.11]. This means in particular that, for any functor betw een small categories u : I / / J , the functor u ∗ has a left adjoint (3.2.13.2) L u ] : Ho ( V )( I ) / / Ho ( V )( J ) as w ell as a r ight adjoint (3.2.13.3) R u ∗ : Ho ( V )( I ) / / Ho ( V )( J ) (in the case where J = e is the ter minal categor y , then L u ] is the homotopy colimit functor , while R u ∗ is the homotopy limit functor). If V and V 0 are tw o model categor ies, a morphism of derivat ors Φ : Ho ( V ) / / Ho ( V 0 ) is simply a mor phism of 2 -functors, that is the data of functors Φ I : Ho ( V )( I ) / / Ho ( V 0 )( I ) together with coherent isomor phisms u ∗ ( Φ J ( F )) ' Φ I ( u ∗ ( F )) f or any functor u : I / / J and any presheaf F on J with values in V (see [ Cis03 , p. 210] f or a precise deﬁnition). Such a morphism Φ is said to be continuousmor phism!continuous if, f or an y functor u : I / / J , and any object F of Ho ( V )( I ) , the canonical map (3.2.13.4) Φ J R u ∗ ( F ) / / R u ∗ Φ I ( F ) is an isomor phism. One can check that a mor phism of derivators Φ is continuous if and onl y if it commutes with homotop y limits (i.e. if and only if the maps ( 3.2.13.4 ) are isomor phisms in the case where J = e is the ter minal category); see [ Cis08 , Prop. 2.6]. For instance, the total r ight derived functor of any r ight Quillen functor deﬁnes a continuous mor phism of der iv ators; see [ Cis03 , Prop. 6.12]. Dually a morphism Φ of der ivators is cocontinuous if, for any functor u : I / / J , and an y object F of Ho ( V )( I ) , the canonical map (3.2.13.5) L u ! Φ I ( F ) / / Φ J L u ! ( F ) is an isomor phism. 3 Descent in P -ﬁbred model categor ies 105 3.2.14 W e shall say that a stable model categor y V is Q -linear if all the objects of the triangulated category Ho ( V ) are uniquely divisible. Theorem 3.2.15 Let V be a model categor y (r esp. a stable model categor y , r esp. a Q -linear stable model category), and denot e by S the model category of simplicial sets (resp. the stable model category of S 1 -spectra, r esp. the Q -linear stable model category of complexes of Q -vector spaces). Denote by 1 the unit object of the closed symmetric monoidal categor y Ho ( S ) . Then, f or each object E of Ho ( V ) , ther e exists a unique continuous morphism of derivat ors R Hom ( E , −) : Ho ( V ) / / Ho ( S ) suc h that, f or any object F of Ho ( V ) , there is a functorial bi jection Hom Ho ( S ) ( 1 , R Hom ( E , F )) ' Hom Ho ( V ) ( E , F )) . Proof N ote that the stable Q -linear case f ollo ws from the stable case and from the fact that the der iv ator of complex es of Q -vector spaces is (equivalent to) the full subderivator of the der iv ator of S 1 -spectra spanned by uniquely divisible objects. It thus remains to prov e the theorem in the case where V be a model categor y (resp. a stable model categor y) and S is the model category of simplicial sets (resp. the stable model categor y of S 1 -spectra). The exis tence of R Hom ( E , −) follo ws then from [ Cis03 , Prop. 6.13] (resp. [ CT11 , Lemma A.6]). For the unicity , as we don ’ t really need it here, we shall onl y sketch the proof (the case of simplicial sets is done in [ Cis03 , Rem. 6.14]). One uses the univ ersal property of the der iv ator Ho ( S ) : b y vir tue of [ Cis08 , Cor . 3.26] (resp. of [ CT11 , Thm. A.5]), f or an y model category (resp. stable model categor y) V 0 there is a canonical equiv alence of categor ies betw een the categor y of cocontinous mor phisms from Ho ( S ) to Ho ( V 0 ) and the homotop y categor y Ho ( V ) . As a consequence, the derivator Ho ( S ) admits a unique closed symmetric monoidal structure, and any derivator (resp. tr iangulated der ivator) is naturally and uniquel y enr iched in Ho ( S ) ; see [ Cis08 , Thm. 5.22]. More concretely , this univ ersal proper ty giv es, f or any object E in Ho ( V 0 ) , a unique cocontinuous mor phism of derivators Ho ( S ) / / Ho ( V 0 ) , K  / / K ⊗ E such that 1 ⊗ E = E . For a ﬁx ed K in Ho ( S )( I ) , this deﬁnes a cocontinuous morphism of der iv ators Ho ( V 0 ) / / Ho ( V 0 I op ) , E  / / K ⊗ E which has a right adjoint Ho ( V 0 I op ) / / Ho ( V 0 ) , F  / / F K . Let R Hom ( E , −) : Ho ( V ) / / Ho ( S ) 106 Fibred categories and the six functors formalism be a continuous mor phism such that, f or any object F of V , there is a functor ial bi jection i F : Hom Ho ( S ) ( 1 , R Hom ( E , F )) ' Hom Ho ( V ) ( E , F )) . Then, f or any object K of Ho ( S )( I ) , and any object F of Ho ( V )( I ) a canonical isomorphism R Hom ( E , F K ) ' R Hom ( E , F ) K which is completel y deter mined by being the identity f or K = 1 (this requires the full univ ersal proper ty of Ho ( S ) giv en by b y [ Cis08 , Thm. 3.24] (resp. by the dual v ersion of [ CT11 , Thm. A.5])). W e thus g et from the functor ial bijections i F the natural bi jections: Hom Ho ( S )( I ) ( K , R Hom ( E , F )) ' Hom Ho ( S ) ( 1 , R Hom ( E , F ) K ) ' Hom Ho ( S ) ( 1 , R Hom ( E , F K )) ' Hom Ho ( V ) ( E , F K ) ' Hom Ho ( V )( I ) ( K ⊗ E , F ) . In other w ords, R Hom ( E , −) has to be a r ight adjoint to (−) ⊗ E .  Remar k 3.2.16 The preceding theorem mostl y holds f or abstract der iv ators. The only problem is f or the e xistence of the morphism R Hom ( E , −) (the unicity is alwa ys clear). How ev er , this problem disappears f or derivators which hav e a Quillen model (as we ha ve seen abo ve), as w ell as f or triangulated derivators (see [ CT11 , Lemma A.6]). Hence Theorem 3.2.15 holds in fact f or any triangulated Grothendieck derivator . In the case when V is a combinatorial model categor y (which, in practice, will essentially alwa ys be the case), the enrichment ov er simplicial sets (resp, in the stable case, ov er spectra) can be constructed via Quillen functors b y Dugger’ s presentation theorems [ Dug01 ] (resp. [ Dug06 ]). Corollary 3.2.17 Let M be a P -ﬁbr ed combinatorial model category (r esp. a stable P -ﬁbr ed combinatorial model categor y , resp. a Q -linear stable P -ﬁbr ed combinatorial model category) ov er S , and S the model categor y of simplicial sets (resp. the stable model category of S 1 -spectra, r esp. the Q -linear stable model category of complexes of Q -vect or spaces). Consider an object S of S , a morphism f : X / / S , and a morphism of S - diagr ams p : ( X , I ) / / X ov er S . Then, for an object M of Ho ( M )( S ) , the follo wing conditions ar e equivalent. (a) The map R f ∗ L f ∗ ( M ) / / R f ∗ R p ∗ L p ∗ L f ∗ ( M ) is an isomorphism in Ho ( M )( S ) . (b) The map R Γ ge om ( X , M ) / / R lim o o i ∈ I op R Γ ge om ( X i , M ) 3 Descent in P -ﬁbred model categor ies 107 is an isomorphism in Ho ( M )( S ) . (c) F or any object E of Ho ( M )( S ) , the map R Hom ( E , R Γ ge om ( X , M )) / / R lim o o i ∈ I op R Hom ( E , R Γ ge om ( X i , M )) is an isomorphism in Ho ( S ) . Proof The equiv alence between (a) and (b) follo ws from Propositions 3.1.15 and 3.1.16 , which giv e the f ormula R f ∗ R p ∗ L p ∗ L f ∗ ( M ) ' R lim o o i ∈ I op R Γ ge om ( X i , M ) . The identiﬁcation Hom Ho ( S ) ( 1 , R Hom ( E , F )) ' Hom Ho ( M )( S ) ( E , F ) and the Y oneda Lemma sho w that a map in Ho ( M )( S ) is an isomorphism if and only its image by R Hom ( E , −) is an isomor phism for an y object E of Ho ( M )( S ) . Moreo v er , as R Hom ( E , −) is continuous, f or an y small categor y I and any presheaf F on I with values in M ( S ) , there is a canonical isomor phism R Hom ( E , R lim o o i ∈ I op F i )) ' R lim o o i ∈ I op R Hom ( E , F i )) . This pro v es the equivalence between conditions (b) and (c).  Corollary 3.2.18 Under the assumptions of Corollary 3.2.17 , given an object S of S , an object M of Ho ( M )( S ) satisﬁes t -descent if and only if, for any object E of Ho ( M )( S ) the pr esheaf of simplicial sets (resp. of S 1 -spectra, r esp. of complexes of Q -v ector spaces) R Hom ( E , R Γ ge om (− , M )) satisﬁes t -descent ov er S / S . Proof This follo ws from the preceding corollar y , using the f ormula giv en by Propo- sition 3.2.4 .  Remar k 3.2.19 W e need the category S to be small in some sense to apply the tw o preceding corollaries because we need to make sense of the projective model category structure of Proposition 3.2.10 . Ho w ev er , we can use these corollar ies ev en if the site S is not small as w ell: w e can either use the theor y of univ erses, or appl y these corollar ies to all the adequate small subsites of S . As a consequence, we shall f eel free to use Corollaries 3.2.17 and 3.2.18 for non necessarily small sites S , lea ving to the reader the task to av oid set-theoretic diﬃculties according to her/his taste. 108 Fibred categor ies and the six functors f or malism Deﬁnition 3.2.20 For an S 1 -spectrum E and an integ er n , w e deﬁne its n th coho- mology group H n ( E ) by the formula H n ( E ) = π − n ( E ) , where π i stands for the i th stable homotop y group functor . Let M be a monoidal P -ﬁbred stable combinatorial model category ov er S . Giv en an object S of S as well as an object M of Ho ( M )( S ) , we deﬁne the presheaf of absolute deriv ed g lobal sections of M o v er S by the f or mula R Γ (− , M ) = R Hom ( 1 S , R Γ ge om (− , M )) . For a map X / / S of S , we thus hav e the absolute cohomology of X with coeﬃcients in M , R Γ ( X , M ) , as well as the cohomology gr oups of X with coeﬃcients in M : H n ( X , M ) = H n ( R Γ ( X , M )) . W e hav e canonical isomor phisms of abelian groups H n ( X , M ) ' Hom Ho ( M )( S ) ( 1 S , R f ∗ L f ∗ ( M )) ' Hom Ho ( M )( X ) ( 1 X , L f ∗ ( M )) . Note that, if moreov er M is Q -linear , the presheaf R Γ (− , M ) can be considered as a presheaf of complex es of Q -vector spaces on S / S . 3.3 Descent o ver schemes The aim of this section is to giv e natural suﬃcient conditions f or M to satisfy descent with respect to various Grothendieck topologies 58 3.3.a Localization and Nisnevic h descent 3.3.1 Recall from example 2.1.11 that a Nisnevich distinguished square is a pullback square of schemes 58 In fact, using remark 3.2.16 , all of this section (results and proofs) holds f or an abstract algebraic prederivator in the sense of A y oub [ A yo07a , Def. 2.4.13] without any chang es (note that the results of 3.1.b are in fact a proof that (stable) combinator ial ﬁbred model categor ies o v er S giv e r ise to algebraic prederivators). The only interest of considering a ﬁbred model categor y ov er S is that it allow s f or mulating things in a little more naive wa y . Of course, the optimal setting in which to formulate descent theor y is the one of ∞ -categor ies. How ev er, restricting to presentable ∞ - categories, using Dugger ’s presentation theorem [ Dug01 ], as well as rectiﬁcation theorems such as [ Cis19 , Thm. 7.5.30 and 7.9.8] as well as those from [ Bal19 ], we can see that the case of model categories remains meaningful. 3 Descent in P -ﬁbred model categor ies 109 V l / / g   Y f   U j / / X (3.3.1.1) in which f is étale, j is an open immersion with reduced complement Z and the induced mor phism f − 1 ( Z ) / / Z is an isomorphism. For any scheme X in S , we denote b y X Nis the small Nisnevic h site of X . Theorem 3.3.2 (Morel- V oe v odsky) Let V be a (combinatorial) model cat egor y and T a scheme in S . F or a pr esheaf F on T Nis with values in V , the follo wing conditions ar e equivalent. (i) F ( ∅ ) is a terminal object in Ho ( V ) , and for any Nisnevich distinguished square ( 3.3.1.1 ) in T Nis , the squar e F ( X ) / /   F ( Y )   F ( U ) / / F ( V ) is a homotopy pullback squar e in V . (ii) The pr esheaf F satisﬁes Nisnevic h descent on T Nis . Proof By vir tue of corollaries 3.2.17 and 3.2.18 , it is suﬃcient to prov e this in the case where V is the usual model categor y of simplicial sets, in which case this is precisely Morel and V oev odsky’s theorem; see [ MV99 , V oe10b , V oe10c ].  3.3.3 Consider a Nisnevic h distinguished square ( 3.3.1.1 ) and put a = j g = f l . A ccording to our general assumption 3.0.1 , the maps a , j and f are P -mor phisms. For any object M of M ( X ) , w e obtain a commutativ e square in M (which is w ell-deﬁned as an object in the homotopy of commutative squares in M ( X ) ): L a ] a ∗ M / /   L f ] f ∗ ( M )   L j ] j ∗ ( M ) / / M . (3.3.3.1) W e also obtain another commutativ e square in M b y applying the functor R Hom X (− , 1 X ) : M / /   R f ∗ f ∗ ( M )   R j ∗ j ∗ ( M ) / / R a ∗ a ∗ ( M ) . (3.3.3.2) 110 Fibred categor ies and the six functors f or malism Proposition 3.3.4 If the category Ho ( M ) has the localization property, then for any Nisnevic h distinguished square ( 3.3.1.1 ) and any object M of Ho ( M )( X ) , the squar es ( 3.3.3.1 ) and ( 3.3.3.2 ) are homotopy cartesians. Proof Let i : Z / / X be the complement of the open immersion j ( Z being endo wed with the reduced structure) and p : f − 1 ( Z ) / / Z the map induced b y f . W e hav e onl y to prov e that one of the squares ( 3.3.3.1 ), ( 3.3.3.2 ) are cartesian. W e choose the square ( 3.3.3.1 ). Because the pair of functor ( L i ∗ , j ∗ ) is conservativ e on Ho ( M )( X ) , w e hav e only to check that the pullback of ( 3.3.3.1 ) along j ∗ or L i ∗ is homotopy car tesian. But, using the P -base chang e proper ty , we see that the image of ( 3.3.3.1 ) by j ∗ is (canonically isomor phic to) the commutative square L g ] a ∗ ( M )   L g ] a ∗ ( M )   j ∗ ( M ) j ∗ ( M ) which is obviousl y homotopy car tesian. Using again the P -base chang e proper ty , we obtain that the image of ( 3.3.3.1 ) b y L i ∗ is isomor phic in Ho ( M ) to the square 0 / / p ] p ∗ L i ∗ ( M )   0 / / L i ∗ ( M ) which is ag ain ob viously homotop y cartesian because p is an isomor phism (note f or this last reason, p ] = L p ] ) .  Corollary 3.3.5 If Ho ( M ) has the localization property then it satisﬁes Nisnevich descent. Proof This corollar y thus f ollo ws immediatel y from Corollar y 3.2.17 , Theorem 3.3.2 and Proposition 3.3.4 .  Remar k 3.3.6 Note that using Theorem 3.3.2 , if we assume only that Ho ( M ) satisﬁes Nisnevic h descent, then the squares ( 3.3.3.1 ) and ( 3.3.3.2 ) are homotopy car tesian f or an y Nisnevic h distinguished square ( 3.3.1.1 ). Assume that M is monoidal with g eometr ic sections M . Let S be a base scheme and consider a Nisnevic h distinguished square ( 3.3.1.1 ) of smooth S -schemes. Then the fact that the square ( 3.3.3.1 ) is homotopy cartesian implies there exis ts a canonical distinguished tr iangle: M S ( V ) g ∗ + l ∗ / / M S ( U ) ⊕ M S ( Y ) f ∗ + j ∗ / / M S ( X ) / / M S ( V )[ 1 ] It is called the Mayer - Vie toris triang le associated with the square ( 3.3.1.1 ). 3 Descent in P -ﬁbred model categor ies 111 3.3.b Proper base chang e isomorphism and descent by blow -ups 3.3.7 Recall from e xample 2.1.11 that a c dh -distinguished squar e is a pullback square of schemes T k / / g   Y f   Z i / / X (3.3.7.1) in which f is proper sur jectiv e, i a closed immersion and the induced map f − 1 ( X − Z ) / / X − Z is an isomorphism. Recall from Example 2.1.11 the c dh -topology is the Grothendieck topology on the category of sc hemes g enerated by Nisnevic h co v er ings and b y co verings of shape { Z / / X , Y / / X } f or any c dh -distinguished square ( 3.3.7.1 ). Theorem 3.3.8 (V oev odsky) Let V be a (combinatorial) model categor y . F or a pr esheaf F on S with values in V , the f ollowing conditions ar e equiv alent. (i) The pr esheaf F satisﬁes c dh -descent on S . (ii) The presheaf F satisﬁes Nisnevich descent and, f or any c dh -distinguished squar e ( 3.3.7.1 ) of S , the square F ( X ) / /   F ( Y )   F ( Z ) / / F ( T ) is a homotopy pullback squar e in V . Proof It is suﬃcient to prov e this in the case where V is the usual model categor y of simplicial sets; see corollaries 3.2.17 and 3.2.18 . As the distinguished c dh -squares deﬁne a bounded regular and reduced c d -structure on S , the equiv alence betw een (i) and (ii) follo ws from V oev odsky’ s theorems on descent with respect to topologies deﬁned b y c d -str uctures [ V oe10b , V oe10c ].  3.3.9 Consider a c dh -distinguished square ( 3.3.7.1 ) and put a = i g = f k . For an y object M of M ( X ) , we obtain a commutativ e square in M (which is w ell-deﬁned as an object in the homotopy of commutativ e squares in M ( X ) ): M / /   R f ∗ L f ∗ ( M )   R i ∗ L i ∗ ( M ) / / R a ∗ L a ∗ ( M ) (3.3.9.1) 112 Fibred categor ies and the six functors f ormalism Proposition 3.3.10 Assume Ho ( M ) satisﬁes the localization pr operty and the transv ersality property with r espect to proper morphisms. Then the follo wing condi- tions hold: (i) F or any c dh -distinguished squar e ( 3.3.7.1 ) , and any object M of Ho ( M )( X ) the commutativ e squar e ( 3.3.9.1 ) is homotopy cartesian. (ii) The P -ﬁbred model category Ho ( M ) satisﬁes c dh -descent. Proof W e ﬁrst pro ve (i). Consider a c dh -distinguished square ( 3.3.7.1 ) and let j : U / / X be the complement open immersion of i . As the pair of functor ( L i ∗ , j ∗ ) is conservativ e on Ho ( M )( X ) , we hav e only to chec k that the imag e of ( 3.3.9.1 ) under L i ∗ and j ∗ is homotop y car tesian. Using projective transversality , w e see that the image of ( 3.3.9.1 ) by the functor L i ∗ is (isomor phic to) the homotopy pullback square L i ∗ ( M ) / / R g ∗ L g ∗ L i ∗ ( M ) L i ∗ ( M ) / / R g ∗ L g ∗ L i ∗ ( M ) . Let h : f − 1 ( U ) / / U be the pullback of f ov er U . As j is an open immersion, it is by assumption a P -mor phism and the P -base chang e f or mula implies that the image of ( 3.3.9.1 ) by j ∗ is (isomor phic to) the commutativ e square L j ∗ ( M ) / /   R h ∗ L h ∗ L j ∗ ( M )   0 0 which is obviousl y homotopy car tesian because h is an isomor phism. W e then prov e (ii). W e already kno w that M satisﬁes Nisnevic h descent (Corollary 3.3.5 ). Thus, by virtue of the equivalence between conditions (i) and (ii) of Theorem 3.3.8 , the computation abov e, together with corollar ies 3.2.17 and 3.2.18 impl y that M satisﬁes c dh -descent.  3.3.11 T o an y c dh -distinguished square ( 3.3.7.1 ), one associates a diagram of schemes Y ov er X as follo ws. Let be the category freely generated by the or iented graph a / /   b c (3.3.11.1) Then Y is the functor from to S / X deﬁned by the follo wing diagram. 3 Descent in P -ﬁbred model categor ies 113 T k / / g   Y Z (3.3.11.2) W e then ha v e a canonical map ϕ : Y / / X , and the second asser tion of Theorem 3.3.10 can be ref or mulated b y saying that the adjunction map M / / R ϕ ∗ L ϕ ∗ ( M ) is an isomor phism f or an y object M of Ho ( M )( X ) : indeed, by vir tue of Proposition 3.1.15 , R ϕ ∗ L ϕ ∗ ( M ) is the homotopy limit of the diagram R f ∗ L f ∗ ( M )   R i ∗ L i ∗ ( M ) / / R a ∗ L a ∗ ( M ) in Ho ( M )( X ) . In other w ords, if M has the proper ties of localization and of projec- tiv e transv ersality , then the functor L ϕ ∗ : Ho ( M )( X ) / / Ho ( M )( Y , ) is fully faithful. 3.3.c Proper descent with rational coeﬃcients I: Galois ex cision From now on, we assume that any scheme in S is quasi-e x cellent 59 (in fact, w e shall onl y use the f act that the normalization of a quasi-e x cellent schemes gives r ise to a ﬁnite sur jectiv e mor phism, so that, in fact, universall y japanese schemes w ould be enough). W e ﬁx a scheme S in S , and w e shall work with S -schemes in S (assuming these f orm an essentially small categor y). 3.3.12 The h -topology (resp. the qfh -topology) is the Grothendieck topology on the category of schemes associated to the pretopology whose cov erings are the univ er - sal topological epimor phisms (resp. the quasi-ﬁnite univ ersal topological epimor - phisms). This topology has been introduced and studied b y V oev odsky in [ V oe96 ]. The h -topology is ﬁner than the c dh -topology and, of course, ﬁner than the qfh - topology . The qfh -topology is in turn ﬁner than the étale topology . An interesting f eature of the h -topology (resp. of the qfh -topology) is that any proper (resp. ﬁnite) surjective map is an h -cov er . In fact, the h -topology (resp. the qfh -topology) can be described as the topology g enerated by the Nisnevich cov erings and by the proper (resp. ﬁnite) sur jectiv e maps; see Lemma 3.3.28 (resp. Lemma 3.3.27 ) below f or a precise statement. 59 See 4.1.1 below for a reminder on quasi-ex cellent schemes. 114 Fibred categor ies and the six functors formalism 3.3.13 Consider a mor phism of schemes f : Y / / X . Consider the g roup of auto- morphisms G = Aut Y ( X ) of the X -scheme Y . Assuming X is connected, we say according to [ Gro03 , exp. V] that f is a Galois cov er if it is ﬁnite étale (thus sur jectiv e) and G operates transitiv ely and faithfull y on any (or simply one) of the geometric ﬁbers of Y / X . Then G is called the Galois gr oup of Y / X . 60 When X is not connected, we will still say that f is a Galois cov er if it is so o v er an y connected component of X . Then G will be called the Galois gr oup of X . If ( X i ) i ∈ I is the famil y connected components of X , then G is the product of the Galois groups G i of f × X X i f or each i ∈ I . The g roup G i is equal to the Galois group of an y residual extension ov er a generic point of X i . The f ollo wing deﬁnition is an extension of the deﬁnition 5.5 of [ SV00b ]: Deﬁnition 3.3.14 A pseudo-Galois cov er is a ﬁnite surjective mor phism of schemes f : Y / / X whic h can be factored as Y f 0 / / X 0 p / / X where f 0 is a Galois cov er and p is radicial 61 (such a p is automatically ﬁnite and surjective). Note that the group G deﬁned by the Galois co v er f 0 is independent of the choice of the factorization. In fact, if ¯ X denotes the semi-localization of X at its g eneric points, considering the car tesian squares ¯ Y / /   ¯ X 0 / /   ¯ X   Y f 0 / / X 0 p / / X then G = Aut ¯ X ( ¯ Y ) – for any point y ∈ ¯ Y , x 0 = f 0 ( y ) , x = f ( y ) , κ x 0 / κ x is the maximal radicial sub-extension of the nor mal extension κ y / κ x . It will be called the Galois gr oup of Y / X . Remark also that Y is a G -torsor ov er X locall y f or the qfh -topology (i.e. it is a Galois object of group G in the qfh -topos of X ): this comes from the fact that ﬁnite radicial epimor phisms are isomor phisms locally f or the qfh -topology (an y universal homeomorphism has this proper ty by [ V oe96 , prop. 3.2.5]). Let f : Y / / X be a ﬁnite mor phism, and G a ﬁnite group acting on Y o v er X . Note that, as Y is aﬃne on X , the scheme theoretic quotient Y / G exis ts; see [ Gro03 , Exp. V , Cor . 1.8]. Such scheme-theoretic quotients are stable b y ﬂat pullbacks; see [ Gro03 , Exp. V , Prop. 1.9]. 60 The map f induces a one to one cor respondence betw een the generic points of Y and that of X . For an y generic point y ∈ Y , x = f ( y ) , the residual e xtension κ y / κ x is a Galois extension with Galois group G . 61 See 2.1.6 f or a reminder on radicial morphisms. 3 Descent in P -ﬁbred model categor ies 115 Deﬁnition 3.3.15 Let G be ﬁnite group. A qfh -distinguished square of gr oup G is a pullback square of S -schemes of shape T h / / g   Y f   Z i / / X (3.3.15.1) in which Y is endo wed with an action of G ov er X , and satisfying the follo wing three conditions. (a) The mor phism f is ﬁnite and surjective. (b) The induced mor phism f − 1 ( X − Z ) / / f − 1 ( X − Z )/ G is ﬂat. (c) The mor phism f − 1 ( X − Z )/ G / / X − Z is radicial. Immediate examples of qfh -distinguished squares of tr ivial group are the follo w- ing. The scheme Y might be the nor malization of X , and Z is a no where dense closed subscheme out of which f is an isomorphism; or Y is dense open subscheme of X which is the disjoint union of its ir reducible components; or Y is a closed subscheme of X inducing an isomorphism Y r ed ' X r ed . A qfh -distinguished square of group G ( 3.3.15.1 ) will be said to be pseudo-Galois if Z is nowhere dense in X and if the map f − 1 ( X − Z ) / / X − Z is a pseudo-Galois co v er of group G . The main examples of pseudo-Galois qfh -distinguished squares will come from the f ollo wing situation. Proposition 3.3.16 Consider an irreducible normal scheme X , and a ﬁnite extension L of its ﬁeld of functions k ( X ) . Let K be the inseparable closure of k ( X ) in L , and assume that L / K is a Galois extension of group G . Denote by Y the nor malization of X in L . Then the action of G on k ( Y ) = L ext ends natur ally to an action on Y ov er X . F urthermore, ther e exists a closed subscheme Z of X , suc h that the pullbac k squar e T / /   Y f   Z i / / X is a pseudo-Galois qfh -distinguished square of gr oup G . Proof The action of G on L e xtends naturally to an action on Y o v er X by functor iality . Further more, Y / G is the nor malization of X in K , so that Y / G / / X is ﬁnite radicial and sur jectiv e (see [ V oe96 , Lemma 3.1.7] or [ Bou98 , V , §2, n º 3, lem. 4]). By construction, Y is generically a Galois co v er ov er Y / G , which implies the result (see [ GD67 , Cor . 18.2.4]).  3.3.17 For a giv en S -scheme T , we shall denote b y L ( T ) the cor responding repre- sentable qfh -sheaf of sets (remember that the qfh -topology is not subcanonical, so 116 Fibred categories and the six functors formalism that L ( T ) has to be distinguished from T itself ). Bew are that, in general, there is no reason that, giv en a ﬁnite group G acting on T , the scheme-theoretic quotient L ( T / G ) (whenev er deﬁned) and the qfh -sheaf-theoretic quotient L ( T )/ G w ould coincide. Lemma 3.3.18 Let f : Y / / X be a separ ated mor phism, G a ﬁnite gr oup acting on Y ov er X , and Z a closed subscheme of X such t hat f is ﬁnite and surjective ov er X − Z , and suc h that the quotient map f − 1 ( X − Z ) / / f − 1 ( X − Z )/ G is ﬂat, while the map f − 1 ( X − Z )/ G / / X − Z is radicial. F or g ∈ G , write g : Y / / Y f or the corr esponding automorphism of Y , and deﬁne Y g as the imag e of the diagonal Y / / Y × X Y composed with the automorphism 1 Y × X g : Y × X Y / / Y × X Y . Then, if T = Z × X Y , we g et a qfh -cov er of Y × X Y by closed subschemes: Y × X Y = ( T × Z T ) ∪ Ø g ∈ G Y g . Proof Note that, as f is separated, the diagonal Y / / Y × X Y is a closed embedding, so that the Y g ’ s are closed subschemes of Y × X Y . As the map Y × Y / G Y / / Y × X Y is a univ ersal homeomor phism, w e ma y assume that Y / G = X . It is suﬃcient to pro v e that, if y and y 0 are tw o geometric points of Y whose images coincide in X and do not belong to Z , there e xists an element g of G such that y 0 = g y (which means that the pair ( y, y 0 ) belongs to Y g ). For this pur pose, we may assume, without loss of generality , that Z = ∅ . Then, b y assumption, Y is ﬂat ov er X , from which we g et the identiﬁcation ( Y × X Y )/ G ' Y × X ( Y / G ) ' Y (where the action of G on Y × X Y is tr ivial on the ﬁrst factor and is induced b y the action on Y on the second factor). This achie v es the proof.  Proposition 3.3.19 F or any qfh -distinguished square of gr oup G ( 3.3.15.1 ) , the com- mutativ e squar e L ( T )/ G / /   L ( Y )/ G   L ( Z ) / / L ( X ) is a pullback and a pushout in the category of qfh -sheaves. Moreo v er , if X is nor mal and if Z is nowher e dense in X , then the canonical map L ( Y )/ G / / L ( Y / G ) ' L ( X ) is an isomor phism of qfh -sheav es (which implies that L ( T )/ G / / L ( Z ) is an isomorphism as well). Proof Note that this commutative square is a pullback because it was so before taking the quotients by G (as colimits are universal in any topos). As f is a qfh -co v er , it is suﬃcient to pro v e that L ( T ) × L ( Z ) L ( T )/ G / /   L ( Y ) × L ( X ) L ( Y )/ G   L ( T ) / / L ( Y ) 3 Descent in P -ﬁbred model categor ies 117 is a pushout square. This latter square ﬁts into the follo wing commutative diagram L ( T ) / /   L ( Y )   L ( T ) × L ( Z ) L ( T )/ G / /   L ( Y ) × L ( X ) L ( Y )/ G   L ( T ) / / L ( Y ) in which the tw o vertical composed maps are identities (the vertical maps of the upper commutative square are obtained from the diagonals by taking the quotients under the natural action of G on the right component). It is thus suﬃcient to prov e that the upper square is a pushout. As the lo wer square is a pullbac k, the upper one shares the same property; moreo v er , all the maps in the upper commutative square are monomor phisms of qfh -sheav es, so that it is suﬃcient to prov e that the map ( L ( T ) × L ( Z ) L ( T )/ G ) q L ( Y ) / / L ( Y ) × L ( X ) L ( Y )/ G is an epimor phism of qfh - shea v es. According to Lemma 3.3.18 , this follo ws from the commutativity of the diagram L ( T × Z T ) q  Ý g ∈ G L ( Y g )  / /   L ( Y × X Y )   ( L ( T ) × L ( Z ) L ( T )/ G ) q L ( Y ) / / L ( Y ) × L ( X ) L ( Y )/ G in which the v er tical maps are obviousl y epimor phic. Assume no w that X is nor mal and that Z is no where dense in X , and let us prov e that the canonical map L ( Y )/ G / / L ( X ) is an isomor phism of qfh -shea v es. This is equiv alent to pro ve that, for an y qfh -sheaf of sets F , the map f ∗ : F ( X ) / / F ( Y ) induces a bi jection F ( X ) ' F ( Y ) G . Let F be a qfh -sheaf. The map f ∗ : F ( X ) / / F ( Y ) is injective because f is a qfh -co ver , and it is clear that the image of f ∗ lies in F ( Y ) G . Let a be a section of F o v er Y which is inv ar iant under the action of G . Denote by pr 1 , pr 2 : Y × X Y / / Y the tw o canonical projections. With the notations introduced in Lemma 3.3.18 , we hav e pr ∗ 1 ( a ) | Y g = a = a . g = pr ∗ 2 ( a ) | Y g f or ev ery element g in G . As Z does not contain an y generic point of X , the scheme T × Z T does not contain any g eneric point of Y × X Y neither: as any irreducible component of Y dominates an ir reducible component of X , and, as X is nor mal, the ﬁnite map Y / / X is universall y open; in par ticular , the projection pr 1 : Y × X Y / / Y 118 Fibred categor ies and the six functors f or malism is universall y open, which implies that any generic point of Y × X Y lies ov er a generic point of Y . By vir tue of [ V oe96 , prop. 3.1.4], Lemma 3.3.18 thus giv es a qfh -co v er of Y × X Y by closed subschemes of shape Y × X Y = Ø g ∈ G Y g . This implies that pr ∗ 1 ( a ) = pr ∗ 2 ( a ) . The mor phism Y / / X being a qfh -cov er and F a qfh -sheaf, w e deduce that the section a lies in the image of f ∗ .  Corollary 3.3.20 F or any qfh -distinguished squar e of group G ( 3.3.15.1 ) , we g et a bicartesian squar e of qfh -sheaves of abelian gr oups Z qfh ( T ) G / /   Z qfh ( Y ) G   Z qfh ( Z ) / / Z qfh ( X ) (wher e the subscript G stands for the coinvariants under the action of G ). In ot her w ords, ther e is a canonical short exact sequence of sheaves of abelian gr oups 0 / / Z qfh ( T ) G / / Z qfh ( Z ) ⊕ Z qfh ( Y ) G / / Z qfh ( X ) / / 0 . Proof As the abelianization functor preserves colimits and monomorphisms, the preceding proposition implies f or mally that w e hav e a shor t e xact sequence of shape Z qfh ( T ) G / / Z qfh ( Z ) ⊕ Z qfh ( Y ) G / / Z qfh ( X ) / / 0 , while the left e xactness f ollo ws from the fact that Z / / X being a monomor phism, the map obtained by pullback, L ( T )/ G / / L ( Y )/ G , is a monomor phism as w ell.  3.3.21 Let V be a Q -linear stable model category (see 3.2.14 ). Consider a ﬁnite g roup G , and an object E of V , endow ed with an action of G . By vie wing G as a category with one object we can see E as functor from G to V and take its homotopy limit in Ho ( V ) , which we denote by E h G (in the literature, E h G is called the object of homotopy ﬁxed points under the action of G on E ). One the other hand, the categor y Ho ( V ) is, by assumption, a Q -linear tr iangulated category with small sums, and, in par ticular , a Q -linear pseudo-abelian category so that we can deﬁne E G as the object of Ho ( V ) deﬁned b y (3.3.21.1) E G = Im p , where p : E / / E is the projector deﬁned in Ho ( V ) by the f or mula 3 Descent in P -ﬁbred model categor ies 119 (3.3.21.2) p ( x ) = 1 # G Õ g ∈ G g . x . The inclusion E G / / E induces a canonical isomor phism (3.3.21.3) E G ∼ / / E h G in Ho ( V ) : to see this, by virtue of Theorem 3.2.15 , we can assume that V is the model category of comple xes of Q -vector spaces, in which case it is obvious. Corollary 3.3.22 Let C be a presheaf of complexes of Q -v ector spaces on the category of S -sc hemes. Then, for any qfh -distinguished squar e of gr oup G ( 3.3.15.1 ) , the commutativ e squar e R Γ qfh ( X , C qfh ) / /   R Γ qfh ( Y , C qfh ) G   R Γ qfh ( Z , C qfh ) / / R Γ qfh ( T , C qfh ) G is a homot opy pullback squar e in the derived categor y of Q -v ector spaces. In par - ticular , we g et a long exact sequence of shape · · · / / H n qfh ( X , C qfh ) / / H n qfh ( Z , C qfh ) ⊕ H n qfh ( Y , C qfh ) G / / H n qfh ( T , C qfh ) G / / · · · If furthermore X is nor mal and Z is nowhere dense in X , then the maps H n qfh ( X , C qfh ) / / H n qfh ( Y , C qfh ) G and H n qfh ( Z , C qfh ) / / H n qfh ( T , C qfh ) G ar e isomorphisms f or any integ er n . Proof Let C qfh / / C 0 be a ﬁbrant resolution in the qfh -local injectiv e model cate- gory structure on the categor y of qfh -sheav es of comple xes of Q -vector spaces; see f or instance [ A yo07a , Cor . 4.4.42]. Then f or U = Y , T , we hav e a natural isomor phism of comple xes Hom ( Q qfh ( U ) G , C 0 ) = C 0 ( U ) G which gives an isomor phism R Hom ( Q qfh ( U ) G , C qfh ) ' R Γ qfh ( U , C qfh ) G in the deriv ed categor y of the abelian category of Q -v ector spaces. This corollary thus f ollo ws f or mall y from Corollar y 3.3.20 by ev aluating at the derived functor R Hom (− , C qfh ) . If fur thermore X is normal, then one deduces the isomor phism H n qfh ( X , C qfh ) ' H n qfh ( Y , C qfh ) G from the fact that L ( Y )/ G ' L ( Y / G ) ' X (Proposition 3.3.19 ), which implies that Z qfh ( Y ) G ' Z qfh ( X ) . The isomor phism H n qfh ( Z , C qfh ) ' H n qfh ( T , C qfh ) G then comes as a byproduct of the long ex act sequence abo v e.  120 Fibred categor ies and the six functors f ormalism Theorem 3.3.23 Let X be a scheme, and C be a presheaf of complexes of Q -vect or spaces on the small étale site of X . Then C satisﬁes étale descent if and only if it has the f ollowing properties. (a) The complex C satisﬁes Nisnevich descent. (b) F or any étale X -scheme U and any Galois cov er V / / U of group G , the map C ( U ) / / C ( V ) G is a quasi-isomorphism. Proof These are cer tainl y necessar y conditions. T o prov e that they are suﬃcient, note that the Nisne vich cohomological dimension and the rational étale cohomological dimension of a noether ian sc heme are bounded by the dimension; see [ MV99 , proposition 1.8, page 98] and [ V oe96 , Lemma 3.4.7]. By virtue of [ SV00a , Theorem 0.3], f or τ = Nis , ´ e t , we hav e strongly conv erg ent spectral sequences E p , q 2 = H p τ ( U , H q ( C ) τ ) ⇒ H p + q τ ( U , C τ ) . Condition (a) gives isomor phisms H p + q ( C ( U )) ' H p + q Nis ( U , C Nis ) , so that it is suﬃ- cient to pro v e that, for each of the cohomology presheav es F = H q ( C ) , w e hav e H p Nis ( U , F Nis ) ' H p ´ e t ( U , F ´ e t ) . As the rational étale cohomology of any henselian scheme is trivial in non-zero degrees, it is suﬃcient to prov e that, f or any local henselian scheme U (obtained as the henselisation of an étale X -scheme at some point), F Nis ( U ) ' F ´ e t ( U ) . Let G be the absolute Galois group of the closed point of U . Then w e hav e F Nis ( U ) = F ( U ) and F ´ e t ( U ) = lim / / α F ( U α ) G α , where the U α ’ s run ov er all the Galois co vers of U cor responding to the ﬁnite quotients G / / G α . But it f ollow s from (b) that F ( U ) ' F ( U α ) G α f or any α , so that F Nis ( U ) ' F ´ e t ( U ) .  Lemma 3.3.24 Any qfh -cov er admits a r eﬁnement of the form Z / / Y / / X , where Z / / Y is a ﬁnite surjective mor phism, and Y / / X is an étale cov er . Proof This proper ty being clearl y local on X with respect to the étale topology , w e can assume that X is str ictl y henselian, in which case this f ollow s from [ V oe96 , Lemma 3.4.2].  Theorem 3.3.25 A pr esheaf of complexes of Q -v ector spaces C on the categor y of S -sc hemes satisﬁes qfh -descent if and only if it has the follo wing two properties: (a) the complex C satisﬁes Nisnevich descent ; (b) f or any pseudo-Galois qfh -distinguished squar e of group G ( 3.3.15.1 ) , the com- mutativ e squar e 3 Descent in P -ﬁbred model categor ies 121 C ( X ) / /   C ( Y ) G   C ( Z ) / / C ( T ) G is a homotopy pullback squar e in the derived category of Q -vector spaces. Proof An y complex of presheav es of Q -v ector spaces satisfying qfh -descent satisﬁes properties (a) and (b): proper ty (a) f ollow s from the fact that the qfh -topology is ﬁner than the étale topology; proper ty (b) is Corollary 3.3.22 . Assume no w that C satisﬁes these two proper ties. Let ϕ : C / / C 0 be a mor phism of presheav es of complex es of Q -vector spaces which is a quasi-isomorphism locally f or the qfh -topology , and such that C 0 satisﬁes qfh -descent (such a mor phism e xists thanks to the qfh -local model categor y structure on the categor y of presheav es of comple x es of Q -vector spaces; see Proposition 3.2.10 ). Then the cone of ϕ also satisﬁes conditions (a) and (b). Hence it is suﬃcient to pro v e the theorem in the case where C is acyclic locall y f or the qfh -topology . Assume from now on that C qfh is an acyclic complex of qfh -sheav es, and denote b y H n ( C ) the n th cohomology presheaf associated to C . W e know that the associated qfh -sheav es vanish, and w e want to deduce that H n ( C ) = 0 . W e shall pro v e by induction on d that, for any S -scheme X of dimension d and f or any integer n , the group H n ( C )( X ) = H n ( C ( X )) vanishes. The case where d < 0 f ollow s from the fact, that b y (a), the preshea ves H n ( C ) send ﬁnite sums to ﬁnite direct sums, so that, in par ticular , H n ( C )( ∅ ) = 0 . Before going fur ther , notice that condition (b) implies H n ( C )( X r ed ) = H n ( C )( X ) f or any S -scheme X (consider the case where, in the diag ram ( 3.3.15.1 ), Z = Y = T = X r ed ), so that it is alwa ys har mless to replace X by its reduction. Assume no w that d ≥ 0 , and that the vanishing of H n ( C )( X ) is kno wn whene ver X is of dimension < d and f or an y integ er n . U nder this inductiv e assumption, we hav e the f ollowing reduction principle. Consider a pseudo-Galois qfh -distinguished square of group G ( 3.3.15.1 ). If Z and T are of dimension < d , then by condition (b), the map H n ( C )( X ) / / H n ( C )( Y ) G is an isomor phism: indeed, we hav e an ex act sequence of shape H n − 1 ( C )( T ) G / / H n ( C )( X ) / / H n ( C )( Z ) ⊕ H n ( C )( Y ) G / / H n ( C )( T ) G , which implies our asser tion b y induction on d . W e shall prov e no w the vanishing of H n ( C )( T ) f or nor mal S -schemes T of dimen- sion d . Let a be a section of H n ( C ) o v er such a T . As H n ( C ) qfh ( T ) = 0 , there exis ts a qfh -cov er g : Y / / T such that g ∗ ( a ) = 0 . But, b y vir tue of Lemma 3.3.24 , we can assume g is the composition of a ﬁnite surjective mor phism f : Y / / X and of an étale cov er e : X / / T . W e claim that e ∗ ( a ) = 0 . T o prov e it, as, b y (a), the presheaf H n ( C ) sends ﬁnite sums to ﬁnite direct sums, we can assume that X is normal and connected. R eﬁning f further , w e can assume that Y is the nor malization of X in a ﬁnite e xtension of k ( X ) , and that k ( Y ) is a Galois extension of group G o v er the inseparable closure of k ( X ) in k ( Y ) . By vir tue of Proposition 3.3.16 , we get b y the reduction pr inciple the identiﬁcation H n ( C )( X ) = H n ( C )( Y ) G , whence e ∗ ( a ) = 0 . 122 Fibred categories and the six functors formalism As a consequence, the restriction of the presheaf of comple xes C to the categor y of normal S -schemes of dimension ≤ d is acyclic locally f or the étale topology (note that this is quite meaningful, as any étale scheme o v er a normal scheme is nor mal; see [ GD67 , Prop. 18.10.7]). But C satisﬁes étale descent (by vir tue of Theorem 3.3.23 this follo ws f or mally from proper ty (a) and from proper ty (b) f or Z = ∅ ), so that H n ( C )( T ) = H n ´ e t ( T , C ´ e t ) = 0 f or an y nor mal S -scheme T of dimension ≤ d and any integer n . Consider now a reduced S -scheme X of dimension ≤ d . Let p : T / / X be the normalization of X . As p is birational (see [ GD61 , Cor . 6.3.8]) and ﬁnite surjective (because X is quasi-e x cellent), w e can apply the reduction pr inciple and see that the pullback map p ∗ : H n ( C )( X ) / / H n ( C )( T ) = 0 is an isomor phism for any integer n , which achie ves the induction and the proof.  Lemma 3.3.26 Étale cov erings are ﬁnite étale cov erings locally f or the Nisnevich topology : any étale cover admits a reﬁnement of the form Z / / Y / / X , wher e Z / / Y is a ﬁnite étale cov er and Y / / X is a Nisnevic h cov er . Proof This proper ty being local on X f or the Nisnevic h topology , it is suﬃcient to pro v e this in the case where X is local henselian. Then, by vir tue of [ GD67 , Cor . 18.5.12 and Prop. 18.5.15], w e can e v en assume that X is the spectrum of ﬁeld, in which case this is obvious.  Lemma 3.3.27 Any qfh -cov er admits a r eﬁnement of the form Z / / Y / / X , where Z / / Y is a ﬁnite surjective mor phism, and Y / / X is a Nisnevic h cov er . Proof As ﬁn ite sur jectiv e morphisms are stable b y pullbac k and composition, this f ollow s immediately from lemmata 3.3.24 and 3.3.26 .  Lemma 3.3.28 Any h -cov er of an integ ral scheme X admits a reﬁnement of the form U / / Z / / Y / / X , wher e U / / Z is a ﬁnite surjective mor phism, Z / / Y is a Nisnevic h cov er , Y / / X is a proper surjective bir ational map, and Y is nor mal. Proof By vir tue of [ V oe96 , Theorem 3.1.9], any h -cov er admits a reﬁnement of shape W / / V / / X , where W / / V is a qfh -cov er , and V / / X is a proper surjective birational map. By replacing V by its nor malization Y , w e get a reﬁnement of shape W × V Y / / Y / / X where W × V Y / / Y is a qfh -cov er , and Y / / X is proper sur jectiv e birational map. W e conclude b y Lemma 3.3.27 .  Lemma 3.3.29 Let C be a pr esheaf of complexes of Q -vector spaces on the category of S -sc hemes satisfying qfh -descent. Then, for any ﬁnite surjectiv e mor phism f : Y / / X with X normal, the map f ∗ : H n ( C )( X ) / / H n ( C )( Y ) is a monomorphism. 3 Descent in P -ﬁbred model categor ies 123 Proof It is clearl y suﬃcient to pro v e this when X is connected. Then, up to reﬁnement, w e can assume that f is a map as in Proposition 3.3.16 . In this case, by vir tue of Corollary 3.3.22 , the Q -vector space H n ( C )( X ) ' H n ( C )( Y ) G is a direct f actor of H n ( C )( Y ) .  Theorem 3.3.30 A pr esheaf of complexes of Q -vector spaces on the categor y of S -sc hemes satisﬁes h -descent if and only if it satisﬁes qfh -descent and c dh -descent. Proof This is cer tainl y a necessar y condition, as the h -topology is ﬁner than the qfh -topology and the c dh -topology . For the con verse, as in the proof of Theorem 3.3.25 , it is suﬃcient to prov e that any presheaf of complex es of Q -vector spaces C on the category of S -schemes satisfying qfh -descent and c dh -descent, and which is acy clic locally for the h -topology , is acy clic. W e shall pro v e by noether ian induction that, given such a complex C , f or any integer n , and any S -scheme X , for any section a of H n ( C ) o ver X , there e xists a c dh -cov er X 0 / / X on which a vanishes. In other w ords, w e shall get that C is acyclic locally for the c dh -topology , and, as C satisﬁes c dh -descent, this will imply that H n ( C )( X ) = H n c dh ( X , C c dh ) = 0 f or any integer n and any S -scheme X . Note that the preshea v es H n ( C ) send ﬁnite sums to ﬁnite direct sums (which f ollo ws, for instance, from the fact that C satisﬁes Nisnevich descent). In par ticular , H n ( C )( ∅ ) = 0 f or any integer n . Let X be an S -scheme, and a ∈ H n ( C )( X ) . W e hav e a c dh -co v er of X of shape X 0 q X 00 / / X , where X 0 is the sum of the irreducible components of X r ed and X 00 is a nowhere dense closed subscheme of X , so that we can assume X is integ ral. Let a be a section of the presheaf H n ( C ) o v er X . As H n ( C ) h = 0 , b y vir tue of Lemma 3.3.28 , there e xists a proper sur jectiv e birational map p : Y / / X with Y normal, a Nisnevic h cov er q : Z / / Y , and a sur jectiv e ﬁnite mor phism r : U / / Z suc h that r ∗ ( q ∗ ( p ∗ ( a ))) = 0 in H n ( C )( U ) . But then, Z is nor mal as w ell (see [ GD67 , Prop. 18.10.7]), so that, by Lemma 3.3.29 , we hav e q ∗ ( p ∗ ( a )) = 0 in H n ( C )( Z ) . Let T be a no where dense closed subscheme of X suc h that p is an isomor phism o v er X − T . By noether ian induction, there exis ts a c dh -co ver T 0 / / T such that a | T 0 vanishes. Hence the section a vanishes on the c dh -co v er T 0 q Z / / X .  3.3.d Proper descent with rational coeﬃcients II: separation From no w on, we assume that Ho ( M ) is Q -linear . Proposition 3.3.31 Let f : Y / / X be a morphism of schemes in S , and G a ﬁnite gr oup acting on Y ov er X . Denote by Y the scheme Y considered a functor from G to the categor y of S -sc hemes, and denote by ϕ : ( Y , G ) / / X the mor phism induced by f . Then, for any object M of Ho ( M )( X ) , ther e are canonical isomor phisms ( R f ∗ L f ∗ ( M )) G ' ( R f ∗ L f ∗ ( M )) h G ' R ϕ ∗ L ϕ ∗ ( M ) . (wher e G acts by functoriality of the construction R f ∗ L f ∗ , as expressed by formulas ( 3.2.11.3 ) and ( 3.2.11.4 ) ). 124 Fibred categor ies and the six functors formalism Proof The second isomor phism comes from Proposition 3.1.15 , and the ﬁrst, from ( 3.3.21.3 ).  Theorem 3.3.32 If Ho ( M ) satisﬁes Nisnevich descent, the f ollowing conditions are equiv alent : (i) Ho ( M ) satisﬁes étale descent. (ii) f or any ﬁnite étale co ver f : Y / / X , the functor L f ∗ : Ho ( M )( X ) / / Ho ( M )( Y ) is conser v ative; (iii) f or any ﬁnite Galois cov er f : Y / / X of gr oup G , and f or any object M of Ho ( M )( X ) , the canonical map M / / ( R f ∗ L f ∗ ( M )) G is an isomorphism. Proof The equiv alence between (i) and (iii) f ollo ws from Theorem 3.3.23 by corol- laries 3.2.17 and 3.2.18 , and Proposition 3.2.8 show s that (i) implies (ii). It is thus suﬃcient to pro v e that (ii) implies (iii). Let f : Y / / X be a ﬁnite Galois co v er of group G . As the functor f ∗ = L f ∗ is conser v ativ e by assumption, it is suﬃcient to chec k that the map M / / ( R f ∗ L f ∗ ( M )) G becomes an isomor phism after applying f ∗ . By vir tue of Proposition 3.1.17 , this just means that it is suﬃcient to prov e (iii) when f has a section, i.e. when Y is isomor phic to the tr ivial G -torsor ov er X . In this case, we hav e the (equiv ar iant) identiﬁcation É g ∈ G M ' R f ∗ L f ∗ ( M ) , where G acts on the left ter m by per muting the factors. Hence M ' ( R f ∗ L f ∗ ( M )) G .  Proposition 3.3.33 Assume that Ho ( M ) has the localization property. The f ollow- ing conditions are equivalent : (i) Ho ( M ) is separat ed. (ii) Ho ( M ) is semi-separat ed and satisﬁes étale descent. Proof This f ollo ws from Proposition 2.3.9 and Theorem 3.3.32 .  Corollary 3.3.34 Assume that all the r esidue ﬁelds of S ar e of char acteristic zer o, and that Ho ( M ) has the pr operty of localization. Then the follo wing conditions ar e equiv alent : (i) Ho ( M ) is separat ed. (ii) Ho ( M ) satisﬁes étale descent. Proof Consider a radicial ﬁnite surjectiv e mor phism f : Y / / X in S . T o pro v e that the functor L f ∗ is conser vativ e, as Ho ( M ) has the proper ty of localization, by noetherian induction, we ma y replace X b y an y dense open subscheme U (and Y by U × X Y ). The residue ﬁelds of X being of characteristic zero, this means that w e ma y assume that f induces an isomor phism after reduction Y r ed ' X r ed . But it is 3 Descent in P -ﬁbred model categor ies 125 clear that, b y the localization proper ty , such a mor phism f induces an equivalence of categories L f ∗ , so that Ho ( M ) is automatically semi-separated. W e conclude by Proposition 3.3.33 .  Proposition 3.3.35 Assume that Ho ( M ) is separat ed, satisﬁes the localization property t he proper transv ersality pr oper ty . Then, for any pseudo-Galois cov er f : Y / / X of group G , and for any object M of Ho ( M )( X ) , the canonical map M / / ( R f ∗ L f ∗ ( M )) G is an isomorphism. Proof By Proposition 3.3.33 , this is an easy consequence of Proposition 2.1.9 and of condition (iii) of Theorem 3.3.32 .  3.3.36 From no w on, we assume fur thermore that an y scheme in S is quasi- e x cellent. Theorem 3.3.37 Assume that Ho ( M ) satisﬁes the localization and proper transv er - sality properties. Then the following conditions are equiv alent: (i) Ho ( M ) is separat ed; (ii) Ho ( M ) satisﬁes h -descent; (iii) Ho ( M ) satisﬁes qfh -descent; (iv) f or any qfh -distinguished squar e ( 3.3.15.1 ) of gr oup G , if w e write a = f h = i g : T / / X f or the composed map, then, for any object M of Ho ( M )( X ) , the commutativ e squar e M / /   ( R f ∗ L f ∗ ( M )) G   R i ∗ L i ∗ ( M ) / / ( R a ∗ L a ∗ ( M )) G (3.3.37.1) is homotopy car tesian; (v) the same as condition (iv), but only for pseudo-Galois qfh -distinguished squar es. Proof As M satisﬁes c dh -descent (Theorem 3.3.10 ), the equiv alence between con- ditions (ii) and (iii) f ollow s from Theorem 3.3.30 by Corollar y 3.2.18 . Similarl y , Theorem 3.3.25 and corollar ies 3.3.22 , 3.2.17 and 3.2.18 show that conditions (iii), (iv) and (v) are equivalent. As étale sur jectiv e mor phisms as well as ﬁnite radicial epimorphisms are qfh -cov erings, it f ollow s from Proposition 3.2.8 , Theorem 3.3.32 and Proposition 3.3.33 , that condition (iii) implies condition (i). It thus remains to pro v e that condition (i) implies condition (v). So let us consider a pseudo-Galois qfh -distinguished square ( 3.3.15.1 ) of group G , and prov e that ( 3.3.37.1 ) is homotop y cartesian. Using proper transv ersality , w e see that the image of ( 3.3.37.1 ) by the functor L i ∗ is (isomor phic to) the homotopy pullback square 126 Fibred categories and the six functors formalism L i ∗ ( M ) / / ( R g ∗ L g ∗ L i ∗ ( M )) G L i ∗ ( M ) / / ( R g ∗ L g ∗ L i ∗ ( M )) G . W r ite j : U / / X for the complement open immersion of i , and b : f − 1 ( U ) / / U f or the map induced b y f . As j is étale, w e see, using Proposition 3.1.17 , that the image of ( 3.3.9.1 ) by j ∗ = L j ∗ is (isomor phic to) the square j ∗ ( M ) / /   ( R b ∗ L b ∗ j ∗ ( M )) G   0 0 . in which the upper horizontal map is an isomor phism by Proposition 3.3.35 . Hence it is a homotopy pullback square. Thus, because the pair of functors ( L i ∗ , j ∗ ) is conservativ e on Ho ( M )( X ) , the square ( 3.3.37.1 ) is homotopy cartesian.  Corollary 3.3.38 Assume that all the r esidue ﬁelds of S ar e of c haract eristic zero, and that Ho ( M ) has the localization and proper transv ersality properties. Then Ho ( M ) satisﬁes h -descent if and only if it satisﬁes étale descent. Proof This f ollo ws from Corollar y 3.3.34 and Theorem 3.3.37 .  Corollary 3.3.39 Assume that Ho ( M ) is separated and has the localization and proper transv ersality properties. Let f : Y / / X be a ﬁnite surjective morphism, with X normal, and G a gr oup acting on Y ov er X , such that the map Y / G / / X is g enerically r adicial (i.e. radicial ov er a dense open subscheme of X ). Consider at last a pullbac k squar e of the follo wing shape. Y 0 / / f 0   Y f   X 0 / / X Then, f or any object M of Ho ( M )( X 0 ) , the natur al map M / / ( R f 0 ∗ L f 0∗ ( M )) G is an isomorphism. Proof For an y presheaf C of complex es of Q -v ector spaces on S / X , one has an isomorphism R Γ qfh ( X 0 , C qfh ) ' R Γ qfh ( Y 0 , C qfh ) G . This f ollo ws from the fact that we ha ve an isomor phism of qfh -sheav es of sets L ( Y )/ G ' L ( X ) (the map Y / / Y / G being g ener icall y ﬂat, this is Proposition 3 Descent in P -ﬁbred model categor ies 127 3.3.19 ), which implies that the map L ( Y 0 )/ G / / L ( X 0 ) is an isomor phism of qfh - shea v es (by the univ ersality of colimits in topoi), and implies this asser tion (as in the proof of 3.3.22 ). By virtue of Theorem 3.3.37 , Ho ( M ) satisﬁes qfh -descent, so that the preceding computations imply the result by corollaries 3.2.17 and 3.2.18 .  Corollary 3.3.40 Assume that Ho ( M ) is separated and has the localization and proper transv ersality pr operties. Then for any ﬁnite surjective mor phism f : Y / / X with X normal, the morphism M / / R f ∗ L f ∗ ( M ) is a monomor phism and admits a functorial splitting in Ho ( M )( X ) . F ur thermor e, this r emains true af ter base c hang e by any map X 0 / / X . Proof It is suﬃcient to treat the case where X is connected. W e ma y replace Y b y a normalization of X in a suitable ﬁnite extension of its ﬁeld of function, and assume that a ﬁnite group G acts on Y o v er X , so that the proper ties descr ibed in the preceding corollary are fulﬁlled (see 3.3.16 ).  Remar k 3.3.41 The condition (iv) of Theorem 3.3.37 can be reformulated in a more global w ay as f ollo ws (this won ’ t be used in these notes, but this might be useful f or the reader who might want to formulate all this in ter ms of (pre-)algebraic der iv ators [ A y o07a , Def. 2.4.13]). Giv en a qfh -distinguished square ( 3.3.15.1 ) of group G , w e can f or m a functor F from category I = ( 3.3.11.1 ) to the category of diagrams of S -schemes cor responding to the diag ram of diagrams of S -schemes ( T , G ) ( h , 1 G ) / / g   ( Y , G ) Z in which T and Y cor respond to T anf Y respectiv ely , seen as functor from G to S / X . The construction of 3.1.22 giv es a diag ram of X -schemes ( ∫ F , I F ) which can be descr ibed e xplicitly as follo ws. The categor y I F is the coﬁbred categor y o v er associated to the functor from to the categor y of small categor ies deﬁned by the diagram G 1 G / /   G e in which e stands f or the ter minal categor y , and G f or the categor y with one object associated to G . It has thus three objects a , b , c (see ( 3.3.11.1 )), and the mor phisms are determined b y 128 Fibred categor ies and the six functors f ormalism Hom I F ( x , y ) =          * if y = c ; ∅ if x , y and x = b , c ; G otherwise. The functor F sends a , b , c to T , Y , Z respectivel y , and simply encodes the fact that the diagram T h / / g   Y Z is G -equiv ar iant, the action on Z being trivial. No w , by propositions 3.1.23 and 3.3.31 , if ϕ : ( F , I F ) / / ( X , ) denotes the canonical map, f or any object M of Ho ( M )( X ) , the object R ϕ ∗ L ϕ ∗ ( M ) is the functor from = op to M ( X ) corresponding to the diagram below (of course, this is well deﬁned onl y in the homotop y categor y of the category of functors from to M ( X ) ). ( R f ∗ L f ∗ ( M )) G   R i ∗ L i ∗ ( M ) / / ( R a ∗ L a ∗ ( M )) G As a consequence, if ψ : ( ∫ F , I F ) / / X denotes the structural map, the object R ψ ∗ L ψ ∗ ( M ) is simply the homotopy limit of the diag ram of M ( X ) abo ve, so that condition (iv) of Theorem 3.3.37 can now be ref or mulated by saying that the map M / / R ψ ∗ L ψ ∗ ( M ) is an isomor phism, i.e. that the functor L ψ ∗ : Ho ( M )( X ) / / Ho ( M )( ∫ F , I F ) is fully faithful. 4 Constructible motiv es 4.0.1 Consider as in 2.0.1 a base scheme S and a sub-categor y S of the categor y of S -schemes. In section 4.4 , and for the main theorem of section 4.2 , we will assume: (a) An y scheme in S is quasi-e xcellent. 62 62 See Paragraph 4.1.1 . The reader can saf ely restrict his attention to the more classical notion of an e xcellent scheme ([ GD67 , IV , 7.8.5]). 4 Constr uctible motives 129 Apart in Deﬁnition 4.3.2 and the subsequent proposition, where w e will consider an abstract situation, we will be concer ned with the study of a ﬁx ed premotivic triangulated categor y T ov er S (recall Deﬁnition 2.4.45 ) such that: (b) T is motivic (see Deﬁnition 2.4.45 ). (c) T is endo w ed with a set of twists τ (see Paragraph 1.4.4 ) which is stable under T ate twists 1 ( p )[ q ] , f or p , q ∈ Z . (d) T is the homotop y categor y associated with a stable combinator ial Sm -ﬁbred model category M o v er S . 63 As usual, the geometric sections of T will be denoted by M . U nless e xplicitly referr ing to the underlying model categor y M , w e will not indicate in the notation of the six operations that the functors are der iv ed functors. 4.1 Resolution of singularities The aim of this subsection is to gather the results from the theor y of resolution of singularities that will be used subsequently . 4.1.1 In [ GD67 , IV , 7.8.2], Grothendieck deﬁned the notion of an excellent ring . Matsumura introduced in [ Mat70 ] the weak er notion of a quasi-excellent r ing A . Recall A is quasi-ex cellent if the follo wing conditions hold: 1. A is noetherian. 2. For an y pr ime ideal p , ˆ A p being the completion of A at p , the canonical mor phism A / / ˆ A p is regular (see 4.1.4 below). 3. For any A -alg ebra B of ﬁnite type, the regular locus of Sp ec ( B ) is open. Then a ring A is e x cellent if it is quasi-e x cellent and universall y catenary . Follo wing Gabber , we sa y a scheme X is quasi-excellent ( excellent ) if it admits an open cov er b y aﬃne schemes whose r ings are quasi-e xcellent (ex cellent, respectivel y). Theorem 4.1.2 (Gabber’s w eak local uniformization) Let X be a quasi-excellent sc heme, and Z ⊂ X a nowhere dense closed subsc heme. Then ther e exists a ﬁnite h -cov er { f i : Y i / / X } i ∈ I suc h that for all i in I , f i is a mor phism of ﬁnite type, the sc heme Y i is regular , and f − 1 i ( Z ) is either empty or the support of a strict nor mal crossing divisor in Y i . See [ ILO14 ] for a proof. Note that, if w e are onl y interested in schemes of ﬁnite type o v er Spec ( R ) , for R either a ﬁeld, a complete discrete valuation r ing, or a Dedekind domain whose ﬁeld of functions is a global ﬁeld, this is an immediate consequence of de Jong’ s resolution of singular ities by alterations; see [ dJ96 ]. One can also deduce the case of schemes of ﬁnite type o v er an e xcellent noether ian scheme of dimension lesser or equal to 2 from [ dJ97 ]; see Theorem 4.1.10 and Corollar y 4.1.11 belo w f or a precise statement. 63 W e need this assumption to apply descent theor y as descr ibed in section 3.3 . 130 Fibred categor ies and the six functors f or malism Remar k 4.1.3 This theorem will be used in the proof of Lemma 4.2.14 which is the ke y point for the proof of Theorem 4.2.16 . 4.1.4 Recall that a mor phism of r ings u : A / / B is regular if it is ﬂat, and if, f or any prime ideal p in A , with residue ﬁeld κ ( p ) , the κ ( p ) -algebra κ ( p ) ⊗ A B is geometricall y regular (equivalentl y , this means that, f or an y pr ime ideal q of B , the A -alg ebra B q is formally smooth f or the q -adic topology). W e recall the f ollowing great generalization of Neron ’ s desingularization theorem: Theorem 4.1.5 (Popescu-Spiv ak ov sky) A mor phism of noetherian rings u : A / / B is r egular if and only if B is a ﬁlter ed colimit of smooth A -alg ebras of ﬁnite type. Proof See [ Spi99 , theorems 1.1 and 1.2].  4.1.6 Recall that an alter ation is a proper surjective mor phism p : X 0 / / X which is genericall y ﬁnite, i.e. such that there exis ts a dense open subscheme U ⊂ X o v er which p is ﬁnite. Deﬁnition 4.1.7 (de Jong) Let X be a noether ian scheme endow ed with an action of a ﬁnite g roup G . A Galois alt eration of the couple ( X , G ) is the data of a ﬁnite group G 0 , of a surjective mor phism of groups G 0 / / G , of an alteration X 0 / / X , and of an action of G 0 on X 0 , such that: (i) the map X 0 / / X is G 0 -equiv ariant; (ii) f or an y ir reducible component T of X , there e xists a unique irreducible compo- nent T 0 of X 0 o v er T , and the cor responding ﬁnite ﬁeld extension k ( T ) G ⊂ k ( T 0 ) G 0 is purely inseparable. In practice, we shall keep the mor phism of groups G 0 / / G implicit, and we shall sa y that ( X 0 / / X , G 0 ) is a Galois alteration of ( X , G ) . Giv en a noether ian scheme X , a Galois alter ation of X is a Galois alteration ( X 0 / / X , G ) of ( X , e ) , where e denotes the tr ivial group. In this case, we shall sa y that X 0 / / X is a Galois alter ation of X of gr oup G . Remar k 4.1.8 If p : X 0 / / X is a Galois alteration of group G ov er X , then, if X and X 0 are nor mal, ir reducible and quasi-e x cellent, p can be factored as a radicial ﬁnite surjective mor phism X 00 / / X , f ollo wed b y a Galois alteration X 0 / / X 00 of group G , such that k ( X 00 ) = k ( X 0 ) G (just deﬁne X 00 as the nor malization of X in k ( X 0 ) G ). In other w ords, up to a radicial ﬁnite sur jectiv e morphism, X is generically the quotient of X 0 under the action of G . Deﬁnition 4.1.9 A noether ian scheme S admits canonical dominant r esolution of singularities up to quotient singularities if, f or an y Galois alteration S 0 / / S of group G , and for any G -equivariant no where dense closed subscheme Z 0 ⊂ S 0 , there e xists a Galois alteration ( p : S 00 / / S 0 , G 0 ) of ( S 0 , G ) , such that S 00 is regular and projectiv e ov er S , and such that the in v erse image of Z 0 in S 00 is contained in a 4 Constr uctible motives 131 G 0 -equiv ariant strict nor mal crossing divisor (i.e. a strict normal crossing divisor whose ir reducible components are stable under the action of G 0 ). A noetherian sc heme S admits canonical resolution of singularities up to quo- tient singularities if any integ ral closed subscheme of S admits canonical dominant resolution of singular ities up to quotient singularities. A noether ian scheme S admits wide r esolution of singularities up to quotient singularities if, for an y separated S -scheme of ﬁnite type X , and any nowhere dense closed subscheme Z ⊂ X , there e xists a projectiv e Galois alteration p : X 0 / / X of group G , with X 0 regular , such that, in each connected component of X 0 , Z 0 = p − 1 ( Z ) is either empty or the suppor t of a str ict normal crossing divisor . Theorem 4.1.10 (de Jong) If an excellent noetherian sc heme of ﬁnite dimension S admits canonical r esolution of singularities up to quotient singularities, then any separat ed S -scheme of ﬁnite type admits canonical resolution of singularities up to quotient singularities. Proof Let X be an integral separated S -scheme of ﬁnite type. There e xists a ﬁnite morphism S 0 / / S , with S 0 integral, an integ ral dominant S 0 -scheme X 0 and a radicial extension X 0 / / X ov er S , such that X 0 has a geometrically ir reducible generic ﬁber ov er S 0 . It follo ws then from (the proof of ) [ dJ97 , theorem 5.13] that X 0 admits canonical dominant resolution of singularities up to quotient singularities, which implies that X has the same proper ty .  Corollary 4.1.11 (de Jong) Let S be an excellent noetherian scheme of dimension lesser or equal to 2 . Then any separat ed scheme of ﬁnite type ov er S admits canonical r esolution of singularities up to quotient singularities. In particular , S admits wide r esolution of singularities up to quotient singularities. Proof See [ dJ97 , corollar y 5.15].  4.2 Finiteness theorems The aim of this section is to study the notion of τ -constructibility in the triangulated motivic case and to study its stability proper ties under Grothendieck six operations. Recall the follo wing par ticular case of Deﬁnition 1.4.9 : Deﬁnition 4.2.1 For a scheme X in S , w e denote b y T c ( X ) the thick tr iangulated sub-category of T ( X ) generated by premotives of the f orm M X ( Y ) { i } f or a smooth X - scheme Y and a twist i ∈ τ . W e will say that a premotiv e in T c ( X ) is τ -constructible , or , simply , constructible . Remar k 4.2.2 Let us mention that our main e xamples: • the stable homotop y category SH (cf. Example 1.4.3 ), • the category of V oev odsky motives DM (cf. Deﬁnition 11.1.1 ), • the category of Beilinson motives DM B (cf. Deﬁnition 14.2.1 ) 132 Fibred categor ies and the six functors f ormalism are all generated b y the T ate twists ( i.e. τ = Z ). R ecall also Proposition 1.4.11 : it applies to all these ex amples so that constructible premotiv es coincides with compact objects. 64 Proposition 4.2.3 If M and N are constructible in T ( X ) , so is M ⊗ X N . Proof For a ﬁxed M , the full subcategory of T ( X ) spanned b y objects such that M ⊗ X N is constructible is a thic k triangulated subcategor y of T ( X ) . In the case M is of shape M X ( Y ) { n } f or Y smooth o v er X and n ∈ τ , this pro v es that M ⊗ X N is constructible whenev er N is. By the same argument, using the symmetry of the tensor product, we get to the general case.  Similarl y , one has the f ollo wing conser v ation proper ty . Proposition 4.2.4 F or any mor phism f : X / / Y of schemes, the functor f ∗ : T ( Y ) / / T ( X ) pr eser v es constructible objects. If moreo ver f is smooth, the functor f ] : T ( X ) / / T ( Y ) also pr eser v es constructible objects. Corollary 4.2.5 The categories T c ( X ) form a thic k triangulated monoidal Sm - ﬁbr ed subcategory of T . Proposition 4.2.6 Let X a scheme, and X = Ð i ∈ I U i a cov er of X by open sub- sc hemes. An object M of T ( X ) is constructible if and only if its r estriction to U i is constructible in T ( U i ) f or all i ∈ I . Proof This is a necessar y condition by 4.2.4 . For the conv erse, as X is noether ian, it is suﬃcient to treat the case where I is ﬁnite. Proceeding by induction on the cardinal of I it is suﬃcient to treat the case of a cov er by tw o open subschemes X = U ∪ V . For an open immersion j : W / / X , wr ite M W = j ] j ∗ ( M ) . If the restrictions of M to U and V are constructible, then so is its restriction to U ∩ V . According to Proposition 3.3.4 , we get a distinguished tr iangle M U ∩ V / / M U ⊕ M V / / M / / M U ∩ V [ 1 ] in which M W is constructible f or W = U , V , U ∩ V (using 4.2.4 again). Thus the premotiv e M is constr uctible.  Corollary 4.2.7 F or any scheme X and any vector bundle E ov er X , the functors T h ( E ) and T h (− E ) preserve constructible objects in T ( X ) . 64 Notice how ev er this fact is not true for étale motivic complex es. 4 Constr uctible motives 133 Proof T o prov e that T h ( E ) and T h (− E ) preser v es constr uctible objects, by vir tue of the preceding proposition, w e ma y assume that E is trivial of rank r . It is thus suﬃcient to prov e that M ( r ) is constr uctible whene v er M is so f or any integ er r . For we may assume that M = 1 X { n } f or some n ∈ τ (using 4.2.4 ), this is tr ue by assumption on τ ; see 4.0.1 (c).  Corollary 4.2.8 Let f : X / / Y a morphism of ﬁnit e type. The pr operty that the functor f ∗ : T ( X ) / / T ( Y ) pr eser v es constructible objects is local on Y with r espect to the Zariski topology . Proof Consider a ﬁnite Zar iski co v er { v i : Y i / / Y } i ∈ I , and wr ite f i : X i / / Y i f or the pullbac k of f along v i f or each i in I . Assume that the functors f i , ∗ preserves constructible objects; we shall pro v e that f ∗ has the same proper ty . Let M be a con- structible object in T ( X ) . Then for i ∈ I , using the smooth base chang e isomorphism (f or open immersions), w e see that the restriction of f ∗ ( M ) to Y i is isomorphic to the imag e b y f i , ∗ of the restriction of M to X i , hence is constructible. The preceding proposition thus implies that f ∗ ( M ) is constructible.  Proposition 4.2.9 F or any closed immersion i : Z / / X , the functor i ∗ : T ( Z ) / / T ( X ) pr eser v es constructible objects. Proof It is suﬃcient to pro v e that for an y smooth Z -scheme Y 0 and any twist n ∈ τ , the premotiv e i ∗ ( M Z ( Y 0 ){ n } ) is constructible in T ( X ) . A ccording to the May er - Vietoris tr iangle (see R emark 3.3.6 ), this asser tion is local in X . Thus we can assume there exis ts a smooth X -scheme Y such that Y 0 = Y × X Z (apply [ GD67 , 18.1.1]). Put U = X − Z and let j : U / / X be the obvious open immersion. From the localization property , w e get a distinguished tr iangle M X ( Y × X U ) { n } / / M X ( Y ) { n } / / i ∗ ( M Z ( Y 0 ){ n } ) / / M X ( Y × X U ) { n }[ 1 ] and this concludes.  Corollary 4.2.10 Let i : Z / / X be a closed immersion with open complement j : U / / X . an object M of T ( X ) is constructible if and only if j ∗ ( M ) and i ∗ ( M ) ar e constructible in T ( U ) and T ( Z ) respectiv ely. Proof W e ha v e a distinguished tr iangle j ] j ∗ ( M ) / / M / / i ∗ i ∗ ( M ) / / j ] j ∗ ( M )[ 1 ] . Hence this asser tion f ollow s from propositions 4.2.4 and 4.2.9 .  Proposition 4.2.11 If f : X / / Y is proper , then the functor 134 Fibred categor ies and the six functors formalism f ∗ : T ( X ) / / T ( Y ) pr eser v es constructible objects. Proof W e shall ﬁrs t consider the case where f is projectiv e. As this proper ty is local on Y (Corollary 4.2.8 ), w e may assume that f f actors as a closed immersion i : X / / P n Y f ollow ed by the canonical projection p : P n Y / / Y . By vir tue of Proposition 4.2.9 , w e can assume that f = p . In this case, the functor p ∗ is isomorphic to p ] composed with the quasi-in v erse of the Thom endofunctor associated to the cotangent bundle of p ; see 2.4.50 (3). Theref ore, the functor p ∗ preserves constructible objects b y vir tue of Proposition 4.2.4 and of Corollar y 4.2.7 . The case where f is proper f ollow s easily from the projectiv e case, using Cho w’ s lemma and c dh -descent (the homotop y pullback squares ( 3.3.9.1 )), b y induction on the dimension of X .  Corollary 4.2.12 If f : X / / Y is separated of ﬁnite type, then the functor f ! : T ( X ) / / T ( Y ) pr eser v es constructible objects. Proof It is suﬃcient to treat the case where f is either an open immersion, either a proper mor phism, which follo ws respectivel y from 4.2.4 and 4.2.11 .  Proposition 4.2.13 Let X be a sc heme. The category of constructible objects in T ( X ) is the smallest thick triangulated subcategory whic h contains the objects of shape f ∗ ( 1 X 0 { n } ) , wher e f : X 0 / / X is a (strictly) projectiv e morphism, and n ∈ τ . Proof Let T p ( X ) be the smallest thick tr iangulated subcategor y which contains the objects of shape f ∗ ( 1 X 0 { n } ) , where f : X 0 / / X is a (strictly) projectiv e mor phism, and n ∈ τ . Proposition 4.2.11 sho ws that T p ( X ) ⊂ T c ( X ) , to that it is suﬃcient to pro v e the re v erse inclusion. Note that, f or an y separated smooth mor phism f , locall y f or the Zar iski topology , f ] coincides with f ! up to a T ate twist. In other words, it is suﬃcient to pro v e that, f or any separated mor phism of ﬁnite type f : Y / / X , f ! ( 1 Y ) belongs to T p ( X ) . If we f actor f into an open immersion j : Y / / X 0 f ollow ed by a proper mor phism p : X 0 / / X , w e see that is suﬃcient to prov e that j ] ( 1 Y ) belongs to T p ( X 0 ) . This f ollo ws straight aw ay from the localization proper ty .  The f ollo wing lemma is the k e y geometrical point f or the ﬁniteness Theorem 4.2.16 Lemma 4.2.14 Let j : U / / X be a dense open immersion such that X is quasi- excellent. Then, ther e exists the f ollowing data: (i) a ﬁnite h -cov er { f i : Y i / / X } i ∈ I suc h that for all i in I , f i is a mor phism of ﬁnite type, the scheme Y i is regular , and f − 1 i ( U ) is either Y i itself or the complement of a strict normal crossing divisor in Y i ; w e shall write f : Y = Þ i ∈ I Y i / / X f or the induced g lobal h -cov er; 4 Constr uctible motives 135 (ii) a commutativ e diagr am X 000 g / / q   Y f   X 00 u / / X 0 p / / X (4.2.14.1) in which: p is a proper bir ational mor phism, X 0 is nor mal, u is a Nisnevich cov er , and q is a ﬁnite surjectiv e mor phism. Let T (resp. T 0 ) be a closed subsc heme of X (resp. X 0 ) and assume that for any irr educible component T 0 of T , the f ollowing inequality is satisﬁed: (4.2.14.2) co dim X 0 ( T 0 ) ≥ co dim X ( T 0 ) , Then, possibly after shrinking X in an open neighborhood of the g eneric points of T in X , one can r eplace X 00 by an open cov er and X 000 by its pullbac k along this cov er , in suc h a way that we hav e in addition the follo wing properties: (iii) p ( T 0 ) ⊂ T and the induced map T 0 / / T is ﬁnite and pseudo-dominant; 65 (iv) if w e write T 00 = u − 1 ( T 0 ) , the induced map T 00 / / T 0 is an isomorphism. Proof The exis tence of f : Y / / X as in (i) follo ws from Gabber’ s w eak uniformiza- tion theorem (see 4.1.2 ), while the commutative diagram ( 4.2.14.1 ) satisfying prop- erty (ii) is ensured by Lemma 3.3.28 . Consider moreo v er closed subschemes T ⊂ X and T 0 ⊂ X 0 satisfying ( 4.2.14.2 ). W e ﬁrst sho w that, by shrinking X in an open neighborhood of the generic points of T and by replacing the diagram ( 4.2.14.1 ) by its pullback ov er this neighborhood, w e can assume that condition (iii) is satisﬁed. Note that shrinking X in this wa y does not change the condition ( 4.2.14.2 ) because co dim X ( T 0 ) does not chang e and co dim X 0 ( T 0 ) can only increase. 66 Note ﬁrst that, b y shrinking X , w e can assume that an y irreducible component T 0 0 of T 0 dominates an ir reducible component T 0 of T . In fact, giv en an ir reducible component T 0 0 which does not satisfy this condition, p ( T 0 0 ) is a closed subscheme of X disjoint from the set of generic points of T and replacing X by X − f ( T 0 0 ) , w e can thro w out T 0 0 . Further, shrinking X ag ain, we can assume that f or an y pair ( T 0 0 , T 0 ) as in the preceding parag raph, p ( T 0 0 ) ⊂ T 0 . In fact, in any case, as p ( T 0 0 ) is closed w e get that T 0 ⊂ p ( T 0 0 ) . Let Z be the closure of p ( T 0 0 ) − T 0 in X . Then Z does not contain any generic point of T (because p ( T 0 0 ) is ir reducible), and p ( T 0 0 ) ∩ ( X − Z ) ⊂ T 0 . Thus it is suﬃcient to replace X b y X − Z to ensure this assumption. Consider again a pair ( T 0 0 , T 0 ) as in the two preceding paragraphs and the induced commutativ e square: 65 Recall from 8.1.3 that this means that any ir reducible component of T 0 dominates an ir reducible component of T . 66 Remember that f or any scheme X , co dim X ( ∅ ) = + ∞ . 136 Fibred categories and the six functors formalism T 0 0 / / p 0   X 0 p   T 0 / / X (4.2.14.3) W e sho w that the map p 0 is generically ﬁnite. In fact, this will conclude the ﬁrst step, because if it is tr ue for any ir reducible component T 0 0 of T 0 , w e can shrink X again so that the dominant mor phism p 0 : T 0 0 / / T 0 becomes ﬁnite. Denote by c 0 (resp. c ) the codimension of T 0 in X (resp. T 0 0 in X 0 ). Note that ( 4.2.14.2 ) giv es the inequality c 0 ≥ c . Let t 0 be the g ener ic point of T 0 , Ω the localization of X at t 0 and consider the pullback of ( 4.2.14.3 ): W 0 / / q 0   Ω 0 q   { t 0 } / / Ω . (4.2.14.4) W e hav e to pro ve that dim ( W 0 ) = 0 . Consider an ir reducible component Ω 0 0 of Ω 0 containing W 0 . As q is s till proper birational, Ω 0 0 corresponds to a unique ir re- ducible component Ω 0 of Ω such that q induces a proper birational map Ω 0 0 / / Ω 0 . A ccording to [ GD67 , 5.6.6], we g et the inequality dim ( Ω 0 0 ) ≤ dim ( Ω 0 ) . Thus, w e obtain the follo wing inequalities: dim ( W 0 ) ≤ dim ( Ω 0 0 ) − co dim Ω 0 0 ( W 0 ) ≤ dim ( Ω 0 ) − co dim Ω 0 0 ( W 0 ) ≤ dim ( Ω ) − codim Ω 0 0 ( W 0 ) . As this is tr ue f or an y ir reducible component Ω 0 0 of Ω 0 , w e ﬁnally obtain: dim ( W 0 ) ≤ dim ( Ω ) − co dim Ω 0 ( W 0 ) ≤ c − c 0 and this concludes the ﬁrst step. Keeping T 0 and T as abo ve, as the map from T 00 to T 0 is a Nisnevich co v er , it is a split epimor phism in a neighborhood of the g eneric points of T 0 in X 0 . Hence, as the map X 0 / / X is proper and birational, w e can ﬁnd a neighborhood of the g ener ic points of T in X o v er which the map T 00 / / T 0 admits a section s : T 0 / / T 00 . Let S be a closed subset of X 00 such that T 00 = s ( T 0 ) q S (which exis ts because X 00 / / X 0 is étale). The map ( X 00 − T 00 ) q ( X 00 − S ) / / X 0 is then a Nisne vich cov er . Replacing X 00 b y ( X 00 − T 00 ) q ( X 00 − S ) (and X 000 b y the pullbac k of X 000 / / X 00 along ( X 00 − T 00 ) q ( X 00 − S ) / / X 0 ), we ma y assume that the induced map T 00 / / T 0 is an isomor phism, without modifying further the data f , p , T and T 0 . This giv es property (iv) and ends the proof the lemma.  4 Constr uctible motives 137 4.2.15 Let T 0 be a full Op en -ﬁbred subcategor y of T (where Op en stands f or the class of open immersions). W e assume that T 0 has the f ollo wing proper ties: (a) f or any scheme X in S , T 0 ( X ) is a thick subcategor y of T ( X ) which contains the objects of the form 1 X { n } , n ∈ τ ; (b) f or an y separated mor phism of ﬁnite type f : X / / Y in S , T 0 is stable under f ! ; (c) f or any dense open immersion j : U / / X , with X regular, which is the com- plement of a strict nor mal crossing divisor , j ∗ ( 1 U { n } ) is in T 0 ( U ) for any n ∈ τ . Properties (a) and (b) ha ve the follo wing consequences: an y constr uctible object be- longs to T 0 ; given a closed immersion i : Z / / X with complement open immersion j : U / / X , an object M of T ( X ) belongs to T 0 ( X ) if and only if j ∗ ( M ) and i ∗ ( M ) belongs to T 0 ( U ) and T 0 ( Z ) respectiv ely ; f or any scheme X in S , the condition that an object of T ( X ) belongs to T 0 ( X ) is local on X f or the Zar iski topology . Theorem 4.2.16 Consider the abov e hypothesis and assume that T is Q -linear and separat ed. Let Y be a quasi-excellent sc heme and f : X / / Y be a morphism of ﬁnite type. Then f or any constructible object M of T ( X ) , the object f ∗ ( M ) belongs to T 0 ( Y ) . Proof It is suﬃcient to pro v e that, for an y dense open immersion j : U / / X , and f or any n ∈ τ , the object j ∗ ( 1 U { n } ) is in T 0 . Indeed, assume this is kno wn. W e want to prov e that f ∗ ( M ) is in T 0 ( Y ) whenev er M is constructible. W e deduce from property (b) of 4.2.15 and from Proposition 4.2.13 that it is suﬃcient to consider the case where M = 1 X { n } , with n ∈ τ . Then, as this property is assumed to be kno wn for dense open immersions, by an easy May er - Vietoris argument, w e see that the condition that f ∗ ( 1 X { n } ) belongs to T 0 is local on X with respect to the Zariski topology . Therefore, we may assume that f is separated. Consider a compactiﬁcation of f , i.e. a commutative diag ram Y j / / f   ¯ Y ¯ f   X with j a dense open immersion, and ¯ f proper . By property (b) of 4.2.15 , w e ma y assume that f = j is a dense open immersion. Let j : U / / X be a dense open immersion. W e shall pro v e b y induction on the dimension of X that , f or any n ∈ τ , the object j ∗ ( 1 U { n } ) is in T 0 . The case where X is of dimension ≤ 0 f ollo ws from the fact the map j is then an isomor phism, which implies that j ] ' j ∗ , and allo ws to conclude (because T 0 is Op en -ﬁbred). Assume that dim X > 0 . Follo wing an argument used b y Gabber [ ILO14 ] in the conte xt of ` -adic sheav es, we shall prov e by induction on c ≥ 0 that there exis ts a closed subscheme T ⊂ X of codimension > c such that, f or any n ∈ τ , the restriction of j ∗ ( 1 U { n } ) to X − T is in T 0 ( X − T ) . As X is of ﬁnite dimension, this will obviousl y pro v e Theorem 4.2.16 . 138 Fibred categor ies and the six functors f or malism The case where c = 0 is clear: w e can choose T suc h that X − T = U . If c > 0 , w e choose a closed subscheme T of X , of codimension > c − 1 , such that the restr iction of j ∗ ( 1 U { n } ) to X − T is in T 0 . It is then suﬃcient to ﬁnd a dense open subscheme V of X , which contains all the generic points of T , and such that the restriction of j ∗ ( 1 U { n } ) to V is in T 0 : f or such a V , w e shall obtain that the restriction of j ∗ ( 1 U { n } ) to V ∪ ( X − T ) is in T 0 , the complement of V ∪ ( X − T ) being the suppor t of a closed subscheme of codimension > c in X . In par ticular , using the smooth base chang e isomor phism (f or open immersions), w e can alw a ys replace X by a generic neighborhood of T . It is suﬃcient to pro v e that, possibly after shr inking X as abo v e, the pullback of j ∗ ( 1 U { n } ) along T / / X is in T 0 (as w e already kno w that its restriction to X − T is in T 0 ). W e may assume that T is purely of codimension c . W e may assume that w e hav e data as in points (i) and (ii) of Lemma 4.2.14 . W e let j 0 : U 0 / / X 0 denote the pullback of j along p : X 0 / / X . Then, w e can ﬁnd, by induction on c , a closed subscheme T 0 in X 0 , of codimension > c − 1 , such that the restriction of j 0 ∗ ( 1 U 0 { n } ) to X 0 − T 0 is in T 0 . By shrinking X , w e may assume that conditions (iii) and (iv) of Lemma 4.2.14 are fulﬁlled as well. For an X -scheme w : W / / X and a closed subscheme Z ⊂ W , we shall wr ite ϕ ( W , Z ) = w ∗ i ∗ i ∗ j W , ∗ j ∗ W ( 1 W { n } ) , where i : Z / / W denotes the inclusion, and j W : W U / / W stands for the pullback of j along w . This construction is functorial with respect to mor phisms of pairs of X -schemes: if W 0 / / W is a morphism of X -schemes, with Z 0 and Z two closed subschemes of W 0 and W respectiv ely , such that Z 0 is sent to Z , then w e get a natural map ϕ ( W , Z ) / / ϕ ( W 0 , Z 0 ) . Remember that w e want to pro v e that ϕ ( X , T ) is in T 0 . This will be done via the f ollo wing lemmas (which hold assuming all the conditions stated in Lemma 4.2.14 as well as our inductiv e assumptions). Lemma 4.2.17 The cone of the map ϕ ( X , T ) / / ϕ ( X 0 , T 0 ) is in T 0 .  The map ϕ ( X , T ) / / ϕ ( X 0 , T 0 ) factors as ϕ ( X , T ) / / ϕ ( X 0 , p − 1 ( T )) / / ϕ ( X 0 , T 0 ) . By the octahedral axiom, it is suﬃcient to prov e that each of these two maps has a cone in T 0 . W e shall pro v e ﬁrst that the cone of the map ϕ ( X 0 , p − 1 ( T )) / / ϕ ( X 0 , T 0 ) is in T 0 . Giv en an immersion a : S / / X 0 , w e shall wr ite M S = a ! a ∗ ( M ) . W e then ha v e distinguished triangles M p − 1 ( T )− T 0 / / M p − 1 ( T ) / / M T 0 / / M p − 1 ( T )− T 0 [ 1 ] . For M = j 0 ∗ ( 1 U 0 { n } ) (recall j 0 is the pullback of j along p ) the imag e of this tr iangle b y p ∗ giv es a distinguished triangle 4 Constr uctible motives 139 p ∗ ( M p − 1 ( T )− T 0 ) / / ϕ ( X 0 , p − 1 ( T )) / / ϕ ( X 0 , T 0 ) / / p ∗ ( M p − 1 ( T )− T 0 )[ 1 ] . As the restriction of M = j 0 ∗ ( 1 U 0 { n } ) to X 0 − T 0 is in T 0 b y assumption on T 0 , the object M p − 1 ( T )− T 0 is in T 0 as well (by proper ty (b) of 4.2.15 and because T 0 is Op en -ﬁbred), from which w e deduce that p ∗ ( M p − 1 ( T )− T 0 ) is in T 0 (using condition (iii) of Lemma 4.2.14 and proper ty (b) of 4.2.15 ). Let V be a dense open subscheme of X such that p − 1 ( V ) / / V is an isomor phism. W e ma y assume that V ⊂ U , and write i : Z / / U f or the complement closed immersion. Let p U : U 0 = p − 1 ( U ) / / U be the pullback of p along j , and let ¯ Z be the reduced closure of Z in X . W e thus get the commutative squares of immersions belo w , Z k / / i   ¯ Z l   U j / / X and Z 0 k 0 / / i 0   ¯ Z 0 l 0   U 0 j 0 / / X 0 where the square on the r ight is obtained from the one on the left by pulling back along p : X 0 / / X . As p is an isomor phism o ver V , we get by c dh -descent (Proposition 3.3.10 ) the homotopy pullback square belo w . 1 U { n } / /   p U , ∗ ( 1 U 0 { n } )   i ∗ i ∗ ( 1 Z { n } ) / / i ∗ i ∗ p U , ∗ ( 1 U 0 { n } ) If a : T / / X denotes the inclusion, applying the functor a ∗ a ∗ j ∗ to the commutative square abov e, w e see from the proper base chang e f ormula and from the identiﬁcation j ∗ i ∗ ' l ∗ k ∗ that w e get a commutativ e square isomorphic to the f ollo wing one ϕ ( X , T ) / /   ϕ ( X 0 , p − 1 ( T ))   ϕ ( ¯ Z , ¯ Z ∩ T ) / / ϕ ( ¯ Z 0 , p − 1 ( ¯ Z ∩ T )) , which is thus homotopy car tesian as well. It is suﬃcient to prov e that the tw o objects ϕ ( ¯ Z , ¯ Z ∩ T ) and ϕ ( ¯ Z 0 , p − 1 ( ¯ Z ∩ T )) are in T 0 . It f ollo ws from the proper base chang e f or mula that the object ϕ ( ¯ Z , ¯ Z ∩ T ) is canonically isomorphic to the restriction to T of l ∗ k ∗ ( 1 Z { n } ) . As dim ¯ Z < dim X , we know that the object k ∗ ( 1 Z { n } ) is in T 0 . By proper ty (b) of 4.2.15 , we obtain that ϕ ( ¯ Z , ¯ Z ∩ T ) is in T 0 . Similarly , the object ϕ ( ¯ Z 0 , p − 1 ( ¯ Z ∩ T )) is canonically isomorphic to the restr iction of p ∗ l 0 ∗ k 0 ∗ ( 1 Z 0 { n } ) to T , and, as dim ¯ Z 0 < dim X 0 (because, p being an isomor phism o v er the dense open subscheme V of X , ¯ Z 0 does not contain any generic point of X 0 ), k 0 ∗ ( 1 Z 0 { n } ) is in 140 Fibred categor ies and the six functors formalism T 0 . W e deduce again from proper ty (b) of 4.2.15 that ϕ ( ¯ Z 0 , p − 1 ( ¯ Z ∩ T )) is in T 0 as w ell, which achie ves the proof of the lemma. Lemma 4.2.18 The map ϕ ( X 0 , T 0 ) / / ϕ ( X 00 , T 00 ) is an isomor phism in T ( X ) .  Condition (iv) of Lemma 4.2.14 can be ref or mulated by saying that we ha v e the Nisnevic h distinguished square below . X 00 − T 00 / /   X 00 v   X 0 − T 0 / / X 0 This lemma f ollow s then by Nisne vich e xcision (Proposition 3.3.4 ) and smooth base chang e (f or étale maps). Lemma 4.2.19 Let T 000 be the pullback of T 00 along the ﬁnite surjectiv e mor - phism X 000 / / X 00 . The map ϕ ( X 00 , T 00 ) / / ϕ ( X 000 , T 000 ) is a split monomor phism in T ( X ) .  W e hav e the f ollo wing pullback squares T 000 t / / r   X 000 q   U 000 j 0 00 o o q U   T 00 s / / X 00 U 0 j 0 0 o o in which j 00 and j 000 denote the pullback of j along pu and pu q respectivel y , while s and t are the inclusions. By the proper base chang e f or mula applied to the left-hand square, we see that the map ϕ ( X 00 , T 00 ) / / ϕ ( X 000 , T 000 ) is isomor phic to the image of the map j 00 ∗ ( 1 U 0 0 { n } ) / / q ∗ q ∗ j 00 ∗ ( 1 U 0 0 { n } ) / / q ∗ j 000 ∗ ( 1 U 0 00 { n } ) . b y f ∗ s ∗ , where f : T 00 / / T is the map induced b y p (note that f is proper as T 00 ' T 0 b y assumption). As q ∗ j 000 ∗ ' j 00 ∗ q U , ∗ , we are thus reduced to prov e that the unit map 1 U 0 0 { n } / / q U , ∗ ( 1 U 0 00 { n } ) is a split monomor phism. As X 00 is nor mal (because X 0 is so by assumption, while X 00 / / X 0 is étale), this f ollo ws immediately from Corollary 3.3.40 . No w , we can ﬁnish the proof of Theorem 4.2.16 . Consider the V erdier quotient D = T ( X )/ T 0 ( X ) . W e want to pro ve that, under the conditions stated in Lemma 4.2.14 , w e hav e ϕ ( X , T ) ' 0 in D . Let π : T 000 / / X be the map induced by pu q : X 000 / / X . If 4 Constr uctible motives 141 a : T 000 / / Y denotes the map induced by g : X 000 / / Y , and j Y : Y U / / Y the pullback of j b y f , we hav e the commutativ e diagram below . ϕ ( X , T ) / / ' ' ϕ ( X 000 , T 000 ) π ∗ a ∗ j Y , ∗ ( 1 Y U { n } ) 6 6 By vir tue of lemmas 4.2.17 , 4.2.19 , and 4.2.18 , the horizontal map is a split monomor- phism in D . It is thus suﬃcient to pro ve that this map vanishes in D , f or whic h it will be suﬃcient to prov e that π ∗ a ∗ j Y , ∗ ( 1 Y U { n } ) is in T 0 . The morphism π is ﬁnite (b y construction, the map T 00 / / T 0 is an isomor phism, while the maps T 000 / / T 00 and T 0 / / T are ﬁnite). U nder this condition, T 0 is stable under the operations π ∗ and a ∗ . T o ﬁnish the proof of the theorem, it remains to check that j Y , ∗ ( 1 Y U { n } ) is in T 0 , which f ollow s from proper ty (c) of 4.2.15 (and additivity).  Deﬁnition 4.2.20 W e shall say that T is τ -compatible if it satisﬁes the f ollowing tw o conditions. (a) For an y closed immersion i : Z / / X between regular schemes in S , the image of 1 X { n } , n ∈ τ , by the ex ceptional in v erse image functor i ! : T ( X ) / / T ( Z ) is constructible. (b) For an y scheme X , any n ∈ τ , and any constructible object M in T ( X ) , the object Hom X ( 1 X { n } , M ) is constructible. As usual, when τ is the monoid generated by the T ate twist, w e say compatible with T ate twists . Remar k 4.2.21 Condition (b) of the deﬁnition abov e will come essentially f or free if the objects 1 X { n } are ⊗ -inv er tible with constr uctible ⊗ -quasi-in verse (which will hold in practice, essentially by deﬁnition). Example 4.2.22 In practice, condition (a) of the deﬁnition abo v e will be a con- sequence of the absolute purity theor em . In par ticular , the categor y of Beilinson motiv es DM B is compatible with T ate twist as a corollar y of the fact the T ate twist is in vertible and Theorem 14.4.1 . Lemma 4.2.23 Assume that T is τ -compatible. Let i : Z / / X be a closed immer - sion, with X regular , and Z the support of a strict nor mal cr ossing divisor . Then i ! ( 1 X { n } ) is constructible f or any n ∈ τ . As a consequence, if j : U / / X denotes the complement open immersion, then j ∗ ( 1 U { n } ) is constructible for any n ∈ τ . Proof The ﬁrst asser tion f ollo ws easily by induction on the number of ir reducible components of Z , using Proposition 4.2.6 . The second assertion f ollo ws from the distinguished tr iangles i ∗ i ! ( M ) / / M / / j ∗ j ∗ ( M ) / / i ∗ i ! ( M )[ 1 ] and from Lemma 4.2.9 .  142 Fibred categor ies and the six functors formalism Theorem 4.2.24 Assume that T is Q -linear , separated, and τ -compatible. Then, f or any morphism of ﬁnite type f : X / / Y suc h that Y is quasi-excellent, the functor f ∗ : T ( X ) / / T ( Y ) pr eser v es constructible objects. Proof By vir tue of propositions 4.2.4 and 4.2.11 as well as of Lemma 4.2.23 , if T is τ -compatible, w e can appl y Theorem 4.2.16 , where T stands for the subcategor y of constructible objects.  Corollary 4.2.25 Under t he assump tions of t he abo v e theor em, f or any quasi- excellent scheme X , and for any couple of constructible objects M and N in T ( X ) , the object Hom X ( M , N ) is constructible. Proof It is suﬃcient to treat the case where M = f ] ( 1 Y { n } ) , for n ∈ τ and f : Y / / X a smooth morphism. But then, we hav e, by transposition of the Sm -projection f or mula, a natural isomor phism: Hom X ( M , N ) ' f ∗ Hom ( 1 Y { n } , f ∗ ( N )) . This corollar y follo ws then immediately from Proposition 4.2.4 and from Theorem 4.2.24 .  Corollary 4.2.26 Under the assumptions of the abov e theor em, f or any closed im- mersion i : Z / / X suc h that X is quasi-excellent, the functor i ! : T ( X ) / / T ( Z ) pr eser v es constructible objects. Proof Let j : U / / X be the complement open immersion. For an object M of T ( X ) , we ha v e the f ollo wing distinguished triangle. i ∗ i ! ( M ) / / M / / j ∗ j ∗ ( M ) / / i ∗ i ! ( M )[ 1 ] . By vir tue of Proposition 4.2.6 and Theorem 4.2.24 , if M is constructible, then j ∗ j ∗ ( M ) ha v e the same proper ty , which allo ws us to conclude.  Lemma 4.2.27 Let f : X / / Y be a separ ated mor phism of ﬁnite type. The con- dition that the functor f ! pr eser v es constructible objects in T is local ov er X and ov er Y for the Zariski topology. Proof If u : X 0 / / X is a Zariski co ver , then w e hav e, b y deﬁnition, u ∗ = u ! , so that, b y Proposition 4.2.6 , the condition that f ! preserves τ -constructibility is equivalent to the condition that the functors u ∗ f ! ' ( f u ) ! preserve τ -constr uctibility . Let v : Y 0 / / Y be a Zar iski cov er , and consider the follo wing pullback square. 4 Constr uctible motives 143 X 0 u / / g   X f   Y 0 v / / Y W e then hav e a natural isomor phism u ∗ f ! ' g ! v ∗ , and, as u is still a Zariski cov er , w e deduce again from Proposition 4.2.6 that, if g ! preserves τ -constr uctibility , so does f ! .  Corollary 4.2.28 Under the assumptions of the abov e theor em, f or any separ ated morphism of ﬁnite type f : X / / Y , the functor f ! : T ( Y ) / / T ( X ) pr eser v es constructible objects. Proof By virtue of the preceding lemma, w e ma y assume that f is aﬃne. W e can then f actor f as an immersion i : X / / A n Y f ollow ed by the canonical projection p : A n Y / / Y . The case of an immersion is reduced to the case of an open immersion ( 4.2.4 ) and to the case of a closed immersion ( 4.2.26 ). Thus w e ma y assume that f = p , in which case p ! ' p ∗ (−)( n )[ 2 n ] (according to point (3) of Theorem 2.4.50 ), so that we conclude by 4.2.4 and 4.2.9 .  In conclusion, we hav e prov ed the follo wing ﬁniteness theorem: Theorem 4.2.29 Assume the motivic triangulated category T is Q -linear , separ ated and τ -compatible. 67 Then constructible objects of T ar e closed under the six oper ations of Gr othendiec k when restrict ed to the subcategor y S 0 of S made of quasi-excellent schemes and morphisms of ﬁnite type. In particular , T c is a τ -g enerated motivic category o ver S 0 . 4.3 Continuity 4.3.1 For the ne xt deﬁnition, w e consider an admissible class P of mor phisms in S and an abstract symmetr ic monoidal P -ﬁbred category T o ver S . Let ( S α ) α ∈ A be a projectiv e sys tem of schemes in S , with aﬃne transition maps, and such that S = lim o o α ∈ A S α is representable in S (we assume that A is a partially ordered set to keep the notations simple). For each index α , we denote b y p α : S / / S α the canonical projection. Giv en an inde x α 0 ∈ A and an object E α 0 in T ( S α 0 ) , we wr ite E α f or the pullback of E α 0 along the map S α / / S α 0 , and put E = L p ∗ α ( E α ) . W e will sa y that ( S α ) α ∈ A is dominant if eac h transition map is further more dominant. 67 Remember also that T is associated with a combinatorial stable premotivic model category . 144 Fibred categor ies and the six functors formalism Deﬁnition 4.3.2 Consider the assumptions abov e and let τ be a set of twists of T . W e say that T is τ -continuous (resp. weakly τ -continuous ), or simply continuous (resp. weakly continuous ) if τ is clearly speciﬁed by the context, if it is τ -generated and if, given any projectiv e system (resp. dominant projective sy stem) of schemes ( S α ) as abov e, f or any inde x α 0 , any object E α 0 in T ( S α 0 ) , and any twist n ∈ τ , the canonical map lim / / α ≥ α 0 Hom T ( S α ) ( 1 S α { n } , E α ) / / Hom T ( S ) ( 1 S { n } , E ) , is bi jectiv e. Example 4.3.3 The main ex amples of τ -continuous categor ies will be seen after - wards: • the A 1 -derived categor y D A 1 , Λ (Example 6.1.13 ); • the motivic categor y DM B of Beilinson motiv es (Proposition 14.3.1 ). The tr iangulated motivic categor y of motivic comple xes DM Λ , as well as its eﬀective counterpar t DM eﬀ Λ , is weakl y continuous (Theorem 11.1.24 ). W e are onl y able to prov e it is continuous in some special cases (namely when it compares to Beilinson motiv es, see Theorem 16.1.4 ). The interest of the continuity proper ty is to allow a description of constructible objects o v er S in ter ms of constructible objects o v er the S α ’ s. Proposition 4.3.4 Assume, under the hypothesis of 4.3.1 , that T is τ -continuous (r esp. weakly τ -continuous). Consider a sc heme S in S , as well as a pr ojectiv e sys tem of sc hemes ( S α ) α ∈ A in S with aﬃne (resp. aﬃne dominant) transition maps and suc h that S = lim o o α S α . Then, f or any index α 0 , and for any objects C α 0 and E α 0 in T ( S α 0 ) , if C α 0 is constructible, then the canonical map (4.3.4.1) lim / / α ≥ α 0 Hom T ( S α ) ( C α , E α ) / / Hom T ( S ) ( C , E ) is bi jective. Moreo v er , the canonical functor (4.3.4.2) 2 - lim / / α T c ( S α ) / / T c ( S ) is an equiv alence of monoidal triangulated categories. Proof T o prov e the ﬁrst assertion, w e may assume, without loss of g enerality , that C α 0 = M S α 0 ( X α 0 ){ n } f or some smooth S α 0 -scheme of ﬁnite type X α 0 , and n ∈ τ . Consider an object E α 0 in T ( S α 0 ) . For α ≥ α 0 , wr ite X α (resp. E α ) for the pullback of X α 0 (resp. of E α 0 ) along the map S α / / S α 0 . Similarl y , wr ite X (resp. E ) f or the pullback of X α 0 (resp. of E α 0 ) along the map S / / S α 0 . W e shall also wr ite E 0 α (resp. E 0 ) for the pullback of E α (resp. E ) along the smooth map X α / / S α (resp. 4 Constr uctible motives 145 X / / S ). Then, ( X α ) is a projective sy stem of schemes in S , with aﬃne transition maps, such that X = lim o o α X α . Note that if ( S α ) is dominant in the sense of Paragraph 4.3.1 , then ( X α ) is dominant, as dominant mor phisms are stable under smooth base chang e. Then, b y continuity (resp. w eak continuity), we ha v e the follo wing natural isomorphism, which pro ves the ﬁrst asser tion. lim / / α Hom T ( S α ) ( M S α ( X α ){ n } , E α ) ' lim / / α Hom T ( X α ) ( 1 X α { n } , E 0 α ) ' Hom T ( X ) ( 1 X { n } , E 0 ) ' Hom T ( S ) ( M S ( X ){ n } , E ) Note that the ﬁrst asser tion implies that the functor ( 4.3.4.2 ) is fully faithful. Pseudo- abelian triangulated categor ies are stable under ﬁltered 2 -colimits. In par ticular , the source of the functor ( 4.3.4.2 ) can be seen as a thick subcategory of T ( S ) . The essential surjectivity of ( 4.3.4.2 ) f ollo ws from the fact that, f or an y smooth S -scheme of ﬁnite type X , there exis ts some index α , and some smooth S α -scheme X α , such that X ' S × S α X α ; see [ GD67 , 8.8.2 and 17.7.8]: this implies that the essential image of the fully faithful functor ( 4.3.4.2 ) contains all the objects of shape M S ( X ){ n } f or n ∈ τ and X smooth o ver S , so that it contains T c ( S ) , by deﬁnition.  4.3.5 Bef ore sho wing how the assumption of weak continuity can be used in the case of motivic categories, w e s tate a proposition which later on will allow us to sho w continuity or w eak continuity in concrete cases. Let M be a symmetric monoidal P -ﬁbred model categor y M ov er S . W e consider again the assumptions and notations of 4.3.1 , assuming the transition maps of the pro-scheme ( S α ) are P -morphisms, with T = Ho ( M ) . For each index α ∈ A , we c hoose a small set I α (resp. J α ) of g enerating coﬁbrations (resp. of generating trivial coﬁbration) in Ho ( M )( S α ) . W e also choose a small set I (resp. J ) of generating coﬁbrations (resp. of generating tr ivial coﬁbration) in Ho ( M )( S ) . Consider the f ollo wing assumptions: (a) W e hav e I ⊂ Ð α ∈ A p ∗ α ( I α ) and J ⊂ Ð α ∈ A p ∗ α ( J α ) . (b) For an y index α 0 , if C α 0 and E α 0 are two objects of M ( S α 0 ) , with C α 0 either a source or a targ et of a map in I α 0 ∪ J α 0 , the natural map lim / / α ∈ A Hom M ( S α ) ( C α , E α ) / / Hom M ( S ) ( C , E ) is bi jectiv e. Proposition 4.3.6 Under the assumptions of 4.3.5 , for any index α 0 ∈ A , the pull- bac k functor p ∗ α 0 : M ( S α 0 ) / / M ( S ) preserves ﬁbrations and trivial ﬁbrations. Mor eov er , giv en an index α 0 ∈ A , as well as tw o objects C α 0 and E α 0 in M ( S α 0 ) , if C α 0 belongs to the smallest full subcategory of T ( S α 0 ) which is closed under ﬁnite homotopy colimits and whic h contains the source and targ ets of I α 0 , then, the canonical map 146 Fibred categories and the six functors formalism lim / / α ∈ A Hom Ho ( M )( S α ) ( C α , E α ) / / Hom Ho ( M )( S ) ( C , E ) is bi jective. Proof W e shall pro v e ﬁrst that, f or an y inde x α 0 ∈ A , the pullback functor p ∗ α 0 preserves ﬁbrations and tr ivial ﬁbrations. By assumption, f or an y α ≥ α 0 , the pullback functor along the P -mor phism S α / / S α 0 is both a left Quillen functor and a r ight Quillen functor . Let E α 0 / / F α 0 be a tr ivial ﬁbration (resp. a ﬁbration) of M ( S α 0 ) . Let i : C / / D a generating coﬁbration (resp. a g enerating trivial coﬁbration) in M ( S ) . By condition (a) of 4.3.5 , we ma y assume that there e xists α 1 ∈ A , a coﬁbration (resp. a trivial coﬁbration) i α 1 : C α 1 / / D α 1 , such that i = p ∗ α 1 ( i α 1 ) . W e want to prov e that the map Hom ( D , E ) / / Hom ( C , E ) × Hom ( C , F ) Hom ( D , F ) is surjectiv e. But, by condition (b) of 4.3.5 , this map is isomor phic to the ﬁltered colimit of the sur jectiv e maps Hom ( D α , E α ) / / Hom ( C α , E α ) × Hom ( C α , F α ) Hom ( D α , F α ) with α ≥ sup ( α 0 , α 1 ) , which pro ves the ﬁrst asser tion. T o prov e the second asser tion, w e may assume that C α 0 is coﬁbrant and that E α 0 if ﬁbrant. The set of maps from a coﬁbrant object to a ﬁbrant object in the homotop y category of a model categor y can be described as homotop y classes of maps. Therefore, using the fact that p ∗ α 0 preserves coﬁbrations and ﬁbrations, as w ell as the tr ivial ones, w e see it is suﬃcient to pro v e that the map lim / / α ∈ A Hom M ( S α ) ( C α , E α ) / / Hom M ( S ) ( C , E ) is bijectiv e f or some nice coﬁbrant replacement of C α 0 . But the assumptions on C α 0 imply that it is weakl y equivalent to an object C 0 α 0 such that the map ∅ / / C 0 α 0 belongs to the smallest class of maps in M ( S α 0 ) , which contains I α 0 , and which is closed under pushouts and (ﬁnite) compositions. W e ma y thus assume that C α 0 = C 0 α 0 . In that case, C α 0 is in par ticular contained in the smallest full subcategor y of M ( S α 0 ) which is stable b y ﬁnite colimits and which contains the source and targ ets of I α 0 . As ﬁltered colimits commute with ﬁnite limits in the category of sets, w e conclude b y using again condition (a) of 4.3.5 .  W e now go back to the situation of a motivic tr iangulated categor y T satisfying our general assumptions 4.0.1 on page 128 Lemma 4.3.7 Let a : X / / Y be a morphism in S . Assume that X = lim o o α X α , wher e ( X α ) α ∈ A is a projectiv e syst em of smooth aﬃne Y -schemes. If T is τ -continuous, then, for any objects E and F in T ( Y ) , with E constructible, then the exchang e mor phism 4 Constr uctible motives 147 a ∗ Hom Y ( E , F ) ' Hom X ( a ∗ ( E ) , a ∗ ( F )) , deﬁned in P aragr aph 1.1.33 , is an isomorphism. The same conclusion holds if T is weakly τ -continuous and the transition maps of ( X α ) ar e dominant. Proof W e ha v e a ∗ Hom X ( a ∗ ( E ) , a ∗ ( F )) ' Hom Y ( E , a ∗ a ∗ ( F )) , so that the map F / / a ∗ a ∗ ( F )) induces a map Hom Y ( E , F ) / / a ∗ Hom X ( a ∗ ( E ) , a ∗ ( F )) , hence, b y adjunction, a map a ∗ Hom Y ( E , F ) / / Hom X ( a ∗ ( E ) , a ∗ ( F )) . W e already kno w that the later is an isomor phism whene v er a is smooth. Let us wr ite a α : X α / / Y f or the str uctural maps. Let C be a constructible object in T ( X ) . By Proposition 4.3.4 , w e ma y assume that there e xists an index α 0 , and a constructible object C α 0 in T ( X α 0 ) , such that, if we wr ite C α f or the pullback of C α 0 along the map X α / / X α 0 f or α ≥ α 0 , w e hav e isomor phisms: Hom ( C , a ∗ Hom Y ( E , F )) ' lim / / α Hom ( C α , a ∗ α Hom Y ( E , F )) ' lim / / α Hom ( C α , Hom X ( a ∗ α ( E ) , a ∗ α ( F ))) ' lim / / α Hom ( C α ⊗ X α a ∗ α ( E ) , a ∗ α ( F )) ' Hom ( C ⊗ X a ∗ ( E ) , a ∗ ( F )) ' Hom ( C , Hom X ( a ∗ ( E ) , a ∗ ( F ))) . As constructible objects generate T ( X ) , this pro v es the ﬁrst asser tion. The second assertion obviousl y follo ws from the same argument.  4.3.8 Let X be a scheme in S . Assume that, for any point x of X , the cor responding morphism i x : Sp ec  O h X , x  / / X is in S (where O h X , x denotes the henselization of O X , x ). Consider at last a scheme of ﬁnite type Y o v er X , and wr ite a x : Y x = Sp ec  O h X , x  × X Y / / Y f or the mor phism obtained by pullback. Finally , f or an object E of T ( Y ) , let us write E x = a ∗ x ( E ) . 148 Fibred categor ies and the six functors formalism Proposition 4.3.9 If the motivic categor y T is weakly τ -continuous, then the family of functors T ( Y ) / / T ( Y x ) , E  / / E x , x ∈ X , is conser v ative. Proof Let E be an object of T ( Y ) such that E x ' 0 f or an y point x of X . F or an y constructible object C of T ( Y ) , w e hav e a presheaf of S 1 -spectra on the small Nisnevic h site of X : F : U  / / F ( U ) = Hom ( M Y ( U × X Y ) , Hom Y ( C , E )) . It is suﬃcient to pro v e that F ( X ) is acy clic. As T satisﬁes Nisne vich descent ( 3.3.4 ), it is suﬃcient to pro v e that F is acyclic locally f or the Nisnevic h topology , i.e. that, for an y point x of X , the spectrum F ( Sp ec  O h X , x  ) is acyclic. W riting Sp ec  O h X , x  as the projectiv e limit of the Nisnevic h neighborhoods of x in X , w e see easil y , using Proposition 4.3.4 and Lemma 4.3.7 , that, for any integer i , π i ( F ( Sp ec  O h X , x  ) ' Hom ( C x , E x [ i ]) ' 0 .  Proposition 4.3.10 Let S be a quasi-excellent noetherian and henselian scheme. W rite ˆ S f or its completion along its closed point, and assume that both S and ˆ S ar e in S . Consider an S -scheme of ﬁnite type X , and write i : ˆ S × S X / / X for the induced map. If T is τ -continuous, then the pullbac k functor i ∗ : T ( X ) / / T ( ˆ S × S X ) is conser v ative. Proof As S is quasi-e xcellent, the map ˆ S / / S is regular . By Popescu ’ s theorem, w e can then wr ite ˆ S = lim o o α S α , where { S α } is a projective sys tem of schemes with aﬃne transition maps, and such that each scheme S α is smooth ov er S . Moreo v er , as ˆ S and S hav e the same residue ﬁeld, and as S is henselian, each map S α has a section. W r ite X α = S α × S X , so that we hav e X = lim o o α X α . Consider a constr uctible object C and an object E in T ( X ) . Then, as the maps X α / / X hav e sections, it f ollo ws from the ﬁrst assertion of Proposition 4.3.4 that the map Hom T ( X ) ( C , E ) / / Hom T ( ˆ S × S X ) ( i ∗ ( C ) , i ∗ ( E )) is a monomorphism (as a ﬁltered colimit of suc h things). Hence, if i ∗ ( E ) ' 0 , f or an y constructible object C in T ( X ) , we ha v e Hom T ( X ) ( C , E ) ' 0 . Theref ore, as τ -constructible objects generate T ( X ) , we get E ' 0 .  Proposition 4.3.11 Let a : X / / Y be a r egular morphism in S . If T is τ - continuous, then, f or any objects E and F in T ( Y ) , with E constructible, ther e is a canonical isomor phism a ∗ Hom Y ( E , F ) ' Hom X ( a ∗ ( E ) , a ∗ ( F )) . 4 Constr uctible motives 149 Proof W e want to prov e that the canonical map a ∗ Hom Y ( E , F ) / / Hom X ( a ∗ ( E ) , a ∗ ( F )) is an isomorphism, while we already kno w it is so whenev er a is smooth. Therefore, to prov e the general case, we see that the problem is local on X and on Y with respect to the Zariski topology . In par ticular , we may assume that both X and Y are aﬃne. By Popescu ’ s Theorem 4.1.5 , we thus hav e X = lim o o α X α , where { X α } is a projectiv e sys tem of smooth aﬃne Y -schemes. W e conclude by Lemma 4.3.7 .  4.3.12 Consider the f ollo wing pullback square in S X 0 a / / g   ∆ X f   Y 0 b / / Y and assume that f is separated of ﬁnite type. Then one gets, using the recipe that w e ha v e seen sev eral times bef ore, the f ollo wing e x chang e transf or mation: E x ( ∆ ∗ ! ) : a ∗ f ! a d ( b ∗ , b ∗ ) / / a ∗ f ! b ∗ b ∗ [ E x ( ∆ ! ∗ )] − 1 / / a ∗ a ∗ g ! b ∗ a d 0 ( a ∗ , a ∗ ) / / g ! b ∗ where E x ( δ ! ∗ ) is the ex chang e isomor phism of Theorem 2.4.50 , point (4). Proposition 4.3.13 Consider the previous notations and assume that b is regular and T is τ -continuous. Then the exc hang e transf or mation deﬁned abov e E x ( ∆ ∗ ! ) : a ∗ f ! / / g ! b ∗ is an isomorphism. Proof The ex chang e transf or mation E x ( ∆ ∗ ! ) is inv er tible whenev er b is smooth: this is ob vious in the case of an open immersion, so that, b y Zariski descent, it is suﬃcient to treat the case where b is smooth with trivial cotang ent bundle of rank d ; in this case, by relative pur ity ( 2.4.50 (3)), this reduces to the canonical isomor phism a ! f ! ' g ! b ! ev aluated at E (− d )[− 2 d ] . T o prov e the g eneral case, as the condition is local on X and on Y f or the Zariski topology , w e ma y assume that f factors as an immersion X / / P n Y , follo w ed b y the canonical projection P n Y / / Y . W e deduce from there that it is suﬃcient to treat the case where f is either a closed immersion, either a smooth mor phism of ﬁnite type. The case where f (hence also g ) is smooth f ollow s by relative purity ( 2.4.50 ): we can then replace f ! and g ! b y f ∗ and g ∗ respectiv ely , and the formula follo ws from the fact that a ∗ f ∗ ' g ∗ b ∗ . W e ma y thus assume that f is a closed immersion. As g is a closed immersion as w ell, the functor g ! is conser vativ e (it is fully faithful). Theref ore, it is suﬃcient to pro v e that the map b ∗ f ! f ! ( E ) ' g ! a ∗ f ! ( E ) / / g ! g ! b ∗ ( E ) 150 F ibred categor ies and the six functors formalism is inv er tible. Then, using Proposition 4.3.11 (which makes sense because the functor f ! preserves τ -constructibility by 4.2.11 ), and the projection f or mula, w e ha v e b ∗ f ! f ! ( E ) ' b ∗ Hom Y ( f ! ( 1 X ) , E ) ' Hom Y 0 ( b ∗ f ! ( 1 X ) , b ∗ ( E )) ' Hom Y 0 ( g ! ( 1 X 0 ) , b ∗ ( E )) ' g ! g ! b ∗ ( E ) , which achiev es the proof.  Lemma 4.3.14 Let f : X / / Y be a morphism in S . Assume that X = lim o o α X α and Y = lim o o α Y α , wher e { X α } and { Y α } are projectiv e sys tems of schemes with aﬃne (resp. aﬃne and dominant) transition maps, while f is induced by a system of morphisms f α : X α / / Y α . Let α 0 be some index, C α 0 a constructible object of T ( Y α 0 ) , and E α 0 an object of T ( X α 0 ) . If T is τ -continuous (resp. w eakly τ - continuous), then w e have a natural isomorphism of abelian gr oups lim / / α ≥ α 0 Hom T ( Y α ) ( C α , f α , ∗ ( E α )) ' Hom T ( Y ) ( C , f ∗ ( E )) . Proof By vir tue of Proposition 4.3.4 , we hav e a natural isomor phism lim / / α ≥ α 0 Hom T ( X α ) ( f ∗ α ( C α ) , E α ) ' Hom T ( Y ) ( f ∗ ( C ) , E ) . The e xpected f or mula f ollow s by adjunction.  Proposition 4.3.15 Consider the f ollowing pullback squar e in S . X 0 a / / g   X f   Y 0 b / / Y with b regular . If T is τ -continuous, then, for any object E in T ( X ) , there is a canonical isomorphism in T ( Y 0 ) : b ∗ f ∗ ( E ) ' g ∗ a ∗ ( E ) . Proof This proposition is tr ue in the case where b is smooth (by deﬁnition of Sm - ﬁbred categor ies), from which we deduce, by Zar iski separation, that this proper ty is local on Y and on Y 0 f or the Zariski topology . In par ticular , we may assume that both Y and Y 0 are aﬃne. Then, b y P opescu ’ s Theorem 4.1.5 , we may assume that Y 0 = lim o o α Y 0 α , where { Y 0 α } is a projective sys tem of smooth Y -algebras. Then, using the preceding lemma as well as Proposition 4.3.4 , we reduce easily the proposition to the case where b is smooth.  4 Constr uctible motives 151 Proposition 4.3.16 Assume that T is w eakly τ -continuous, Q -linear and semi- separat ed, and consider a ﬁeld k , with inseparable closur e k 0 , such that both Sp ec ( k ) and Spec ( k 0 ) ar e in S . Given a k -scheme X write X 0 = k 0 ⊗ k X , and f : X 0 / / X f or the canonical projection. Then the functor f ∗ : T ( X ) / / T ( X 0 ) is an equiv alence of categories. Proof Note that X 0 is a projectiv e limit of k -schemes with aﬃne and dominant (ev en ﬂat) transition maps. Thus, it follo ws from w eak τ -continuity , Proposition 4.3.4 and Proposition 2.1.9 that the functor f ∗ : T c ( X ) / / T c ( X 0 ) is an equivalence of categor ies. Similarl y , for an y objects C and E in T ( X ) , if C is constructible, the map Hom T ( X ) ( C , E ) / / Hom T ( X ) ( f ∗ ( C ) , f ∗ ( E )) is bi jectiv e. As constructible objects generate T ( X ) , this implies that the functor f ∗ : T ( X ) / / T ( X 0 ) is fully faithful. As the latter is essentiall y surjective on a set of g enerators, this implies that it is an equivalence of categories (see 1.3.20 ).  Proposition 4.3.17 Assume that T is weakly τ -continuous. Then, for any sc heme X in S , the family of functors induced by its points x ∗ : T ( X ) / / T ( Sp ec ( κ ( x ) ) , x ∈ X , is conser v ative. Proof W e proceed by induction on the dimension d of X . If d ≤ 0 , this is tr ivial. If d > 0 , using Proposition 4.3.9 , w e may assume that X is local. By induction, the proposition is tr ue on the complement of the closed point of x . Therefore, Proposition 2.3.6 achie v es the proof.  4.4 Duality The aim of this section is to pro ve a local duality theorem in T (see 4.4.21 and 4.4.24 ). If we w ork with rational coeﬃcients, resolution of singularities up to quotient singularities is almost as good as classical resolution of singularities: we hav e the f ollowing replacement of the blow -up formula. 152 Fibred categor ies and the six functors formalism Theorem 4.4.1 Assume that T is Q -linear and separated. Let X be a scheme in S . Consider a proper surjectiv e morphism p : X 0 / / X and a ﬁnite g roup G acting on X 0 ov er X . Assume that ther e is a closed subsc heme Z ⊂ X suc h that U = X − Z is normal, while the induced map p U : U 0 = p − 1 ( U ) / / U is ﬁnite, and the map U 0 / G / / U is generically radicial (i.e. is radicial ov er an open dense subsc heme of U ) — e.g. this situation occurs when p is a Galois alter ation. Then the pullback squar e Z 0 i 0 / / q   X 0 p   Z i / / X (4.4.1.1) induces an homotopy pullback squar e M / /   ( R p ∗ L p ∗ ( M )) G   R i ∗ L i ∗ ( M ) / / ( R i ∗ R q ∗ L q ∗ L i ∗ ( M )) G (4.4.1.2) f or any object M of T ( X ) . Proof W e already kno w that, f or any object N of T ( U ) , the map N / / ( R p U ∗ L p ∗ U ( N )) G is an isomor phism (Corollary 3.3.39 ). The proof is then similar to the proof of condition (iv) of Theorem 3.3.37 .  Remar k 4.4.2 Under the assumptions of the preceding theorem, applying the total derived functor R Hom X (− , E ) to the homotopy pullback square ( 4.4.1.2 ) for M = 1 X , w e obtain the homotopy pushout square ( i ! q ! q ! i ! ( E )) G / /   ( p ! p ! ( E )) G   i ! i ! ( E ) / / E (4.4.2.1) f or an y object E of T ( X ) . Corollary 4.4.3 Assume that T is Q -linear and separated. Let B be a sc heme in S , admitting wide r esolution of singularities up to quo tient singularities. Consider a separat ed B -scheme of ﬁnite type S , endow ed with a closed subsc heme T ⊂ S . The category of constructible objects in T ( S ) is the smallest thic k triangulated subcategory which contains the objects of shape R f ∗ ( 1 X { n } ) f or n ∈ τ , and for f : 4 Constr uctible motives 153 X / / S a projectiv e morphism, with X r egular and connected, suc h that f − 1 ( T ) r ed is either empty , either X itself or the support of a strict normal crossing divisor . Proof Let T ( S ) 0 be the smallest thick triangulated subcategory of T ( S ) which contains the objects of shape R f ∗ ( 1 X { n } ) f or n ∈ τ and f : X / / S a projectiv e morphism with X regular and connected, while f − 1 ( T ) r ed is empty , or X itself, or the suppor t of a s tr ict normal crossing divisor . W e clearl y hav e T ( S ) 0 ⊂ T c ( S ) (Proposition 4.2.11 ). T o prov e the rev erse inclusion, by virtue of Proposition 4.2.13 , it is suﬃcient to pro ve that, f or an y n ∈ τ , and any projective mor phism f : X / / S , the object R f ∗ ( 1 X { n } ) belongs to T ( S ) 0 . W e shall proceed b y induction on the dimension of X . If X is of dimension ≤ 0 , we may replace it b y its reduction, which is regular . If X is of dimension > 0 , b y assumption on B , there e xists a Galois alteration p : X 0 / / X of group G , with X 0 regular and projectiv e ov er S (and in which T becomes either empty , either X 0 itself, either the suppor t of a strict nor mal crossing divisor , in each connected component of X 0 ). Choose a closed subscheme Z ⊂ X , such that U = X − Z is a nor mal dense open subscheme, and such that the induced map r : U 0 = p − 1 ( U ) / / U is a ﬁnite mor phism, and consider the pullback square ( 4.4.1.1 ). As Z and Z 0 = p − 1 ( Z ) are of dimension smaller than the dimension of X , we conclude from the homotop y pullback square obtained by applying the functor R f ∗ to ( 4.4.1.2 ) f or M = 1 X { n } , n ∈ τ .  Deﬁnition 4.4.4 Let S be a scheme in S . An object R of T ( S ) is τ -dualizing if it satisﬁes the f ollo wing conditions. (i) The object R is constructible. (ii) For any constr uctible object M of T ( S ) , the natural map M / / R Hom S ( R Hom S ( M , R ) , R ) is an isomor phism. Remar k 4.4.5 If T is τ -compatible, Q -linear and separated, then, in par ticular , the six operations of Grothendieck preser v e τ -constructibility in T ( 4.2.29 ). Under this assumption, f or any scheme X in S , and any ⊗ -inv er tible object U in T ( X ) which is constructible, its quasi-in v erse is cons tr uctible: the quasi-inv erse of U is simply its dual U ∧ = R Hom ( U , 1 X ) , which is constructible by vir tue of 4.2.25 . Proposition 4.4.6 Assume that T is τ -compatible, Q -linear and separat ed, and consider a scheme X in S . (i) Let R be a τ -dualizing object, and U be a constructible ⊗ -invertible object in T ( X ) . Then U ⊗ L S R is τ -dualizing. (ii) Let R and R 0 be tw o τ -dualizing objects in T ( X ) . Then the evaluation map R Hom S ( R , R 0 ) ⊗ L S R / / R 0 is an isomorphism. Proof This f ollo ws immediately from [ A y o07a , 2.1.139].  154 F ibred categories and the six functors formalism Proposition 4.4.7 Consider an open immersion j : U / / X in S . If R is a τ - dualizing object in T ( X ) , then j ! ( R ) is τ -dualizing in T ( U ) . Proof If M is a constr uctible object in T ( U ) , then j ! ( M ) is constructible, and the map (4.4.7.1) j ! ( M ) / / R Hom X ( R Hom X ( j ! ( M ) , R ) , R ) is an isomor phism. Using the isomor phisms of type M ' j ∗ j ! ( M ) = j ! j ! ( M ) and j ∗ R Hom X ( A , B ) ' R Hom U ( j ∗ ( A ) , j ∗ ( B )) , w e see that the imag e of the map ( 4.4.7.1 ) b y the functor j ∗ = j ! is isomor phic to the map (4.4.7.2) M / / R Hom U ( R Hom U ( M , j ! ( R )) , j ! ( R )) , which prov es the proposition.  Proposition 4.4.8 Let X be a scheme in S , and R an object in T ( X ) . Assume there exists an open cov er X = Ð i ∈ I U i suc h that the r estriction of R on each of the open subsc hemes U i is τ -dualizing in T ( U i ) . Then R is τ -dualizing. Proof W e already kno w that the property of τ -constructibility is local with respect to the Zar iski topology ( 4.2.6 ). Denote by j i : U i / / X the cor responding open immersions, and put R i = j ! i ( R ) . Let M be a constructible object in T ( X ) . Then, f or all i ∈ I , the image b y j ∗ i = j ! i of the map M / / R Hom X ( R Hom X ( M , R ) , R ) is isomor phic to the map j ∗ i ( M ) / / R Hom U i ( R Hom U i ( j ∗ i ( M ) , R i ) , R i ) . This proposition thus f ollo ws from the proper ty of separation with respect to the Zariski topology .  Corollary 4.4.9 Let f : X / / Y be a separated morphism of ﬁnite type in S . Given an object R of T ( Y ) , the property for f ! ( R ) of being a τ -dualizing object in T ( X ) is local ov er X and ov er Y f or the Zariski topology. Proposition 4.4.10 Assume that T is τ -compatible. Let i : Z / / X be a closed immersion and R be a τ -dualizing object in T ( X ) . Then i ! ( R ) is τ -dualizing in T ( Z ) . Proof As T is τ -compatible, w e already kno w that i ! ( R ) is constructible. F or any objects M and R of T ( Z ) and T ( X ) respectivel y , we hav e the identiﬁcation: i ! R Hom Z ( M , i ! ( R )) ' R Hom X ( i ! ( M ) , R ) . 4 Constr uctible motives 155 Let j : U / / X be the complement immersion. Then we hav e j ! R Hom X ( i ! ( M ) , R ) ' R Hom U ( j ∗ i ! ( M ) , j ! ( R )) ' 0 , so that R Hom X ( i ! ( M ) , R ) ' i ! L i ∗ R Hom X ( i ! ( M ) , R ) . As i ! is fully faithful, this pro vides a canonical isomor phism L i ∗ R Hom X ( i ! ( M ) , R ) ' i ! R Hom X , ( i ! ( M ) , R ) . U nder this identiﬁcation, we see easil y that the map i ! ( M ) / / R Hom X ( R Hom X ( i ! ( M ) , R ) , R ) is isomor phic to the image by i ! of the map M / / R Hom Z ( R Hom Z ( M , i ! ( R )) , i ! ( R )) . As i ! is fully faithful, it is conservativ e, and this ends the proof.  Proposition 4.4.11 Assume that T is τ -compatible, Q -linear and separat ed, and consider a sc heme B in S which admits wide resolution of singularities up to quotient singularities. Consider a separ ated B -sc heme of ﬁnite type S , and a constructible object R in T ( S ) . The follo wing conditions are equivalent. (i) F or any separat ed morphism of ﬁnite type f : X / / S , the object f ! ( R ) is τ -dualizing. (ii) F or any projectiv e morphism f : X / / S , the object f ! ( R ) is τ -dualizing. (iii) F or any pr ojective morphism f : X / / S , with X r egular , the object f ! ( R ) is τ -dualizing. (iv) F or any pr ojective morphism f : X / / S , with X r egular , and for any n ∈ τ , the map (4.4.11.1) 1 X { n } / / R Hom X ( R Hom X ( 1 X { n } , f ! ( R )) , f ! ( R )) is an isomorphism in T ( X ) . If, furthermore, f or any r egular separat ed B -sc heme of ﬁnite type X , and for any n ∈ τ , the object 1 X { n } is ⊗ -invertible, then these conditions are equivalent to the f ollowing one. (v) F or any projectiv e morphism f : X / / S , with X r egular , the map (4.4.11.2) 1 X / / R Hom X ( f ! ( R ) , f ! ( R )) is an isomorphism in T ( X ) . Proof It is clear that (i) implies (ii), which implies (iii), which implies (iv). Let us chec k that condition (ii) also implies condition (i). Let f : X / / S be a mor phism 156 Fibred categor ies and the six functors f or malism of separated B -schemes of ﬁnite type, with S regular . W e want to prov e that f ! ( 1 S ) is τ -dualizing, while w e already kno w it is tr ue whenev er f is projective. In the general case, by vir tue of Corollar y 4.4.9 , we may assume that f is quasi-projectiv e, so that f = p j , where p is projectiv e, and j is an open immersion. As f ! ' j ! p ! , w e conclude with Proposition 4.4.7 . Under the additional assumption, the equivalence betw een (iv) and (v) is ob vious. It thus remains to pro ve that (iv) implies (ii). It is in fact suﬃcient to prov e that, under condition (iv), the object R itself is τ -dualizing. T o pro v e that the map (4.4.11.3) M / / R Hom X ( R Hom X ( M , R ) , R ) is an isomor phism f or an y constructible object M of T ( S ) , it is suﬃcient to consider the case where M = R f ∗ ( 1 X { n } ) = f ! ( 1 X { n } ) , where n ∈ τ and f : X / / S is a projectiv e mor phism with X regular (Corollar y 4.4.3 ). For an y object A of T ( X ) , w e hav e canonical isomor phisms R Hom S ( f ! ( A ) , R ) ' R f ∗ R Hom X ( A , f ! ( R )) = f ! R Hom X ( A , f ! ( R )) , from which w e get a natural isomor phism: R Hom S ( R Hom S ( f ! ( A ) , R ) , R ) ' f ! R Hom X ( R Hom X ( A , f ! ( R )) , f ! ( R )) . U nder these identiﬁcations, the map ( 4.4.11.3 ) for M = f ! ( 1 X { n } ) is the image of the map ( 4.4.11.1 ) b y the functor f ! . As ( 4.4.11.1 ) is in v er tible b y assumption, this prov es that R is τ -dualizing.  Lemma 4.4.12 Let X be a scheme in S , and R be an object of T ( X ) . The property f or R of being ⊗ -inv er tible is local ov er X with respect to the Zariski topology . Proof Let R ∧ = R Hom ( R , 1 X ) be the dual of R . The object R is ⊗ -in v er tible if and only if the ev aluation map R ∧ ⊗ L X R / / 1 X is inv er tible. Let j : U / / X be an open immersion. Then, f or an y objects M and N in T ( X ) , w e hav e the identiﬁcation j ∗ R Hom X ( M , N ) ' R Hom U ( j ∗ ( M ) , j ∗ ( N )) . In par ticular , we ha v e j ∗ ( R ∧ ) ' j ∗ ( R ) ∧ . As j ∗ is monoidal, the lemma follo ws from the fact that T has the proper ty of separation with respect to the Zar iski topology .  Deﬁnition 4.4.13 W e shall sa y that T is τ -dualizable if it satisﬁes the f ollo wing conditions: (i) T is τ -compatible ( 4.2.20 ); (ii) f or any closed immersion between regular sc hemes i : Z / / S in S , the ob- ject i ! ( 1 S ) is ⊗ -in vertible (i.e. the functor i ! ( 1 S ) ⊗ L S (−) is an equivalence of categories); 4 Constr uctible motives 157 (ii) f or an y regular scheme X in S , and f or any n ∈ τ , the map 1 X { n } / / R Hom X ( R Hom X ( 1 X { n } , 1 X ) , 1 X ) is an isomor phism. As in other similar situations, we simply say dualizable with respect to T ate twist when the set of twists τ is generated by the T ate twist. Example 4.4.14 In practice, the property of being dualizable with respect to T ate twist is a consequence of the absolute purity theorem. Our main e xample is the motivic category DM B of Beilinson motiv es o ver e x cellent noether ian schemes, as a consequence of Theorem 14.4.1 . Remar k 4.4.15 Note that, whenev er the set of twists τ consists of rigid objects (which will be the case in practice), conditions (i) and (ii) of the preceding deﬁnition are equiv alent to the condition that i ! ( 1 X ) is constructible and ⊗ -in v er tible for any closed immersion i between regular separated schemes in S , while condition (iii) is then automatic. This principle gives easily the proper ty of τ -purity when S is made of schemes of ﬁnite type o ver some per f ect ﬁeld: Proposition 4.4.16 Assume that S consists exactly of schemes of ﬁnite type ov er a ﬁeld k . If the objects 1 { n } are rigid with constructible duals in T ( Sp ec ( k ) ) for all n ∈ τ , then T is τ -dualizable. Proof For an y k -scheme of ﬁnite type f : X / / Sp ec ( k ) , as the functor L f ∗ is symmetric monoidal, the objects 1 X { n } are r igid in T ( X ) for all n ∈ τ . Theref ore, as stated in remark 4.4.15 , we ha v e only to pro v e that, f or any closed immersion i : Z / / X between regular k -schemes of ﬁnite type, the object i ! ( 1 X ) is ⊗ -inv er tible and constr uctible. Note that, as k is perfect, X and Z are in fact smooth. Using 4.4.12 and 4.2.6 , w e may also assume that X is quasi-projectiv e and that Z is purely of codimension c in X , while the normal bundle of i is tr ivial. This proposition is then a consequence of relativ e purity ( 2.4.50 ), which gives a canonical isomorphism i ! ( 1 X ) ' 1 Z (− c )[− 2 c ] .  Proposition 4.4.17 Assume that S consists of schemes of ﬁnite type ov er a ﬁeld k and that T has the f ollowing pr operties: (a) it is τ -dualizable; (b) f or any n ∈ τ , 1 { n } is rigid; (c) either k is per fect, either T is continuous. Then, any constructible object of T ( k ) is rigid. Proof By 4.3.16 , it is suﬃcient to treat the case where k is per f ect. It is well-kno wn that rigid objects form a thick subcategory of T . Thus, we conclude easily from Corollary 4.4.3 and Proposition 2.4.31 .  158 F ibred categor ies and the six functors formalism Lemma 4.4.18 Assume that T is τ -dualizable. Then, for any pr ojective mor phism f : X / / S betw een regular sc hemes in S , t he object f ! ( 1 S ) is ⊗ -invertible and constructible. Proof As, f or an y open immersion j : U / / X , one has j ∗ = j ! , w e deduce easily from Lemma 4.4.12 (resp. Proposition 4.2.6 ) that the proper ty f or f ! ( 1 S ) of being ⊗ -in v er tible (resp. constructible) is local on S f or the Zar iski topology . Theref ore, w e ma y assume that S is separated o v er B and that f factors as a closed immersion i : X / / P n S f ollow ed b y the canonical projection p : P n S / / S . Using relative pur ity f or p , we ha v e the f ollo wing computations: f ! ( 1 S ) ' i ! p ! ( 1 S ) ' i ! ( 1 P n S ( n )[ 2 n ]) ' i ! ( 1 P n S )( n )[ 2 n ] . As i is a closed immersion between regular schemes, the object i ! ( 1 P n S ) is ⊗ -inv er tible and constructible by assumption on T , which implies that f ! ( 1 S ) is ⊗ -in v er tible and constructible as well.  Deﬁnition 4.4.19 Let B a scheme in S . W e shall say that local duality holds ov er B in T if, f or any separated mor phism of ﬁnite type f : X / / S , with S regular and of ﬁnite type ov er B , the object f ! ( 1 S ) is τ -dualizing in T ( X ) . Remar k 4.4.20 By deﬁnition, if T is τ -compatible, and if local duality holds ov er B in T , then the restriction of T to the categor y of B -schemes of ﬁnite type is τ -dualizable. A conv enient suﬃcient condition f or local duality to hold in T is the f ollowing (in particular , using the result below as w ell as Proposition 4.4.16 , local duality holds almost sy stematicall y o v er ﬁelds). Theorem 4.4.21 Assume that T is τ -dualizable, Q -linear and separated, and con- sider a sc heme B in S whic h admits wide resolution of singularities up to quotient singularities (e.g. B might be any sc heme which is separat ed and of ﬁnite type ov er an excellent noetherian scheme of dimension lesser or equal to 2 in S ; see 4.1.11 ). Then local duality holds ov er B in T . Proof Let S be a regular separated B -scheme of ﬁnite type. Then, f or any separated morphism of ﬁnite type f : X / / S , the object f ! ( 1 S ) is τ -dualizing: Lemma 4.4.18 implies immediately condition (iv) of Proposition 4.4.11 . The general case (without the separation assumption on S ) f ollo ws easily from Corollar y 4.4.8 .  Proposition 4.4.22 Consider a sc heme B in S . Assume that T is τ -dualizable, and that local duality holds ov er B in T . Consider a regular B -scheme of ﬁnite type S . (i) An object of T ( S ) is τ -dualizing if and only if it is constructible and ⊗ -inv ertible. (ii) F or any separated morphism of S -schemes of ﬁnit e type f : X / / Y , and for any τ -dualizing object R in T ( Y ) , the object f ! ( R ) is τ -dualizing in T ( X ) . Proof As the unit of T ( S ) is τ -dualizing b y assumption, Proposition 4.4.6 implies that an object of T ( S ) is τ -dualizing if and only if it is constructible and ⊗ -inv er tible. 4 Constr uctible motives 159 Consider a regular B -scheme of ﬁnite type S , as well as a separated mor phism of S -schemes of ﬁnite type f : X / / Y , as w ell as a τ -dualizing object R in T ( Y ) . T o pro v e that f ! ( R ) is τ -dualizing, by vir tue of Corollar y 4.4.8 , we may assume that Y is separated ov er S . Denote by u and v the structural maps from X and Y to S respectiv ely . As w e already know that v ! ( 1 S ) is τ -dualizing, b y vir tue of Proposition 4.4.6 , there e xists a constructible and ⊗ -inv er tible object U in T ( Y ) such that U ⊗ L Y R ' v ! ( 1 S ) . As the functor L f ∗ is symmetric monoidal, it preser v es ⊗ -in vertible objects and their duals, from which w e deduce the follo wing isomor phisms: u ! ( 1 S ) ' f ! v ! ( 1 S ) ' f ! ( U ⊗ L Y R ) ' f ! R Hom Y ( U ∧ , R ) ' R Hom X ( L f ∗ ( U ∧ ) , f ! ( R )) ' R Hom X ( L f ∗ ( U ) ∧ , f ! ( R )) ' L f ∗ ( U ) ⊗ L X f ! ( R ) . The object a ! ( 1 S ) being τ -dualizing, while L f ∗ ( U ) is constructible and inv er tible, w e deduce from Proposition 4.4.6 that f ! ( R ) is τ -dualizing as well.  4.4.23 Assume that T is τ -dualizable, Q -linear and separated. Consider a scheme B in S , such that local duality holds o ver B in T — this is the case if B admits wide resolution of singular ities up to quotient singular ities according to the abo v e Theorem. Consider a ﬁxed regular B -scheme of ﬁnite type S , as well as a constructible and ⊗ -inv er tible object R in T ( S ) (in the case S is of pure dimension d , it might be wise to consider R = 1 S ( d )[ 2 d ] , but an arbitrar y R as abov e is eligible by 4.4.22 ). Then, f or any separated S -scheme of ﬁnite type f : X / / S , w e deﬁne the local duality functor D X : T ( X ) op / / T ( X ) b y the f or mula D X ( M ) = R Hom X ( M , f ! ( R )) . This functor D X is right adjoint to itself. Corollary 4.4.24 Under the abov e assumptions, w e hav e the following properties of the motivic triangulated category T : (a) F or any separated S -scheme of ﬁnite type X , the functor D X pr eser v es con- structible objects. (b) F or any separated S -sc heme of ﬁnite type X , the natural map M / / D X ( D X ( M )) is an isomorphism f or any constructible object M in T ( X ) . 160 Fibred categories and the six functors formalism (c) F or any separ ated S -sc heme of ﬁnite type X , and for any objects M and N in T ( X ) , if N is constructible, then w e have a canonical isomorphism D X ( M ⊗ L X D X ( N )) ' R Hom X ( M , N ) . (d) F or any morphism betw een separ ated S -schemes of ﬁnite type f : Y / / X , we hav e natural isomor phisms D Y ( f ∗ ( M )) ' f ! ( D X ( M )) f ∗ ( D X ( M )) ' D Y ( f ! ( M )) D X ( f ! ( N )) ' f ∗ ( D Y ( N )) f ! ( D Y ( N )) ' D X ( f ∗ ( N )) f or any constructible objects M and N in T ( X ) and T ( Y ) respectiv ely. This corollar y sums up what must be called the Gro thendieck duality property for the motivic tr iangulated category T with respect to the set of twists τ . Proof Asser tions (a) and (b) are only stated for the record 68 ; see 4.2.25 . T o prov e (c), w e see that we ha v e an obvious isomor phism D X ( M ⊗ L X P ) ' R Hom X ( M , D X ( P )) f or any objects M and P . If N is constructible, we may replace P by D X ( N ) and get the expected formula using (b). The identiﬁcation D Y f ∗ ' f ! D X is a special case of the f ormula R Hom Y ( f ∗ ( A ) , f ! ( B )) ' f ! R Hom X ( A , B ) . Theref ore, w e also get: f ∗ D X ' D 2 Y f ∗ D X ' D Y f ! D 2 X ' D Y f ! . The tw o other f or mulas of (d) f ollo w by adjunction.  Theorem 4.4.25 Assume that S consists of sc hemes of ﬁnite type o v er a ﬁeld k . W e consider a τ 0 -g enerated motivic triangulated categor y T 0 ov er S as well as a pr emotivic morphism ϕ ∗ : T / / T 0 . W e suppose that the f ollowing pr operties hold: (a) T is τ -dualizable, Q -linear and separat ed; (b) T 0 is Q -linear and separat ed; 68 W e hav e put to a lot of assumptions here: in f act, if T is τ -dualizable and if local duality holds o ver B in T , the six Grothendieck operations preserve constructible objects on the restr iction of T to B -schemes of ﬁnite type; we leav e this as a formal e xercise f or the reader . 4 Constr uctible motives 161 (b) the object 1 { i } is rigid in T ( k ) for any i ∈ τ . Then, the pr emotivic morphism ϕ ∗ : T c / / T 0 commutes with the six operations. Remar k 4.4.26 Remark that, as a corollar y , we obtain immediately , under the as- sumptions of the theorem that T 0 is ϕ ∗ ( τ ) -dualizable and that the functor ϕ ∗ com- mutes with the duality functors on T and T 0 , respectiv ely obtained b y applying the abo v e corollar y in the case B = Sp ec ( k ) . Proof Giv en a mor phism of ﬁnite type f : X / / Sp ec ( k ) , let us consider the f ollowing proper ty . (∗) f F or any constructible object M in T ( X ) , the natur al exchang e map ϕ ∗ f ∗ ( M ) / / f ∗ ϕ ∗ ( M ) is inv ertible. W e will ﬁrst prov e the theorem assuming that proper ty (∗) f holds f or an y f . Let u : X / / Y be a k -mor phism of ﬁnite type. W e claim that the e x chang e map ϕ ∗ u ∗ ( M ) / / u ∗ ϕ ∗ ( M ) is in vertible f or an y τ -constructible object M of T ( X ) . It is suﬃcient to prov e that, for any smooth separated k -morphism of ﬁnite type g : T / / X , any constr uctible object M in T ( X ) and any twist i in τ 0 , the natural map Hom T 0 ( X ) ( g ] ( 1 T ){ i } , ϕ ∗ u ∗ ( M )) / / Hom T 0 ( X ) ( g ] ( 1 T ){ i } , u ∗ ϕ ∗ ( M )) is bi jective. Consider the f ollo wing commutativ e diagram of morphisms of schemes: V v / / h   T g   X a u / / Y b ~ ~ Sp ec ( k ) in which the square is car tesian. Recall that the functor v ∗ preserves constructible objects b y vir tue of Theorem 4.2.16 . Then w e conclude b y the computations belo w: 162 Fibred categories and the six functors formalism Hom T 0 ( Y ) ( g ] ( 1 T ){ i } , ϕ ∗ u ∗ ( M )) = Hom T 0 ( T ) ( 1 T { i } , g ∗ ϕ ∗ u ∗ ( M )) = Hom T 0 ( T ) ( 1 T { i } , ϕ ∗ g ∗ u ∗ ( M )) = Hom T 0 ( T ) ( g ∗ b ∗ ( 1 k ){ i } , ϕ ∗ g ∗ u ∗ ( M )) = Hom T 0 ( T ) ( g ∗ b ∗ ( 1 k ){ i } , ϕ ∗ v ∗ h ∗ ( M )) = Hom T 0 ( k ) ( 1 k { i } , ( b g ) ∗ ϕ ∗ v ∗ h ∗ ( M )) = Hom T 0 ( k ) ( 1 k { i } , ϕ ∗ ( b g ) ∗ v ∗ h ∗ ( M )) (b y (∗) b g ) = Hom T 0 ( k ) ( 1 k { i } , ( b g v ) ∗ ϕ ∗ h ∗ ( M )) (b y (∗) b g v ) = Hom T 0 ( k ) ( 1 k { i } , ( b g ) ∗ g ∗ u ∗ ϕ ∗ ( M )) = Hom T 0 ( Y ) ( g ] ( 1 T ){ i } , u ∗ ϕ ∗ ( M )) From there, we see that, f or any k -scheme of ﬁnite type X and any τ -constr uctible objects M and N of T ( X ) , the natural map ϕ ∗ ( Hom X ( M , N )) / / Hom X ( ϕ ∗ ( M ) , ϕ ∗ ( N )) is inv er tible in T 0 ( X ) . For this, we may assume that M = f ] ( 1 Y { i } ) f or a smooth morphism of ﬁnite type f : Y / / X and a twist i , in which case we hav e ϕ ∗ ( Hom X ( M , N )) = ϕ ∗ f ∗ f ∗ ( N ) ' f ∗ f ∗ ϕ ∗ ( N ) = Hom X ( ϕ ∗ ( M ) , ϕ ∗ ( N )) . It remains to prov e that f or any separated k -morphism f : X / / Y of ﬁnite type and an y constructible object N in T ( X ) , the e x chang e map: ϕ ∗ f ! ( N ) / / f ! ϕ ∗ ( N ) is an isomorphism. It is easy to see that this proper ty is local for the Zar iski topology , both on X and on Y , so that we may assume that the mor phism f is aﬃne. Therefore, it is suﬃcient to consider the situation where f i either a closed immersion or a separated smooth map. In the smooth case, as the functor f ! is of the form f ∗ ( d )[ 2 d ] , this is obvious. If f = i is a closed immersion with open complement j , as w e already kno w that ϕ ∗ commutes with u ∗ f or any mor phism u , this proper ty follo ws straight a wa y from the localization distinguished triangles i ∗ i ! / / 1 / / j ∗ j ∗ / / . It remains to prov e proper ty (∗) f f or an y mor phism f of ﬁnite type. W e claim it is suﬃcient to pro v e that, for an y k -scheme of ﬁnite type X with structural mor phism f , the follo wing proper ty holds: (∗∗) X F or any twist i ∈ τ , the natural exc hang e map ϕ ∗ f ∗ ( 1 X { i } ) / / f ∗ ϕ ∗ ( 1 X { i } ) is inv ertible. 4 Constr uctible motives 163 Indeed, by vir tue of Theorem 4.2.13 , we may assume that M = w ∗ ( 1 W { i } ) for w : W / / X a projective k -morphism, and i ∈ τ . As the ex change map ϕ ∗ w ∗ / / w ∗ ϕ ∗ is inv er tible (Proposition 2.4.53 ), we see that we may assume that M = 1 X { i } f or some twist i . Let us prov e proper ty (∗∗) X in the case X is in addition smooth o v er k . As ϕ ∗ is monoidal, f or an y r igid object M of T ( k ) , w e get the identiﬁcation: ϕ ∗ ( M ∨ ) = ϕ ∗ ( M ) ∨ . On the other hand, according to assumption (b), the object f ] ( 1 X ) is r igid in T ( k ) as well as in T 0 ( k ) (because the functor ϕ ∗ is symmetr ic monoidal and commutes with the operations of the form f ] f or f smooth). Thus we g et: f ∗ ( 1 X { i } ) = Hom k ( f ] ( 1 X ) , 1 k { i } ) = f ] ( 1 X ) ∨ { i } . Then proper ty (∗∗) X readily f ollo ws. W e ﬁnally prov e proper ty (∗∗) X f or an y algebraic k -scheme X . W e will proceed b y induction on the dimension of X . In case dim ( X ) < 0 , the result is obvious. Let us assume dim ( X ) ≥ 0 . A ccording to the localization proper ty , we can assume that X is reduced. Let ¯ k be an inseparable closure of k and ¯ X = X ⊗ k ¯ k . According to De Jong theorem applied to ¯ X (see Th. 4.1.10 f or S = Spec  ¯ k  ), there e xists a Galois alteration ¯ X 0 / / ¯ X of group G such that ¯ X 0 is smooth o v er ¯ k . W e can assume that such a smooth alteration exis ts o v er a ﬁnite inseparable e xtension ﬁeld E / k . Because T (resp. T 0 ) is Q -linear and separated, the base chang e functor π ∗ associated with the ﬁnite morphism π : Sp ec ( E ) / / Sp ec ( k ) and relativ e to the premotivic category T (resp. T 0 ) is an equivalence of categories (see Proposition 2.1.9 ). Thus we can replace k by E and assume that there e xists a Galois alteration p : X 0 / / X of g roup G such that X 0 is a smooth k -scheme. Using the localization proper ty , w e can assume X is reduced. Then there e xists a nowhere dense closed subscheme ν : Z / / X such that U = X − Z is regular (thus normal) and the induced map p | U : p − 1 ( U ) / / U is ﬁnite. Thus w e can apply Theorem 4.4.1 to the car tesian square: Z 0 ν 0 / / q   X 0 p   Z ν / / X and we g et the distinguished tr iangle in T ( X ) (thus in T 0 ( X ) as w ell, as the functor ϕ ∗ is monoidal and commutes with the operations of the f orm u ∗ f or any proper morphism u ) of the f or m: 1 X { i } / / p ∗ ( 1 X 0 { i } ) G ⊕ ν ∗ ( 1 Z { i } ) / / ( ν q ) ∗ ( 1 Z 0 { i } ) G + 1 / / f or any twist i . If we consider the triangles in T ( k ) and T 0 ( k ) obtained by appl ying the functor f ∗ , where f is the structural mor phism of X / k , we deduce that proper ty 164 Fibred categories and the six functors formalism (∗∗) X f ollow s from proper ties (∗∗) X 0 , (∗∗) Z , (∗∗) Z 0 . Thus w e can conclude applying either the case of a smooth k -scheme treated abo ve or the induction hypothesis as dim ( Z ) = dim ( Z 0 ) < dim ( X ) .  P art II Construction of ﬁbred categories 5 Fibred der iv ed categor ies 167 5 Fibred derived categories 5.0.1 In this entire section, w e ﬁx a full subcategor y S of the category of noether ian S -schemes satisfying the f ollo wing proper ties: (a) S is closed under ﬁnite sums and pullback along mor phisms of ﬁnite type. (b) For any scheme S in S , any quasi-projectiv e S -scheme belongs to S . W e ﬁx an admissible class of mor phisms P of S . All our P -premotivic cat- egories ( cf. deﬁnition 1.4.2 ) are deﬁned o ver S . Moreov er , f or an y abelian P - premotivic category A in this section, we assume the f ollowing: (c) A is a Gr othendiec k abelian P -premotivic category (see deﬁnition 1.3.8 and the recall belo w). (d) A is giv en with a generating set of twists τ . W e sometimes refer to it as the twists of A . (e) W e will denote by M S ( X , A ) , or simply by M S ( X ) , the geometric section o v er a P -scheme X / S . Without precision, an y scheme will be assumed to be an object of S . In section 5.2 , ex cept possibly for 5.2.a , we assume further: (f ) P contains the class of smooth mor phisms of ﬁnite type. 5.0.2 W e will sometimes ref er to the canonical dg-structure of the category of comple x es C ( A ) o v er an abelian category A . Recall that to any complex es K and L o v er A , we associate a complex of abelian groups Hom • A ( K , L ) whose component in degree n ∈ Z is Ö p ∈ Z Hom A ( K p , L p + n ) and whose diﬀerential in deg ree n ∈ Z is deﬁned by the formula: ( f p ) p ∈ Z  / /  d L ◦ f p − (− 1 ) n . f p + 1 ◦ d K )  p ∈ Z . In other words, this is the image of the bicomplex Hom A ( K , L ) by the T ot-product functor which we denote b y T ot π . Of course, the associated homotopy categor y is the category K ( A ) of comple x es up to chain homotopy equivalence. 5.1 From abelian premotiv es to triangulated premotiv es 5.1.a Abelian premotiv es: recall and exam ples Consider an abelian P -premotivic category A . According to the conv ention of 5.0.1 , for an y scheme S , A S is a Grothendieck abelian closed symmetric monoidal category . Moreov er , if τ denotes the twists of A , the essentially small famil y 168 Construction of ﬁbred categories  M S ( X ){ i }  X ∈ P / S , i ∈ τ is a famil y of generators of A S in the sense of [ Gro57 ]. Example 5.1.1 Consider a ﬁx ed ring Λ . Let PSh ( P / S , Λ ) be the category of Λ - preshea v es (i.e. presheav es of Λ -modules) on P / S . For an y P -scheme X / S , we let Λ S ( X ) be the free Λ -presheaf on P / S represented b y X . Then PSh ( P / S , Λ ) is a Grothendieck abelian category generated b y the essentially small famil y  Λ S ( X )  X ∈ P / S . There is a unique symmetr ic closed monoidal s tructure on PSh ( P / S , Λ ) such that Λ S ( X ) ⊗ S Λ S ( Y ) = Λ S ( X × S Y ) . Finall y the e xistence of functors f ∗ , f ∗ and, in the case when f is a P -morphism, of f ] , f ollo ws from general sheaf theory ( cf. [ A GV73 ]). Thus, PSh ( P , Λ ) deﬁnes an abelian P -premotivic category . 5.1.2 Consider an abstract abelian P -premotivic categor y A . T o any premotive M of A S , w e can associate a presheaf of abelian groups X  / / Hom A S ( M S ( X ) , M ) which we denote b y γ ∗ ( M ) . This deﬁnes a functor γ ∗ : A S / / PSh ( P / S , Z ) . It admits the f ollo wing left adjoint: γ ∗ : PSh ( P / S , Z ) / / A S , F  / / lim / / X / F M S ( X , A ) where the colimit r uns o v er the categor y of representable presheav es o v er F . It is now easy to check w e hav e deﬁned a mor phism of (complete) abelian P - premotivic categories: (5.1.2.1) γ ∗ : PSh ( P , Z ) / / o o A : γ ∗ . Moreo v er PSh ( P , Z ) appears as the initial abelian P -premotivic category . Remark that the functor γ ∗ : A S / / PSh ( P / S , Z ) is conser v ativ e if the set of twists τ of A is trivial. Deﬁnition 5.1.3 A P -admissible topology t is a Grothendieck pretopology t on the category S , such that an y t -co v ering famil y consists of P -mor phisms. Note that, f or any scheme S in S , such a topology t induces a pretopology on P / S (which we denote b y the same letter). For any morphism (resp. P -mor phism) f : T / / S , the functor f ∗ (resp. f ] ) preserves t -cov er ing f amilies. As P is ﬁxed in all this section, we will simpl y say admissible f or P -admissible. Example 5.1.4 Let t be an admissible topology . W e denote by Sh t ( P / S , Λ ) the category of t -sheav es of Λ -modules on P / S . Given a P -scheme X / S , we let 5 Fibred der iv ed categor ies 169 Λ t S ( X ) be the free Λ -linear t -sheaf represented b y X . Then, Sh t ( P / S , Λ ) is an abelian Grothendieck categor y with generators ( Λ t S ( X )) X ∈ P / S . As in the preceding ex ample, the categor y Sh t ( P / S , Λ ) admits a unique closed symmetric monoidal structure suc h that Λ t S ( X ) ⊗ S Λ t S ( Y ) = Λ t S ( X × S Y ) . Finall y , f or any mor phism f : T / / S of schemes, the exis tence of functors f ∗ , f ∗ (resp. f ] when f is a P -mor phism) f ollow s from the g eneral theory of shea v es (see ag ain [ A GV73 ]: according to our assumption on t and [ A GV73 , III, 1.6], the functors f ∗ : P / S / / P / T and f ] : P / T / / P / S (f or f in P ) are continuous). Thus, Sh t ( P , Λ ) deﬁnes an abelian P -premotivic category (with trivial set of twists). The associated t -sheaf functor induces a mor phism (5.1.4.1) a ∗ t : PSh ( P , Λ ) / / o o Sh t ( P , Λ ) : a t , ∗ . Remar k 5.1.5 Recall the abelian categor y Sh t ( P / S , Z ) is a localization of the cat- egory PSh ( S , Z ) in the sense of Gabriel-Zisman. In par ticular , giv en an abstract abelian P -premotivic categor y A , the canonical mor phism γ ∗ : PSh ( P / S , Z ) / / o o A S : γ ∗ induces a unique mor phism Sh t ( P / S , Z ) / / o o A S if and only if f or an y presheaf of abelian groups F on P / S such that a t ( F ) = F t = 0 , one has γ ∗ ( F ) = 0 . W e lea v e to the reader the e xercise which consists of f or mulating the univ ersal property of the abelian P -premotivic category Sh t ( P , Z ) . 69 5.1.b The t -descent model category structure 5.1.6 Consider an abelian P -premotivic category A with set of twists τ . W e let C ( A ) be the P -ﬁbred abelian categor y o v er S whose ﬁbers o ver a scheme S is the categor y C ( A S ) of (unbounded) comple x es in A S . For any scheme S , w e let ι S : A S / / C ( A S ) the embedding which sends an object of A S to the corresponding comple x concentrated in deg ree zero. If A is τ -twisted, then the categor y C ( A S ) is obviousl y ( Z × τ ) -twisted. The f ollowing lemma is straightf or w ard: Lemma 5.1.7 Wit h the notations abov e, there is a unique structure of abelian P - pr emotivic category on C ( A ) such that the functor ι : A / / C ( A ) is a mor phism of abelian P -premotivic categories. 69 W e will f ormulate a der ived version in the paragraph on descent proper ties f or derived premotives ( cf. 5.2.9 ). 170 Construction of ﬁbred categories 5.1.8 For a scheme S , let ( P / S ) q be the category introduced in 3.2.1 . The functor M S (−) can be e xtended to ( P / S ) q b y associating to a famil y ( X i ) i ∈ I of P -schemes o v er S the premotive Ê i ∈ I M S ( X i ) . If X is a simplicial object of ( P / S ) q , we denote by M S ( X ) the comple x associated with the simplicial object of A S obtained by applying deg ree wise the abo v e extension of M S (−) . Deﬁnition 5.1.9 Let A be an abelian P -premotivic categor y and t be an admissible topology . Let S be a scheme and C be an object of C ( A S ) : 1. The complex C is said to be local (with respect to the geometric section) if, for an y P -scheme X / S and an y pair ( n , i ) ∈ Z × τ , the canonical mor phism Hom K ( A S ) ( M S ( X ){ i } [ n ] , C ) / / Hom D ( A S ) ( M S ( X ){ i } [ n ] , C ) is an isomor phism. 2. The complex C is said to be t -ﬂasque if f or any t -h yperco v er X / / X in P / S , f or an y ( n , i ) ∈ Z × τ , the canonical mor phism Hom K ( A S ) ( M S ( X ){ i } [ n ] , C ) / / Hom K ( A S ) ( M S ( X ) { i } [ n ] , C ) is an isomor phism. W e say the abelian P -premotivic categor y A satisﬁes cohomological t -descent if f or an y t -h yperco v er X / / X of a P -scheme X / S , and f or any i ∈ τ , the map M S ( X ) { i } / / M S ( X ){ i } is a quasi-isomorphism (or equiv alently , if any local complex is t -ﬂasque). W e sa y that A is compatible with t if A satisﬁes cohomological t -descent, and if, f or an y scheme S , an y t -ﬂasque complex of A S is local. Example 5.1.10 Consider the notations of 5.1.4 . Consider the canonical dg-structure on C ( Sh t ( P / S , Λ ) ) (see 5.1.1 ). By deﬁnition, f or an y complex es D and C of sheav es, we get an equality : Hom K ( Sh t ( P / S , Λ ) ) ( D , C ) = H 0 ( Hom • Sh t ( P / S , Λ ) ( D , C )) = H 0 ( T ot π Hom Sh t ( P / S , Λ ) ( D , C )) . In the case where D = Λ t S ( X ) (resp. D = Λ t S ( X ) ) f or a P -scheme X / S (resp. a simplicial P -scheme o v er S ) we obtain the follo wing identiﬁcation: Hom K ( Sh t ( P / S , Λ ) ) ( Λ t S ( X ) , C ) = H 0 ( C ( X )) . (resp. Hom K ( Sh t ( P / S , Λ ) ) ( Λ t S ( X ) , C ) = H 0 ( T ot π C ( X )) ) . 5 Fibred der iv ed categor ies 171 Thus, w e get the f ollowing equivalences: C is local ⇔ for any P -scheme X / S , H n t ( X , C ) ' H n ( C ( X )) . C is t -ﬂasque ⇔ f or any t -hyperco ver X / / X , H n ( C ( X )) ' H n ( T ot π C ( X )) . A ccording to the computation of cohomology with h yperco v ers ( cf. [ Bro74 ]), if the complex C is t -ﬂasque, it is local. In other w ords, w e hav e the expected proper ty that the abelian P -premotivic category Sh t ( P , Λ ) is compatible with t . 5.1.11 Consider an abelian P -premotivic categor y A and an admissible topology t . Fix a base scheme S . A mor phism p : C / / D of complex es on A S is called a t -ﬁbration if its kernel is a t -ﬂasque comple x and if f or an y P -scheme X / S , any i ∈ τ and any integ er n ∈ Z , the map of abelian groups Hom A S ( M S ( X ){ i } , C n ) / / Hom A S ( M S ( X ){ i } , D n ) is surjective. For any object A of A S , we let S n A (resp. D n A ) be the comple x with only one non-trivial ter m (resp. tw o non-tr ivial terms) equal to A in degree n (resp. in degree n and n + 1 , with the identity as onl y non-trivial diﬀerential). W e deﬁne the class of coﬁbrations as the smallest class of mor phisms of C ( A S ) which: 1. contains the map S n + 1 M S ( X ){ i } / / D n M S ( X ){ i } f or any P -scheme X / S , an y i ∈ τ , and any integer n ; 2. is stable b y pushout, transﬁnite composition and retract. A complex C is said to be coﬁbrant if the canonical map 0 / / C is a coﬁbration. For instance, for any P -scheme X / S and an y i ∈ τ , the comple x M S ( X ){ i } [ n ] is coﬁbrant. Let G S be the essentiall y small f amily made of premotiv es M S ( X ){ i } for a P - scheme X / S and a twist i ∈ τ , and H S be the famil y of comple xes of the f or m Cone ( M S ( X ) { i } / / M S ( X ){ i } ) f or an y t -hyperco ver X / / X and any twist i ∈ τ . By the very deﬁnition, as A is compatible with t (deﬁnition 5.1.9 ), ( G S , H S ) is a descent structure on A S in the sense of [ CD09 , def. 2.2]. Moreo v er , it is w eakl y ﬂat in the sense of [ CD09 , par . 3.1]. Thus the follo wing proposition is a particular case of [ CD09 , theorem 2.5, proposition 3.2, and corollary 5.5]: Proposition 5.1.12 Let A be an abelian P -pr emotivic categor y, which we as- sume to be compatible with an admissible topology t . Then f or any scheme S , the category C ( A S ) with the pr eceding deﬁnition of ﬁbrations and coﬁbrations, with quasi-isomorphisms as w eak equiv alences is a pr oper symmetric monoidal model category. 5.1.13 W e will call this model structure on C ( A S ) the t -descent model cat egor y structur e (ov er S ). Note that, for an y P -scheme X / S and any twist i ∈ τ , the comple x M S ( X ){ i } concentrated in degree 0 is coﬁbrant by deﬁnition, as well as any 172 Construction of ﬁbred categor ies of its suspensions and twists. The y form a famil y of generators f or the tr iangulated category D ( A S ) . Observe also that the ﬁbrant objects for the t -descent model categor y structure are e xactl y the t -ﬂasque complex es in A S . Moreo v er , essentiall y by deﬁnition, a comple x of A S is local if and only if it is t -ﬂasque (see [ CD09 , 2.5]). 5.1.14 Consider again the notations and hypothesis of 5.1.11 . Consider a mor phism of schemes f : T / / S . Then the functor f ∗ : C ( A S ) / / C ( A T ) sends G S in G T , and H S in H T because the topology t is admissible. This means it satisﬁes descent according to the deﬁnition of [ CD09 , 2.4]. Applying theorem 2.14 of op. cit. , the functor f ∗ preserves coﬁbrations and tr ivial coﬁbrations, i.e. the pair of functors ( f ∗ , f ∗ ) is a Quillen adjunction with respect to the t -descent model category str uctures. Assume that f is a P -mor phism. Then, similarl y , the functor f ] : C ( A T ) / / C ( A S ) sends G S (resp. H S ) in G T (resp. H T ) so that f ] also satisﬁes descent in the sense of op. cit . Theref ore, it preserv es coﬁbrations and tr ivial coﬁbrations, and the pair of adjoint functors ( f ] , f ∗ ) is a Quillen adjunction f or the t -descent model categor y structures. In other w ords, we hav e obtained the f ollo wing result. Corollary 5.1.15 Let A be an abelian P -pr emotivic category compatible with an admissible topology t . The P -ﬁbred categor y C ( A ) with the t -descent model cate- gor y structure deﬁned in 5.1.12 is a symmetric monoidal P -ﬁbr ed model categor y . Mor eov er , it is stable, proper and combinatorial. 5.1.16 Recall the f ollowing consequences of this corollary (see also 1.3.23 for the general theory). Consider a mor phism f : T / / S of schemes. Then the pair of adjoint functors ( f ∗ , f ∗ ) admits total left/r ight derived functors L f ∗ : D ( A S ) / / o o D ( A T ) : R f ∗ . More precisely , f ∗ (resp. f ∗ ) preser v es t -local (resp. coﬁbrant) comple x es. For any comple x K on A S , R f ∗ ( K ) = f ∗ ( K 0 ) (resp. L f ∗ ( K ) = f ∗ ( K 00 ) ) where K 0 / / K (resp. K / / K 00 ) is a t -local (resp. coﬁbrant) resolution of K . 70 When f is a P -mor phism, the functor f ∗ is ev en e xact and thus preser v es quasi- isomorphisms. This implies that L f ∗ = f ∗ . The functor f ] admits a total left der iv ed functor L f ] : D ( A T ) / / o o D ( A S ) : R f ∗ 70 Recall also that ﬁbrant/coﬁbrant resolutions can be made functorially , because our model cate- gories are coﬁbrantly generated, so that the left or right derived functors are in f act deﬁned at the lev el of complex es. 5 Fibred der iv ed categor ies 173 deﬁned b y the f ormula L f ] ( K ) = f ] ( K 00 ) f or a comple x K on A T and a coﬁbrant resolution K 00 / / K . Note also that the tensor product (resp. internal Hom) of C ( A S ) admits a total left der iv ed functor (resp. total r ight derived functor). For any comple xes K and L on A S , this der iv ed functors are deﬁned by the formula: K ⊗ L S L = K 00 ⊗ S L 00 R Hom S ( K , L ) = Hom S ( K 00 , L 0 ) where K / / K 00 and L / / L 00 are coﬁbrant resolutions and L 0 / / L is a t -local resolution. It is now easy to chec k that these functors deﬁne a triangulated P -premotivic category D ( A ) , which is τ -generated according to 5.1.13 . Deﬁnition 5.1.17 Let A be an abelian P -premotivic categor y compatible with an admissible topology t . The triangulated P -premotivic category D ( A ) deﬁned abo v e is called the deriv ed P -premo tivic category associated with A . 71 The g eometr ic section of a P -scheme X / S in the categor y D ( A ) is the comple x concentrated in degree 0 equal to the object M S ( X ) . The triangulated P -ﬁbred category is τ -g enerated and w ell generated in the sense of 1.3.15 . Recall this means that D ( A S ) is equal to the localizing 72 subcategor y generated by the famil y (5.1.17.1) { M S ( X ){ i } ; X / S P -scheme , i ∈ τ } . Example 5.1.18 Giv en any admissible topology t , the abelian P -premotivic categor y Sh t ( P , Λ ) introduced in e xample 5.1.4 is compatible with t ( cf. 5.1.10 ) and deﬁnes the derived P -premotivic categor y D ( Sh t ( P , Λ ) ) . Remark also that the abelian P -premotivic categor y PSh ( P , Λ ) introduced in e xample 5.1.1 is compatible with the coarse topology and gives the der iv ed P - premotivic category D ( PSh ( P , Λ )) . Remar k 5.1.19 Recall from 5.0.2 there exis ts a canonical dg-structure on C ( A S ) . Then we can deﬁne a der iv ed dg-structure by deﬁning f or an y comple x es K and L of A S , the comple x of mor phisms: R Hom A S ( K , L ) = Hom • A S ( Q ( K ) , R ( L )) where R and Q are respectivel y some ﬁbrant and coﬁbrant (functor ial) resolutions f or the t -descent model structure. The homotop y categor y associated with this ne w dg-structure on C ( A S ) is the derived categor y D ( A S ) . Moreo v er , f or any morphism (resp. P -morphism) of schemes f , the pair ( L f ∗ , R f ∗ ) (resp. ( L f ] , f ∗ ) ) is a dg- adjunction. The same is tr ue f or the pair of bifunctors ( ⊗ L S , R Hom S ) . 71 Indeed remark that D ( A ) does not depend on the topology t . 72 i.e. tr iangulated and stable by sums. 174 Construction of ﬁbred categories 5.1.20 Consider an abelian P -premotivic category A compatible with a topology t . According to section 3.1.b , the 2 -functor D ( A ) can be e xtended to the category of S -diagrams: to any diagram of schemes X : I / / S inde xed by a small categor y I , w e can associate a closed symmetr ic monoidal tr iangulated categor y D ( A )( X , I ) which coincides with D ( A )( X ) when I = e , X = X for a scheme X . Let us be more speciﬁc. The ﬁbred category A admits an e xtension to S - diagrams: a section of A ov er a diagram of schemes X : I / / S , inde xed b y a small category I , is the f ollo wing data: 1. A famil y ( A i ) i ∈ I such that A i is an object of A X i . 2. A famil y ( a u ) u ∈ F l ( I ) such that for an y ar ro w u : i / / j in I , a u : u ∗ ( A j ) / / A i is a mor phism in A X i and this famil y of mor phisms satisﬁes a cocyle condition (see paragraph 3.1.1 ). Then, D ( A )( X , I ) is the deriv ed category of the abelian categor y A ( X , I ) . In particular, objects of D ( A )( X , I ) are comple x es of sections of A o ver ( X , I ) (or , what amount to the same thing, families of comple xes ( K i ) i ∈ I with transition maps ( a u ) as abo v e, relative to the ﬁbred category C ( A ) ). Recall that a morphism of S -diagrams ϕ : ( X , I ) / / ( Y , J ) is giv en by a functor f : I / / J and a natural transf or mation ϕ : X / / Y ◦ f . W e say that ϕ is a P -mor phism if f or any i ∈ I , ϕ i : X i / / Y f ( i ) is a P -mor phism. For any morphism (resp. P -morphism) ϕ , we ha v e deﬁned in 3.1.3 adjunctions of (abelian) categories: ϕ ∗ : A ( Y , J ) / / o o A ( X , I ) : ϕ ∗ resp. ϕ ] : A ( X , I ) / / o o A ( Y , J ) : ϕ ∗ which extends the adjunctions we had on tr ivial diagrams. A ccording to Proposition 3.1.11 , these respectiv e adjunctions admits left/right derived functors as f ollo ws: L ϕ ∗ : D ( A )( Y , J ) / / o o D ( A )( X , I ) : R ϕ ∗ (5.1.20.1) resp. L ϕ ] : D ( A )( X , I ) / / o o D ( A )( Y , J ) : L ϕ ∗ = ϕ ∗ (5.1.20.2) Ag ain, these adjunctions coincide on tr ivial diagrams with the map we already had. Note also that the symmetr ic closed monoidal structure on C ( A ( X , I )) can be de- rived and induces a symmetric monoidal structure on D ( A )( X , I ) (see Proposition 3.1.24 ). 73 Recall from 3.2.5 and 3.2.7 that, giv en a topology t 0 (not necessarily admissible) o v er S , we say that D ( A ) satisﬁes t 0 -descent if f or an y t 0 -h yperco v er p : X / / X (here X is considered as a S -diagram), the functor (5.1.20.3) L p ∗ : D ( A )( X ) / / D ( A )( X ) 73 In fact, D ( A ) is then a monoidal P c art -ﬁbred category ov er the categor y of S -diagrams (remark 3.1.21 ). 5 Fibred der iv ed categor ies 175 is fully faithful (see Corollary 3.2.7 ). Proposition 5.1.21 Consider the notations and hypothesis introduced abov e. Let t 0 be an admissible topology on S . Then the f ollowing conditions ar e equivalent : (i) D ( A ) satisﬁes t 0 -descent, in the sense recalled abov e. (ii) A satisﬁes cohomological t 0 -descent. Proof W e prov e (i) implies (ii). Consider a t 0 -h yperco v er p : X / / X in P / S . This is a P -mor phism. Thus, b y the fully faithfulness of ( 5.1.20.3 ), the counit map L p ] p ∗ / / 1 is an isomor phism. By appl ying the latter to the unit object 1 X of D ( A X ) , w e thus obtain that M X ( X ) / / 1 X is an isomorphism in D ( A X ) . If π : X / / S is the structural P -morphism, by applying the functor L π ] to this isomor phism, w e obtain that M S ( X ) / / M S ( X ) is an isomor phism in D ( A S ) and this concludes. Reciprocall y , to prov e (i), we can restrict to t 0 -h yperco v ers p : X / / X which are P -mor phisms because t 0 is admissible. Because R p ∗ = p ∗ admits a left adjoint L p ] , w e hav e to prov e that the counit L p ] p ∗ / / 1 is an isomor phism. This is a natural transf ormation between triangulated functors which commutes with small sums. Thus, according to ( 5.1.17.1 ), we ha ve only to chec k this is an isomorphism when e valuated at a comple x of the form M X ( Y ) { i } f or a P -scheme Y / X and a twist i ∈ τ . But the resulting mor phism is then M X ( X × X Y ) { i } / / M X ( Y ) { i } and w e can conclude because X × X Y / / Y is a t 0 -h yperco v er in P / S (again because t 0 is admissible).  5.1.22 . Consider the situation of 5.1.20 Let S be a scheme. An interesting par ticular case is giv en f or constant S -diag rams ov er S ; for a small categor y I , w e let I S be the constant S -diagram I / / S , i  / / S , u  / / 1 S . Then the adjunctions ( 5.1.20.1 ) f or this kind of diagrams deﬁne a Gro thendiec k derivat or I  / / D ( A )( I S ) . Recall that, if f : I / / e is the canonical functor to the ter minal category and ϕ = f X : I X / / X the cor responding mor phism of S -diagrams, f or any I -diagram K • = ( K i ) i ∈ I of complex es o v er A S , we get right derived limits and left derived colimits: R ϕ ∗ ( K • ) = R lim o o i ∈ I K i . L ϕ ] ( K • ) = L lim / / i ∈ I K i . 176 Construction of ﬁbred categor ies 5.1.23 The associated der iv ed P -premotivic category is functorial in the f ollowing sense. Consider an adjunction ϕ : A / / o o B : ψ of abelian P -premotivic categor ies. Let τ (resp. τ 0 ) be the set of twists of A (resp. B ), and recall that ϕ induces a mor phism of monoid τ / / τ 0 still denoted by ϕ . Consider tw o topologies t and t 0 such that t 0 is ﬁner than t . Suppose A (resp. B ) is compatible with t (resp. t 0 ) and let ( G A S , H A S ) (resp. ( G B S , H B S ) ) be the descent structure on A S (resp. B S ) deﬁned in 5.1.11 . For any scheme S , consider the evident e xtensions ϕ S : C ( A S ) / / o o C ( B S ) : ψ S of the abov e adjoint functors to comple x es. Recall that for an y P -scheme X / S and an y twist i ∈ τ , ϕ S ( M S ( X , A ) { i }) = M S ( X , B ){ ϕ ( i ) } b y deﬁnition. Thus, ϕ S sends G A S to G A S . Because t 0 is ﬁner than t , it sends also H A S to H B S . In other words, it satisﬁes descent in the sense of [ CD09 , par . 2.4] so that the pair ( ϕ S , ψ S ) is a Quillen adjunction with respect to the respectiv e t -descent and t 0 -descent model structure on C ( A S ) and C ( B S ) . Considering the deriv ed functors, it is now easy to check we hav e obtained a P -premotivic adjunction 74 L ϕ : D ( A ) / / o o D ( B ) : R ψ . Example 5.1.24 Let t be an admissible topology . Consider an abelian P -premotivic category A compatible with t . Then the morphism of abelian P -premotivic cate- gories ( 5.1.2.1 ) induces a mor phism of tr iangulated P -premotivic categor ies: (5.1.24.1) L γ ∗ : D ( PSh ( P , Z )) / / o o D ( A ) : R γ ∗ Similarl y , the morphism ( 5.1.4.1 ) induces a mor phism of tr iangulated P -premotivic categories (5.1.24.2) a ∗ t : D ( PSh ( P , Λ )) / / o o D ( Sh t ( P , Λ ) ) : R a t , ∗ . Note that a ∗ t = L a ∗ t on objects, because the functor a ∗ t is e xact. Example 5.1.25 Consider an admissible topology t . Let ϕ : Λ / / Λ 0 be a mor phism of rings. For any scheme S , it induces a pair of adjoint functors: 74 Remark also that this adjunction extends on S -diag rams consider ing the situation descr ibed in 5.1.20 : f or any diagram X : I / / S , w e get an adjunction L ϕ X : D ( A )( X ) / / o o D ( B )( X ) : R ψ X and this deﬁnes a mor phism of triangulated monoidal P c art -ﬁbred categor ies o v er the S -diag rams ( cf. Proposition 3.1.32 ). 5 Fibred der iv ed categor ies 177 (5.1.25.1) ϕ ∗ : Sh t ( P S , Λ ) / / o o Sh t ( P S , Λ 0 ) : ϕ ∗ such that ϕ ∗ (resp. ϕ ∗ ) is induced b y the ob vious extension (resp. restriction) of scalars functor . By deﬁnition, f or any P -scheme X / S , the functor ϕ ∗ sends the representable sheaf of Λ -modules Λ t S ( X ) to the representable sheaf of Λ 0 -modules Λ 0 t S ( X ) . Thus ( ϕ ∗ , ϕ ∗ ) deﬁnes an adjunction of abelian P -premotivic categor ies. Applying the results of Paragraph 5.1.23 , one deduces a P -premotivic adjunction: L ϕ ∗ : D ( Sh t ( P , Λ ) ) / / o o D ( Sh t ( P , Λ 0 ) ) : R ϕ ∗ . The functor ϕ ∗ is e xact so that R ϕ ∗ = ϕ ∗ . Similarl y when Λ 0 / Λ is ﬂat, L ϕ ∗ = ϕ ∗ . The follo wing result can be used to check the compatibility to a given admissible topology: Proposition 5.1.26 Let t be an admissible topology . Consider a morphism of abelian P -premo tivic categories ϕ : A / / o o B : ψ suc h that : (a) F or any scheme S , ψ S is exact. (b) The morphism ϕ induces an isomor phism of the underlying set of twists of A and B . Accor ding to the last property , w e identify the set of twists of A and B to a monoid τ in such a w ay that ϕ acts on τ by the identity. Assume that A is compatible with t . Then the follo wing conditions are equiv alent: (i) B is compatible with t . (ii) B satisﬁes cohomological t -descent, Proof The fact ( i ) implies ( ii ) is clear from the deﬁnition, and we pro v e the con v erse using the f ollo wing lemma: Lemma 5.1.27 Consider a mor phism of P -premo tivic abelian categories ϕ : A / / o o B : ψ satisfying conditions (a) and (b) of the abov e proposition and a base scheme S . Giv en a simplicial scheme X which is degr ee-wise a sum of P -sc hemes ov er S , a twist i ∈ τ and a complex C ov er B S , w e denote by  X , i , C : Hom C ( B S )  M S ( X , B ) { i } , C  / / Hom C ( A S )  M S ( X , A ){ i } , ψ S ( C )  the adjunction isomorphism obtained for the adjoint pair ( ϕ S , ψ S ) . Then ther e exists a unique isomorphism  0 X , i , C making the follo wing diagr am com- mutativ e: 178 Construction of ﬁbred categories Hom C ( B S )  M S ( X , B ) { i } , C   X , i , C / /   Hom C ( A S )  M S ( X , A ){ i } , ψ S ( C )    Hom K ( B S )  M S ( X , B ) { i } , C   0 X , i , C / / Hom K ( A S )  M S ( X , A ){ i } , ψ S ( C )  . Assume mor eov er that B satisﬁes cohomological t -descent. Then ther e exists an isomor phism  00 X , i , C making the f ollowing diagr am commutativ e: Hom K ( B S )  M S ( X , B ) { i } , C   0 X , i , C / / π B X , i , C   Hom K ( A S )  M S ( X , A ){ i } , ψ S ( C )  π A X , i , C   Hom D ( B S )  M S ( X , B ) { i } , C   0 0 X , i , C / / Hom D ( A S )  M S ( X , A ){ i } , ψ S ( C )  , (5.1.27.1) wher e π A X , i , C and π B X , i , C ar e induced by the obvious localization functors.  The e xistence and unicity of isomor phism  0 X , i , C f ollow s from the fact that the functors ϕ S and ψ S are additive. Indeed, this implies that the isomor phism  X , i , C is compatible with chain homotopies. Consider the injectiv e model structure on C ( A S ) and C ( B S ) (see f or e xample [ CD09 , 1.2] f or the deﬁnition). W e ﬁrst treat the case when C is ﬁbrant f or this model structure on C ( B S ) . Because the premotiv e M S ( X , B ) { i } is coﬁbrant f or the injective model structure, we obtain that the canonical map π B X , i , C is an iso- morphism. This implies there e xists a unique map  00 X , i , C making diag ram ( 5.1.27.1 ) commutativ e. On the other hand, the isomorphism  0 X , i , C obtained previousl y is ob- viously functor ial in X . Thus, because B satisﬁes t -descent, we obtain that ψ S ( C ) is t -ﬂasque. Because A is compatible with t , this implies ψ S ( C ) is t -local, and because M S ( X , B ) { i } is coﬁbrant f or the t -descent model structure on C ( A S ) , this implies π B X , i , C is an isomor phism. Thus ﬁnally ,  00 X , i , C is an isomor phism as required. T o treat the general case, we consider a ﬁbrant resolution C / / D f or the injectiv e model structure on C ( B S ) . Because ψ S is exact, it preser v es isomor phisms. Using the previous case, W e deﬁne  00 X , i , C b y the f ollowing commutative diagram: Hom D ( B S )  M S ( X , B ) { i } , C   0 0 X , i , C / / ∼   Hom D ( A S )  M S ( X , A ){ i } , ψ S ( C )  ∼   Hom D ( B S )  M S ( X , B ) { i } , D   0 0 X , i , D / / Hom D ( A S )  M S ( X , A ){ i } , ψ S ( D )  . The required property f or  00 X , i , C then f ollo ws easily and the lemma is prov ed. T o ﬁnish the proof that (ii) implies (i), w e note the lemma immediately implies, under (ii), that the follo wing tw o conditions are equivalent: 5 Fibred der iv ed categor ies 179 • C is t -ﬂasque (resp. local) in C ( B S ) ; • ψ S ( C ) is t -ﬂasque (resp. local) in C ( A S ) . This concludes.  5.1.c Constructible premotivic complex es Deﬁnition 5.1.28 Let A be an abelian P -premotivic categor y compatible with an admissible topology t . W e will sa y that t is bounded in A if f or an y scheme S , there e xists an essentially small f amily N t S of bounded complex es which are direct factors of ﬁnite sums of objects of type M S ( X ){ i } in each degree, such that, f or an y comple x C of A S , the f ollo wing conditions are equivalent. (i) C is t -ﬂasque. (ii) For any H in N t S , the abelian group Hom K ( A S ) ( H , C ) vanishes. In this case, w e sa y the f amily N t S is a bounded g enerating family for t -hyperco- v erings in A S . Example 5.1.29 1. Assume P contains the open immersions so that the Zar iski topology is admissible. Let M V S to be the famil y of comple xes of the form Λ S ( U ∩ V ) l ∗ − k ∗ / / Λ S ( U ) ⊕ Λ S ( V ) i ∗ + j ∗ / / Λ S ( X ) f or an y open cov er X = U ∪ V , where i , j , k , l denotes the obvious open immer - sions. It f ollow s then from [ BG73 ] that M V S is a bounded generating famil y of Zariski hyperco vers in Sh Zar ( P / S , Λ ) . 2. Assume P contains the étale morphisms so that the Nisnevich topology is admissible. W e let B G S be the famil y of comple x es of the form Λ S ( W ) g ∗ − l ∗ / / Λ S ( U ) ⊕ Λ S ( V ) j ∗ + f ∗ / / Λ S ( X ) f or a Nisnevic h distinguished square in S ( cf. 2.1.11 ) W l / / g   V f   U j / / X . Then, b y applying 3.3.2 , we see that B G S is a bounded generating f amily f or Nisnevic h h yperco v ers in Sh Nis ( P / S , Λ ) . 3. Assume that P = S f t is the class of mor phisms of ﬁnite type in S . W e let PC D H S be the famil y of comple x es of the form Λ S ( T ) g ∗ − k ∗ / / Λ S ( Z ) ⊕ Λ S ( Y ) i ∗ + f ∗ / / Λ S ( X ) f or a c dh -distinguished square in S ( cf. 2.1.11 ) 180 Cons truction of ﬁbred categor ies T k / / g   Y f   Z i / / X . Then, by vir tue of 3.3.8 , C D H S = B G S ∪ P C D H S is a bounded generating famil y for c dh -h yperco vers in Sh c dh  S f t / S , Λ  . 4. The étale topology is not bounded in Sh ´ e t ( Sm , Λ ) f or an arbitrar y r ing Λ . Ho w ev er , if Λ = Q , it is bounded: by vir tue of Theorem 3.3.23 , a bounded generating famil y for étale hyperco vers in Sh ´ e t ( Sm , Q ) S is the union of the class BG S and that of complex es of the form Q S ( Y ) G / / Q S ( X ) for an y Galois co v er Y / / X of g roup G . 5. As in the case of the étale topology , the qfh -topology is not bounded in general, but it is so with rational coeﬃcients. Let P Q F H S be the famil y of comple x es of the f orm Q S ( T ) G g ∗ − k ∗ / / Q S ( Z ) ⊕ Q S ( Y ) G i ∗ + f ∗ / / Q S ( X ) f or a qfh -distinguished square of group G in S ( cf. 3.3.15 ) T k / / g   Y f   Z i / / X . Then, by vir tue of Theorem 3.3.25 , Q F H S = P Q F H S ∪ BG S is a bounded generating famil y f or qfh -hyperco vers in Sh qfh  S f t / S , Q  . 6. Similarl y , b y Theorem 3.3.30 , H S = C D H S ∪ Q F H S is a bounded generating famil y for h -hyperco vers in Sh h  S f t / S , Q  . Proposition 5.1.30 Let A be an abelian P -pr emotivic cat egor y compatible with an admissible topology t . W e make the follo wing assumptions: (a) t is bounded in A ; (b) f or any P -mor phism X / / S and any n ∈ τ , the functor Hom A S ( M S ( X ){ n } , −) pr eser v es ﬁlter ed colimits. Then t -local complexes are stable by ﬁltering colimits. Proof Let N t S is a bounded generating f amily f or t -hyperco v ers in A S . Then a comple x C of A S is t -ﬂasque if and only if for an y H ∈ N t S , the abelian g roup Hom K ( A S ) ( H , C ) is tr ivial. Hence it is suﬃcient to prov e that the functor C  / / Hom K ( A S ) ( H , C ) preserves ﬁlter ing colimits of complex es. This will f ollo w from the fact that the functor C  / / Hom C ( A S ) ( H , C ) preserves ﬁltering colimits. As H a is bounded comple x that is degreewise compact, this latter proper ty is obvious.  5 Fibred der iv ed categor ies 181 5.1.31 Consider an abelian P -premotivic categor y A compatible with an admissi- ble topology t , with g enerating set of twists τ . Assume that t is bounded in A and consider a bounded generating famil y N t S f or t -hyperco vers in A S . Let M ( P / S , A ) be the full subcategory of A S spanned by direct factors of ﬁnite sums of premotives of shape M S ( X ){ i } f or a P -scheme X / S and a twist i ∈ τ . This category is additive and w e can associate with it its categor y of comple x es up to chain homotopy . W e get an obvious triangulated functor (5.1.31.1) K b  M ( P / S , A )  / / D ( A S ) . Then the previous functor induces a tr iangulated functor K b  M ( P / S , A )  / N t S / / D ( A S ) where the left hand side stands f or the V erdier quotient of K b  M ( P / S , A )  b y the thick subcategor y generated by N t S . The category K b  M ( P / S , A )  / N t S ma y not be pseudo-abelian while the aim of the previous functor is. Thus we can consider its pseudo-abelian env elope and the induced functor (5.1.31.2)  K b  M ( P / S , A )  / N t S  \ / / D ( A S ) . A ccording to Deﬁnition 1.4.9 , the image of this functor is the subcategor y of τ - constructible premotiv es of the triangulated P -premotivic category D ( A S ) . Then the f ollo wing proposition is a corollar y of [ CD09 , theorem 6.2]: Proposition 5.1.32 Consider the hypot hesis and notations abov e. If A is ﬁnitely τ -presented then D ( A ) is compactly τ -g enerated. Moreo ver , the functor ( 5.1.31.2 ) is fully fait hful. Let us denote b y D c ( A ) the subcategor y of D ( A ) made of τ -constructible premotiv es in the sense of Deﬁnition 1.4.9 . T aking into account Proposition 1.4.11 , the previous proposition admits the f ollo wing corollary: Corollary 5.1.33 Consider the situation of 5.1.31 , and assume that A is ﬁnitely τ - pr esented. F or any premotiv e M in D ( A S ) , the f ollowing conditions are equiv alent: (i) M is compact. (ii) M is τ -constructible. Mor eov er , the functor ( 5.1.31.2 ) induces an equivalence of categories:  K b  M ( P / S , A )  / N t S  \ / / D c ( A S ) . Example 5.1.34 A ccording to ex ample 5.1.29 , we get the f ollo wing e xamples: 1. Let Λ ( Sm / S ) = M ( Sm / S , A ) for A = Sh Nis ( Sm / S , Λ ) . W e obtain a fully faithful functor 182 Construction of ﬁbred categor ies  K b ( Λ ( Sm / S ) ) / BG S  \ / / D  Sh Nis ( Sm / S , Λ )  which is essentially sur jectiv e on compact objects. 2. Let Λ ( S f t / S ) = M ( Sm / S , A ) f or A = Sh c dh  S f t / S , Λ  . W e obtain a fully faithful functor  K b  Λ ( S f t / S )  / BG S ∪ C D H S  \ / / D  Sh c dh ( S f t / S , Λ )  which is essentially sur jectiv e on compact objects. 3. Let Q ´ e t ( Sm / S ) = M ( Sm / S , A ) f or A = Sh ´ e t ( Sm / S , Q ) . W e obtain a fully faithful functor  K b ( Q ´ e t ( Sm / S ) ) / BG S  \ / / D  Sh ´ e t ( Sm / S , Q )  . which is essentially sur jectiv e on compact objects. 5.1.35 Consider an abelian P -premotivic categor y A . W e introduce the follo wing property of A : (C) Consider a projectiv e system ( S α ) α ∈ A of schemes in S with aﬃne transition maps such that S = lim o o α ∈ A S α belongs to S . For an y index α 0 ∈ A , any object A α 0 in A S α 0 , and any twist n ∈ τ , the canonical map lim / / α ∈ A / α 0 Hom A S α ( 1 S α { n } , A α ) / / Hom A S ( 1 S { n } , A ) is an isomor phism where A α (resp. A ) is the pullback of A α 0 along the canonical map S α / / S α 0 (resp. S / / S α 0 ). W e will denote b y (wC) the analogous proper ty when one restricts pro-objects to thus with aﬃne and dominant transition maps. Proposition 5.1.36 Consider an abelian P -pr emotivic category A compatible with an admissible topology t and satisfying the assumption (C) (r esp. (wC)) abov e. Then the deriv ed premotivic category D ( A ) is τ -continuous (resp. weakly τ - continuous) — see Deﬁnition 4.3.2 . Proof W e use Proposition 4.3.6 applied to the t -descent model structure on C ( A T ) f or T = S or T = S α . (see Paragraph 5.1.13 ). R ecall from Paragraph 5.1.11 that this model structure is associated with a descent str ucture. Thus according to [ CD09 , 2.3], there e xists an explicit generating set I (resp. J ) f or coﬁbrations (resp. tr ivial coﬁbrations). Moreo v er , the source or targ et of an y map in I ∪ J is a comple x C satisfying the f ollo wing assumption: (rep) f or any integer i ∈ Z , C i is a sum of premotives of the form M T ( X ){ n } where X / T is a P -scheme and n ∈ τ . 5 Fibred der iv ed categor ies 183 Thus, to check the assumption of 4.3.6 f or C ( A ) , w e ﬁx a projectiv e system ( S α ) α ∈ A satisfying the assumptions of property (C) (resp. (wC)) abov e; w e hav e to prov e that f or any index α 0 ∈ A and any complex es C α 0 and E α 0 such that C α 0 satisﬁes (rep), the natural map: lim / / α ∈ A / α 0 Hom C ( A S α ) ( C α , E α ) / / Hom C ( A S ) ( C , E ) is bi jectiv e. Giv en the deﬁnition of mor phisms in a categor y of complex es, it is suﬃcient to chec k this when the Hom g roups are computed as mor phisms of Z -graded objects. Thus it is suﬃcient to treat the case where C α 0 and E α 0 are concentrated in degree 0 . Thus, as C α 0 satisﬁes proper ty (rep), w e are e xactly reduced to assumption (C) (resp. (wC)) on A .  Example 5.1.37 1. Assume P is contained in the class of mor phisms of ﬁnite type. Then the abelian P -premotivic categor y PSh ( P , Λ ) of e xample 5.1.1 satisﬁes assumption (C). Indeed, proper ty (C) when A is a representable presheaf f ollow s from the assumption on P : P -schemes o ver some base S alwa ys are of ﬁnite presentation ov er S – S is noetherian according to our general assumption 5.0.1 . Then the case of a g eneral presheaf A f ollow s because A is an inductiv e limit of representable presheaf and the global sections functor commutes with inductiv e limit of presheav es. 2. Let S f t be the class of morphisms of ﬁnite type and let t be one of the f ollo wing topologies: Nis , ´ e t , c dh , qfh , h . Then the g eneralized abelian premotivic categor y Sh t  S f t , Λ  of ex ample 5.1.4 satisﬁes assumption (C). Indeed, according to the preceding e xample, we ha v e only to pro v e that f or any morphism f : X / / S , the functor f ∗ : PSh ( S f t S , Λ ) / / PSh ( S f t T , Λ ) preserves the proper ty of being a t -sheaf. If f is a mor phism of ﬁnite type, the functor f ∗ admits as a left adjoint the functor f ] , which preserves t -co v ers. Thus the asser tion is clear in that case. In the general case, w e use the fact that X / S is a projectiv e limit of a projectiv e sys tem ( X α ) α ∈ A where X α is an S -scheme aﬃne and of ﬁnite type ov er S . T o chec k that f or a t -sheaf F o ver S , the presheaf f ∗ ( F ) is a t -sheaf, we ﬁx a t -co v er ( W i ) i ∈ I of X in S f t X . As X is noetherian, w e can assume I is ﬁnite. Moreo v er , there exis ts an index α 0 ∈ A such that f or the t -co v er ( W i ) i ∈ I can be lifted to X α 0 . Then, using proper ty (C) of PSh ( S f t , Λ ) applied to F and ( X α ) , we reduce to chec k that f ∗ α ( F ) is a t -sheaf for α ≥ α 0 . This follo ws from the ﬁrst case treated. 3. Let Sm be the class of smooth mor phisms and t be one of the topologies: Nis , ´ e t . As we will see in Example 6.1.1 , there exis ts a canonical enlarg ement of abelian premotivic categories (see ( 6.1.1.1 )): 184 Cons truction of ﬁbred categor ies ρ ] : Sh t ( Sm , Λ ) / / o o Sh t  S f t , Λ  : ρ ∗ . As the functor ρ ] is fully faithful and commutes with f ∗ f or an y mor phism of schemes f , we deduce from the preceding point that the abelian premotivic category Sh t ( Sm , Λ ) satisﬁes the abo v e condition (C). As an application of the pre vious proposition, w e thus obtain that the derived premotivic category D ( Sh t ( Sm , Λ ) ) is τ -continuous. 5.2 The A 1 -deriv ed premotivic category 5.2.a Localization of triangulated premotivic categories 5.2.1 Let A be an abelian P -premotivic categor y compatible with an admissible topology t and D ( A ) be the associated der iv ed P -premotivic categor y . Suppose giv en an essentiall y small famil y of mor phisms W in C ( A ) which is stable by the operations f ∗ , f ] (in other w ords, W is a sub- P -ﬁbred category of C ( A ) ). R emark that the localizing subcategor y T of D ( A ) generated b y the cones of arrow s in W is again stable b y these operations. Moreo ver , as f or any P -mor phism f : X / / S w e hav e f ] f ∗ = M S ( X ) ⊗ S (−) , the categor y T is stable b y tensor product with a geometric section. W e will say that a complex K o v er A S is W -local if for an y object T of T and an y integer n ∈ Z , Hom D ( A S ) ( T , K [ n ]) = 0 . A mor phism of comple xes p : C / / D o v er A S is a W -equiv alence if for any W -local complex K ov er A S , the induced map Hom D ( A S ) ( D , K ) / / Hom D ( A S ) ( C , K ) is bi jectiv e. A mor phism of complex es o v er A S is called a W -ﬁbration if it is a t -ﬁbration with a W -local kernel. A comple x ov er A S will be called W -ﬁbrant if it is t -local and W -local. As consequence of [ CD09 , 4.3, 4.11 and 5.6], we obtain: Proposition 5.2.2 Let A be an abelian P -premo tivic category compatible with an admissible topology t and W be an essentially small family of mor phisms in C ( A ) stable by f ∗ and f ] . Then the categor y C ( A S ) is a proper closed symmetric monoidal category with the W -ﬁbrations as ﬁbr ations, the coﬁbrations as deﬁned in 5.1.11 , and the W - equiv alences as weak equivalences. The homotopy categor y associated with this model category will be denoted by D ( A S )[ W − 1 S ] . It can be descr ibed as the V erdier quotient D ( A S )/ T S . In f act, the W -local model category on C ( A S ) is nothing else than the left Bousﬁeld localization of the t -local model category str ucture. As a consequence, we obtain an adjunction of tr iangulated categories: 5 Fibred der iv ed categor ies 185 (5.2.2.1) π S : D ( A S ) / / o o D ( A S )[ W − 1 S ] : O S such that O S is fully faithful with essential imag e the W -local complex es. In f act, the model structure gives a functor ial W -ﬁbrant resolution 1 / / R W R W : C ( A S ) / / C ( A S ) , which induces O S . Note that the triangulated categor y D ( A S )[ W − 1 S ] is g enerated b y the comple xes concentrated in deg ree 0 of the f or m M S ( X ){ i } — or , equivalentl y , the W -local comple x es R W ( M S ( X ){ i } ) — for a P -scheme X and a twist i ∈ τ . Remar k 5.2.3 Another very useful property is that W -equiv alences are stable b y ﬁltering colimits; see [ CD09 , prop. 3.8]. 5.2.4 Recall from 5.1.14 that f or an y morphism (resp. P -mor phism) f : T / / S , the functor f ∗ (resp. f ] ) satisﬁes descent; as it also preserves W , it follo ws from [ CD09 , 4.9] that the adjunction f ∗ : C ( A S ) / / C ( A T ) : f ∗ (resp. f ] : C ( A S ) / / C ( A T ) : f ∗ ) is a Quillen adjunction with respect to the W -local model structures. This giv es the f ollowing corollar y . Corollary 5.2.5 The P -ﬁbred category C ( A ) with the W -local model structure on its ﬁbers deﬁned abov e is a monoidal P -ﬁbred model categor y, whic h is moreo v er stable, pr oper and combinatorial. W e will denote by D ( A )[ W − 1 ] the tr iangulated P -premotivic categor y whose ﬁber o v er a scheme S is the homotopy categor y of the W S -local model categor y C ( A S ) . The adjunction ( 5.2.2.1 ) readily deﬁnes an adjuntion of triangulated P -premotivic categories (5.2.5.1) π : D ( A ) / / o o D ( A )[ W − 1 ] : O . The P -ﬁbred categories D ( A ) and D ( A )[ W − 1 ] are both τ -generated (and this adjunction is compatible with τ -twists in a strong sense). Remar k 5.2.6 For an y scheme S , the category D ( A S )[ W − 1 S ] is w ell generated and has a canonical dg-structure (see also 5.1.19 ). 5.2.7 With the notations abo v e, let us put T = D ( A )[ W − 1 ] to clar ify the follo wing notations. As in 5.1.20 , the ﬁbred category T has a canonical e xtension to S - diagrams X : I / / S . If we deﬁne W X as the class of mor phisms ( f i ) i ∈ I in C ( A ( X , I )) such that f or any object i , f i is a W -equivalence, then T ( X ) is the tr iangulated categor y D ( A ( X , I ))[ W − 1 X ] . 186 Construction of ﬁbred categories Ag ain, this tr iangulated categor y is symmetric monoidal closed and f or any morphism (resp. P -mor phism) ϕ : ( X , I ) / / ( Y , J ) , w e get (der iv ed) adjunctions as in 5.1.20 : L ϕ ∗ : T ( Y , J ) / / o o T ( X , I ) : R ϕ ∗ (5.2.7.1) (resp. L ϕ ] : T ( X , I ) / / o o T ( Y , J ) : L ϕ ∗ = ϕ ∗ ) (5.2.7.2) In fact, T is then a complete monoidal P c art -ﬁbred categor y ov er the categor y of diagrams of schemes and the adjunction ( 5.2.5.1 ) e xtends to an adjunction of complete monoidal P c art -ﬁbred categories. Example 5.2.8 Suppose we are under the h ypothesis of Example 5.1.24.2 . Let W t , S denote the family of maps which are of the f or m Λ S ( X ) / / Λ S ( X ) f or a t -hyperco ver X / / X in P / S . Then W t is obviousl y stable by f ∗ and f ] . Recall now that a comple x of t -shea v es on P / S is local if and onl y if its t - h ypercohomology and its h ypercohomology computed in the coarse topology agree ( cf. 5.1.10 ). This readily implies that the adjunction considered in Example 5.1.24.2 a ∗ t : D ( PSh ( P , Λ )) / / o o D ( Sh t ( P , Λ ) ) : R a t , ∗ induces an equiv alence of tr iangulated P -premotivic categor ies D ( PSh ( P , Λ ))[ W − 1 t ] / / o o D ( Sh t ( P , Λ ) ) . Recall R a t , ∗ is fully faithful and identiﬁes D ( Sh t ( S , Λ ) ) with the full subcategory of D ( PSh ( S , Λ )) made by t -local complex es. 5.2.9 A triangulated P -premotivic categor y ( T , M ) such that there exis ts: 1. an abelian P -premotivic category A compatible with an admissible topology t 0 on Sm . 2. an essentially small famil y W of mor phisms in C ( A ) stable by f ∗ and f ] 3. an adjunction of tr iangulated P -premotivic categor ies D ( A )[ W − 1 ] ' T will be called for shor t a derived P -premo tivic category . A ccording to conv ention 5.0.1 (d) and from the abo ve construction, T is τ -generated f or some set of twists τ . 75 Let us denote simply by M S ( X ) the geometric sections of T . In this case, using the mor phisms ( 5.1.24.1 ) and ( 5.2.5.1 ), we get a canonical mor phism of triangulated P -premotivic categor ies: 75 W e will formulate in some remarks below universal proper ties of some derived P -premotivic categories. When doing so, w e will restrict to morphisms of derived P -premotivic categor ies which can be wr itten as L ϕ : D ( A 1 )[ W − 1 1 ] / / D ( A 2 )[ W − 1 2 ] f or a morphism ϕ : A 1 / / A 2 of abelian P -premotivic categor ies compatible with suitable topologies. More natural universal proper ties could be obtained if one considers the framew ork of dg-categories or tr iangulated derivator . 5 Fibred der iv ed categor ies 187 (5.2.9.1) ϕ ∗ : D ( PSh ( P , Z )) / / o o T : ϕ ∗ . By deﬁnition, f or any premotiv e M , any scheme X and any integ er n ∈ Z , we g et a canonical identiﬁcation: (5.2.9.2) Hom T ( S ) ( M S ( X ) , M [ n ]) = H n Γ ( X , ϕ ∗ ( M )) . Giv en an y simplicial scheme X , we put M S ( X ) = ϕ ∗  Z S ( X )  , so that we also obtain: (5.2.9.3) Hom T ( S ) ( M S ( X ) , M [ n ]) = H n  T ot π Γ ( X , R γ ∗ ( M ))  . Proposition 5.2.10 Consider the abov e notations and t an admissible topology. The f ollowing conditions ar e equiv alent. (i) F or any t -hyperco ver X / / X in P / S , the induced map M S ( X ) / / M S ( X ) is an isomorphism in T ( S ) . (i 0 ) F or any t -hyperco ver p : X / / X in P / S , the induced functor L p ∗ : T ( X ) / / T ( X ) is fully faithful. (i 00 ) The triangulated P -premo tivic category T satisﬁes t -descent (Deﬁnition 3.2.5 ). (ii) Ther e exists an essentially unique map ϕ ∗ t : D ( Sh t ( P / S , Z ) ) / / T ( S ) making the f ollowing diagram essentially commutative: D ( PSh ( P / S , Z )) ϕ ∗ / / a t   T ( S ) D ( Sh t ( P / S , Z ) ) ϕ ∗ t 5 5 (ii 0 ) F or any complex C ∈ C ( PSh ( P / S , Z )) suc h that a t ( C ) = 0 , ϕ ∗ ( C ) = 0 . (ii 00 ) F or any map f : C / / D in C ( PSh ( P / S , Z )) such that a t ( f ) is an isomor - phism, ϕ ∗ ( f ) is an isomor phism. (iii) Ther e exists an essentially unique map ϕ t ∗ : T ( S ) / / D ( Sh t ( P / S , Z ) ) making the f ollowing diagram essentially commutative: D ( PSh ( P / S , Z )) T ( S ) ϕ ∗ o o ϕ t ∗ u u D ( Sh t ( P / S , Z ) ) R O t O O (iii 0 ) F or any premo tive M in T ( S ) , the complex ϕ ∗ ( M ) is local. (iii 00 ) F or any premo tive M in T ( S ) , any P -scheme X / S and any integ er n ∈ Z , Hom T ( S ) ( M S ( X ) , M [ n ]) = H n t ( X , ϕ ∗ ( M )) . When these conditions are fulﬁlled f or any scheme S , the functors appearing in (ii) and (iii) induce a mor phism of triangulated P -premo tivic categories: 188 Construction of ﬁbred categories ϕ ∗ t : D ( Sh t ( P , Z ) ) / / o o T : ϕ t ∗ . Proof The equivalence between conditions ( i ) , ( i 0 ) and ( i 00 ) is clear (w e proceed as in the proof of 5.1.21 ). The equiv alences ( ii ) ⇔ ( ii 0 ) ⇔ ( ii 00 ) and ( ii i ) ⇔ ( ii i 0 ) f ollo ws from e xample 5.2.8 and the deﬁnition of a localization. The equiv alence ( i ) ⇔ ( ii 00 ) f ollow s again from loc. cit. The equivalences ( i ) ⇔ ( ii i 0 ) ⇔ ( ii i 00 ) follo ws ﬁnally from ( 5.2.9.2 ), ( 5.2.9.3 ), and the characterization of a local complex of sheav es ( cf. 5.1.10 ).  Remar k 5.2.11 The preceding proposition expresses the fact that the categor y D ( Sh t ( P , Z ) ) is the univ ersal der iv ed P -premotivic categor y satisfying t -descent. 5.2.12 W e end this section b y making explicit tw o particular cases of the descent property f or derived P -premotivic categor ies. Consider a derived P -premotivic categor y T with geometric sections M . Con- sidering an y diag ram X : I / / P / S of P -schemes o v er S , with projection p : X / / S , w e can associate a premotive in T : M S ( X ) = L p ] ( 1 S ) = L lim / / i ∈ I M S ( X i ) . In par ticular , when I is the categor y • / / • , we associate to e v ery S -mor phism f : Y / / X of P -schemes o ver S a canonical 76 bivariant pr emotiv e M S ( X f / / Y ) . When f is an immersion, w e will also write M S ( Y / X ) f or this premotiv e. Note that in an y case, there is a canonical distinguished triangle in T ( S ) : M S ( X ) f ∗ / / M S ( Y ) π f / / M S ( X f / / Y ) ∂ f / / M S ( X )[ 1 ] . This triangle is functor ial in the ar ro w f – with respect to commutativ e squares. Giv en a commutativ e square of P -schemes ov er S B e 0 / / g   Y f   A e / / X (5.2.12.1) w e will say that the image square in T ( S ) 76 In fact, if T = D ( A )[ W − 1 ] f or an abelian P -premotivic categor y A , then we can de- ﬁne M S ( X / / Y ) as the cone of the mor phism of complex es (concentrated in degree 0 ) M S ( X ) f ∗ / / M S ( Y ) . 5 Fibred der iv ed categor ies 189 M S ( B ) e 0 ∗ / / g ∗   M S ( Y ) f ∗   M S ( A ) e ∗ / / M S ( X ) is homotopy car tesian 77 if the premotiv e associated with diag ram 5.2.12.1 is zero. Proposition 5.2.13 Consider a derived P -premo tivic category T . W e assume that P contains the étale morphisms (resp. P = S f t ). Then, with the abov e deﬁnitions, the f ollowing conditions ar e equivalent : (i) T satisﬁes Nisnevich (resp. proper c dh ) descent. (ii) F or any scheme S and any Nisnevich (resp. proper c dh ) distinguished squar e Q of S -schemes, the square M S ( Q ) is homotopy cartesian in T ( S ) . (iii) F or any Nisnevich (resp. proper c dh ) distinguished squar e of shape ( 5.2.12.1 ) , the canonical map M S ( Y / B ) ( f / g ) ∗ / / M S ( X / A ) is an isomorphism. Mor eov er , under these conditions, to any Nisnevic h (resp. proper c dh ) distinguished squar e Q of shape ( 5.2.12.1 ) , we associate a map ∂ Q : M S ( X ) π e / / M S ( X / A ) ( f / g ) − 1 ∗ / / M S ( Y / B ) ∂ e 0 / / M S ( Y )[ 1 ] whic h deﬁnes a distinguished triang le in T ( S ) : M S ( B )  e 0 ∗ − g ∗  / / M Z ( Y ) ⊕ M S ( A ) ( f ∗ , e ∗ ) / / M S ( X ) ∂ Q / / M S ( Y )[ 1 ] . Proof The equiv alence of (i) and (ii) f ollo ws from the theorem of Morel- V oev odsky 3.3.2 (resp. the theorem of V oev odsky 3.3.8 ). T o pro ve the equivalence of (ii) and (iii), we assume T = D ( A )[ W − 1 ] . Then, the homotopy colimit of a square of shape 5.2.12.1 is giv en by the complex Cone  Cone ( M S ( B ) / / M S ( Y )) / / Cone ( M S ( A ) / / M S ( X ))  . This readily prov es the needed equiv alence, together with the remaining asser tion.  Remar k 5.2.14 In the ﬁrst of the respective cases of the proposition, condition (ii) is what w e usually called the Brown-Ger st en proper ty (BG) f or T , whereas condition (iii) can be called the excision property . In the second respectiv e case, condition (ii) will be called the proper c dh proper ty for the g eneralized premo tivic category T . W e sa y also that T satisﬁes the (cdh) proper ty if it satisﬁes condition (ii) with respect to any c dh dis tinguished square Q . 77 If T = D ( A )[ W − 1 ] , this amount to say that the diagram obtained of complex es by applying the functor M S (−) is homotop y car tesian in the W -local model category C ( A ) . 190 Construction of ﬁbred categories 5.2.b The homotop y relation 5.2.15 Let A be an abelian P -premotivic categor y compatible with an admissible topology t . W e consider W A 1 to be the famil y of mor phisms M S ( A 1 X ){ i } / / M S ( X ){ i } f or a P -scheme X / S and a twist i in τ . The f amily W A 1 is ob viously stable by f ∗ and f ] . Deﬁnition 5.2.16 Let A be an abelian P -premotivic categor y compatible with an admissible topology t . With the notation abov e, we deﬁne D eﬀ A 1 ( A ) = D ( A )[ W − 1 A 1 ] and refer to it as the (eﬀectiv e) P -premo tivic A 1 -deriv ed categor y with coeﬃcients in A . By deﬁnition, the categor y D eﬀ A 1 ( A ) satisﬁes the homotopy proper ty (Htp) (see 2.1.3 ). A ccording to the general facts about localization of der iv ed premotivic categor ies, the triangulated premotivic categor y D eﬀ A 1 ( A ) is τ -generated. Example 5.2.17 W e can divide our ex amples into two types: 1) Assume P = Sm : Consider the admissible topology t = Nis . Follo wing F . Morel, w e deﬁne the (eﬀectiv e) A 1 -deriv ed categor y ov er S to be D eﬀ A 1 ( Sh Nis ( Sm / S , Λ )) . Indeed we get a triangulated premotivic categor y (see also the construction of [ A yo07b ]): (5.2.17.1) D eﬀ A 1 , Λ : = D eﬀ A 1 ( Sh Nis ( Sm , Λ )) . W e shall also wr ite its ﬁbers (5.2.17.2) D eﬀ A 1 ( S , Λ ) : = D eﬀ A 1 , Λ ( S ) = D eﬀ A 1 ( Sh Nis ( Sm / S , Λ )) f or a scheme S . For Λ = Z , we shall often write simply (5.2.17.3) D eﬀ A 1 : = D eﬀ A 1 ( Sh Nis ( Sm , Z )) . Another interesting case is when t = ´ e t ; we get a tr iangulated premotivic categor y of eﬀectiv e étale pr emotiv es : D eﬀ A 1 ( Sh ´ e t ( Sm , Λ )) . In each of these cases, we denote by Λ t S ( X ) the premotive associated with a smooth S -scheme X . 2) Assume P = S f t : Consider the admissible topology t = h (resp. t = qfh ). In [ V oe96 ], V oev odsky has introduced the categor y of h -motiv es (resp. qfh -motiv es). In our formalism, one deﬁnes the categor y of eﬀectiv e h -motiv es (resp. eﬀective h -motiv es ) o v er S with coeﬃcients in Λ as: 5 Fibred der iv ed categor ies 191 DM eﬀ h ( S , Λ ) = D eﬀ A 1  Sh h  S f t / S , Λ   , resp. DM eﬀ qfh ( S , Λ ) = D eﬀ A 1  Sh qfh  S f t / S , Λ   . In other words, this is the A 1 -derived category of h -sheav es (resp. qfh -sheav es) of Λ -modules. Moreov er , these categor ies for various schemes S are the ﬁbers of a gen- eralized premotivic tr iangulated categor y . What we ha ve added to the construction of V oev odsky is the functors of the generalized premotivic str ucture. W e will denote simpl y b y Λ t S ( X ) the cor responding premotiv e associated with X in DM eﬀ t ( S , Λ ) . Another interesting case is obtained when t = c dh . W e get an A 1 -derived gen- eralized premotivic categor y D eﬀ A 1  Sh c dh  S f t , Λ   whose premotiv es are simpl y denoted b y Λ c dh S ( X ) f or an y ﬁnite type S -scheme X . 5.2.18 Let C be a comple x with coeﬃcients in A S . According to the general case, w e say that C is A 1 -local if for an y P -scheme X / S and any ( i , n ) ∈ τ × Z , the map induced b y the canonical projection Hom D ( A S ) ( M S ( X ){ i } [ n ] , C ) / / Hom D ( A S ) ( M S ( A 1 X ){ i } [ n ] , C ) is an isomor phism. The adjunction ( 5.2.2.1 ) deﬁnes a mor phism of tr iangulated P -premotivic categor ies D ( A ) / / o o D eﬀ A 1 ( A ) such that f or any scheme S , D eﬀ A 1 ( A S ) is identiﬁed with the full subcategor y of D ( A S ) made of A 1 -local comple xes. Fibrant objects for the model categor y structure on C ( A S ) appearing in Proposi- tion 5.2.2 relativel y to W A 1 , simply called A 1 -ﬁbrant objects, are the t -ﬂasque and A 1 -local comple xes. W e say a mor phism f : C / / D of comple xes of A S is an A 1 -equiv alence if it becomes an isomor phism in D eﬀ A 1 ( A S ) . Consider ing moreov er two mor phisms f , g : C / / D of comple xes of A S , we say they are A 1 -homotopic if there exis ts a morphism of comple xes H : M S ( A 1 S ) ⊗ S C / / D such that H ◦ ( s 0 ⊗ 1 C ) = f and H ◦ ( s 1 ⊗ 1 C ) = g , where s 0 and s 1 are respectivel y induced by the zero and the unit section of A 1 S / S . When f and g are A 1 -homotopic, the y are equal as mor phisms of D eﬀ A 1 ( A S ) . W e sa y the mor phism p : C / / D is a str ong A 1 -equiv alence if there exis ts a morphism q : D / / C such that the morphisms p ◦ q and q ◦ p are A 1 -homotopic to the identity . A complex C is A 1 -contractible if the map C / / 0 is a strong A 1 -equiv alence. As an e xample, f or an y integer n ∈ N , and an y P -scheme X / S , the map p ∗ : M S ( A n X ) / / M S ( X ) 192 Construction of ﬁbred categor ies induced b y the canonical projection is a strong A 1 -equiv alence with inv erse the zero section s 0 , ∗ : M S ( X ) / / M S ( A n X ) . 5.2.19 The category D eﬀ A 1 ( A ) is functor ial in A . Let ϕ : A / / o o B : ψ be an adjunction of abelian P -premotivic categories. Consider tw o topologies t and t ’ such that t 0 is ﬁner than t . Suppose A (resp. B ) is compatible with t (resp. t 0 ). For any scheme S , consider the evident extensions ϕ S : C ( A S ) / / o o C ( B S ) : ψ S of the abo ve adjoint functors to complex es. W e easily check that the functor ψ S preserves A 1 -local complex es. Thus, appl ying 5.1.23 , the pair ( ϕ S , ψ S ) is a Quillen adjunction f or the respectiv e A 1 -localized model structure on C ( A S ) and C ( B S ) ; see [ CD09 , 3.11]. Consider ing the der iv ed functors, it is no w easy to check we ha v e obtained an adjunction L ϕ : D eﬀ A 1 ( A ) / / o o D eﬀ A 1 ( B ) : R ψ of triangulated P -premotivic categor ies. Example 5.2.20 Consider the notations of 5.2.17 . In the case where P = Sm , we get from the adjunction of ( 5.1.24.2 ) the follo wing adjunction of tr iangulated premotivic categories a ∗ ´ e t : D eﬀ A 1 , Λ / / o o D eﬀ A 1 ( Sh ´ e t ( Sm , Λ )) : R a ´ e t , ∗ . Example 5.2.21 Let T be a derived P -premotivic categor y as in 5.2.9 . If T satisﬁes the proper ty (Htp), then the canonical mor phism ( 5.2.9.1 ) induces a mor phism D eﬀ A 1 ( PSh ( P , Z )) / / o o T . If moreo v er T satisﬁes t -descent f or an admissible topology t , we fur ther obtain as in 5.2.10 a mor phism D eﬀ A 1 ( Sh t ( P , Z ) ) / / o o T . Particularl y interesting cases are given b y D eﬀ A 1 (resp. D eﬀ A 1  Sh c dh  S f t , Z   ) which is the univ ersal der iv ed premotivic categor y (resp. generalized premotivic category), i.e. initial premotivic categor y satisfying Nisnevic h descent (resp. c dh descent) and the homotop y proper ty . 5.2.22 As in Example 5.1.25 , let t be an admissible topology and ϕ : Λ / / Λ 0 be an e xtension of r ings. Then, from the P -premotivic adjunction ( 5.1.25.1 ) and according to Paragraph 5.2.19 , w e get an adjunction of tr iangulated P -premotivic categor ies: L ϕ ∗ : D eﬀ A 1  Sh t ( P , Λ )  / / o o D eﬀ A 1  Sh t ( P , Λ 0 )  : R ϕ ∗ . Consider also comple xes C and D of t -shea v es of Λ -modules ov er P S . Then there e xists a canonical mor phism of Λ 0 -modules: (5.2.22.1) Hom D eﬀ A 1 ( Sh t ( P S , Λ ) )  C , D  ⊗ Λ Λ 0 / / Hom D eﬀ A 1 ( Sh t ( P S , Λ 0 ) )  L ϕ ∗ ( C ) , L ϕ ∗ ( D )  5 Fibred der iv ed categor ies 193 There are tw o notable cases where this map is an isomor phism: Proposition 5.2.23 Consider the abov e assumptions. Then the map ( 5.2.22.1 ) is an isomorphism in the two following cases: 1. If Λ 0 is a free Λ -module and C is compact; 2. If Λ 0 is a free Λ -module of ﬁnite rank. Proof Note that in an y case, the functor ϕ ∗ admits a r ight adjoint ϕ ! . 78 W e can assume that Λ 0 = I . Λ f or a set I . In this case, w e get f or an y sheaf F of Λ -modules: ϕ ∗ ϕ ∗ ( F ) = F ⊗ Λ Λ 0 = I . F . Moreo v er , f or an y P -scheme X / S , we get: ϕ ∗ ( Λ 0 t S ( X )) = Λ 0 t S ( X ) = I . Λ t S ( X ) . In par ticular , the functor ϕ ∗ : C ( Sh t ( P S , Λ 0 ) ) / / C ( Sh t ( P S , Λ ) ) satisﬁes descent in the sense of [ CD09 , 2.4] and preser v es the famil y W A 1 . Thus it is a left Quillen functor with respect to the A 1 -local model structures. In par ticular , because it is also a r ight Quillen functor , we get: R ϕ ∗ = ϕ ∗ = L ϕ ∗ . In par ticular , w e get in D eﬀ A 1 ( Sh t ( P S , Λ ) ) : R ϕ ∗ L ϕ ∗ ( D ) = L ϕ ∗ L ϕ ∗ ( D ) = L ( ϕ ∗ ϕ ∗ )( D ) = I . D . Thus the Proposition f ollo ws as the functor Hom ( C , −) commutes with direct sums if C is compact and with ﬁnite direct sums in an y case.  W e remark the f ollo wing useful proper ty . Proposition 5.2.24 Consider a morphism ϕ ∗ : A / / o o B : ϕ ∗ of abelian P -pr emotivic categories suc h that A (resp. B ) is compatible with an admissible topology t (r esp. t 0 ). Assume t 0 is ﬁner than t . Let S be a base scheme. Assume that ϕ ∗ : A S / / B S commutes with colimits 79 . Then ϕ ∗ : C ( A S ) / / C ( B S ) r espects A 1 -equiv alences. In other words, the right der iv ed functor R ϕ ∗ : D eﬀ A 1 ( B S ) / / D eﬀ A 1 ( A S ) satisﬁes the relation R ϕ ∗ = ϕ ∗ . Proof In this proof, we wr ite ϕ ∗ f or ϕ ∗ , S . W e ﬁrst prov e that ϕ ∗ preserves strong A 1 -equiv alences (see 5.2.18 ). 78 It is deﬁned by the f ormula: ϕ ! ( F ) = Hom Λ ( Λ 0 , F ) equipped with its canonical structure of sheaf of Λ 0 -modules. 79 This amounts to ask that ϕ ∗ is e xact and commutes with direct sums. 194 Construction of ﬁbred categories Consider two maps u , v : K / / L in C ( B S ) . T o giv e an A 1 -homotop y H : M S ( A 1 S , B ) ⊗ S K / / L between u and v is equivalent b y adjunction to giv e a map H 0 : K / / Hom B S ( M S ( A 1 S , B ) , L ) which ﬁts into the f ollo wing commutativ e diagram: K H 0   u w w v ' ' L Hom B S ( M S ( A 1 S , B ) , L ) s ∗ 0 o o s ∗ 1 / / L where s 0 and s 1 are the respectiv e zero and unit section of A 1 S / S . Because M S ( A 1 S , B ) = ϕ ∗ S ( M S ( A 1 S , A )) , we g et a canonical isomor phism (see paragraph 1.2.9 ) ϕ ∗ ( Hom B S ( M S ( A 1 S , B ) , L )) ' Hom B S ( M S ( A 1 S , A ) , ϕ ∗ ( L )) . Thus, applying ϕ ∗ to the previous commutativ e diagram and using this identiﬁcation, w e obtain that ϕ ∗ ( u ) is A 1 -homotopic to ϕ ∗ ( v ) . As a consequence, f or an y P -scheme X o v er S , and any B -twist i , the map ϕ ∗ ( M S ( A 1 X , B ){ i } ) / / ϕ ∗ ( M S ( X , B ){ i } ) induced by the canonical projection is a strong A 1 -equiv alence, thus an A 1 - equiv alence. The functor ϕ ∗ : B S / / A S commutes with colimits. Thus it admits a r ight adjoint that w e will denote by ϕ ! . Consider the injective model structure on C ( A S ) and C ( B S ) (see [ CD09 , 2.1]). Because ϕ ∗ is e xact, it is a left Quillen functor f or these model structures. Thus, the right derived functor R ϕ ! is w ell-deﬁned. From the result we just get, we see that R ϕ ! preserves A 1 -local objects, and this readily implies L ϕ ∗ = ϕ ∗ preserves A 1 -equiv alences.  5.2.25 T o relate the category D eﬀ A 1 ( S ) with the homotopy categor y of schemes of Morel and V oev odsky [ MV99 ], we hav e to consider the categor y of simplicial Nisnevic h sheav es of sets denoted b y ∆ o p Sh ( Sm / S ) . Consider ing the free abelian sheaf functor , w e obtain an adjunction of categor ies ∆ o p Sh ( Sm / S ) / / o o C ( Sh ( Sm / S , Z ) ) . If w e consider Blander ’ s projective A 1 -model structure [ Bla03 ] on the categor y ∆ o p Sh ( Sm / S ) , we can easil y see that this is a Quillen pair, so that we obtain a P -premotivic adjunction of simple P -premotivic categor ies N : H / / o o D eﬀ A 1 : K . Note that the functor N sends coﬁber sequences in H ( S ) to distinguished tr iangles in D eﬀ A 1 ( S ) . 5 Fibred der iv ed categor ies 195 5.2.c Explicit A 1 -resolution 5.2.26 Consider an abelian P -premotivic categor y A compatible with an admissi- ble topology t . Consider the canonically split exact sequence 0 / / 1 S s 0 / / M S ( A 1 S ) / / U / / 0 where the map s 0 : 1 S / / M S ( A 1 S ) is induced by the zero section of A 1 . The section corresponding to 1 in A 1 deﬁnes another map s 1 : 1 S / / M S ( A 1 S ) which does not f actor through s 0 , so that w e get canonically a non-tr ivial map u : 1 S / / U . This deﬁnes f or any complex C of A S a map, called the evaluation at 1 , Hom ( U , C ) = 1 S ⊗ S Hom ( U , C ) u ⊗ 1 / / U ⊗ S Hom ( U , C ) ev / / C . W e deﬁne the comple x R ( 1 ) A 1 ( C ) to be R ( 1 ) A 1 ( C ) = Cone  Hom ( U , C ) / / C  . W e hav e by construction a map r C : C / / R ( 1 ) A 1 ( C ) . This deﬁnes a mor phism of functors from the identity functor to R ( 1 ) A 1 . For an integer n ≥ 1 , we deﬁne b y induction a complex R ( n + 1 ) A 1 ( C ) = R ( 1 ) A 1 ( R ( n ) A 1 ( C )) , and a map r R ( n ) A 1 ( C ) : R ( n ) A 1 ( C ) / / R ( n + 1 ) A 1 . W e ﬁnaly deﬁne a comple x R A 1 ( C ) b y the formula R A 1 ( C ) = lim / / n R ( n ) A 1 ( C ) . W e hav e a functorial map C / / R A 1 ( C ) . Lemma 5.2.27 Wit h the abov e hypothesis and notations, the map C / / R A 1 ( C ) is an A 1 -equiv alence. Proof For an y closed symmetr ic monoidal categor y C and an y objects A , B , C and I in C , we ha v e 196 Construction of ﬁbred categor ies Hom ( I ⊗ Hom ( B , C ) , Hom ( A , C )) = Hom ( Hom ( B , C ) , Hom ( I , Hom ( A , C ))) = Hom ( Hom ( B , C ) , Hom ( I ⊗ A , C )) . Hence an y map I ⊗ A / / B induces a map I ⊗ Hom ( B , C ) / / Hom ( A , C ) f or any object C . If we apply this to C = C ( A S ) and I = M S ( A 1 ) , we see immediately that the functor Hom (− , C ) preserves strong A 1 -homotop y equivalences. In par ticular , f or any comple x C , the map C / / Hom ( M S ( A 1 X ) , C ) is a strong A 1 -homotop y equiv alence. This implies that Hom ( U , C ) / / 0 is an A 1 -equiv alence, so that the map r C is an A 1 -equiv alence as well. As A 1 -equiv alences are stable by ﬁltering colimits, this implies our result.  Proposition 5.2.28 Consider the abov e notations and hypothesis, and assume that t is bounded in A . F or any t -ﬂasque complex C of A S , the complex R A 1 ( C ) is t -ﬂasque and A 1 -local. Mor eov er , the morphism C / / R A 1 ( C ) is an A 1 -equiv alence. If furthermore C is t -ﬂasque, so is R A 1 ( C ) . Proof The last asser tion is a particular case of Lemma 5.2.27 . The functor R ( 1 ) A 1 pre- serves t -ﬂasque complex es. By virtue of 5.1.30 , the functor R A 1 has the same gentle property . It thus remains to pro v e that the functor R A 1 sends t -ﬂasque comple xes on A 1 -local ones. W e shall use that the derived categor y D ( A S ) is compactly generated; see 5.1.30 . Let C be a t -ﬂasque complex of A S . T o pro ve R A 1 ( C ) is A 1 -local, w e are reduced to pro v e that the map R A 1 ( C ) / / Hom ( M S ( A 1 X ) , R A 1 ( C )) is a quasi-isomorphism, or , equiv alently , that the comple x Hom ( U , R A 1 ( C )) is acyclic. As U is a direct f actor of M S ( A 1 X , A ) , f or any P -scheme X ov er S and any i in I , the object Z S ( X ; A ){ i } ⊗ S U is compact. This implies that the canonical map lim / / n Hom ( U , R ( n ) A 1 ( C )) / / Hom ( U , R A 1 ( C )) is an isomor phism of complex es. As ﬁlter ing colimits preserve quasi-isomor phisms, the complex Hom ( U , R A 1 ( C )) (resp. R A 1 ( C ) ) can be considered as the homotop y colimit of the complex es Hom ( U , R ( n ) A 1 ( C )) (resp. R ( n ) A 1 ( C ) ). In par ticular , f or any compact object K of D ( A S ) , the canonical mor phisms lim / / n Hom ( K , Hom ( U , R ( n ) A 1 ( C ))) / / Hom ( K , Hom ( U , R A 1 ( C ))) lim / / n Hom ( K , R ( n ) A 1 ( C )) / / Hom ( K , R A 1 ( C )) are bi jectiv e. By construction, we hav e distinguished tr iangles 5 Fibred der iv ed categor ies 197 Hom ( U , R ( n ) A 1 ( C )) / / R ( n ) A 1 ( C ) / / R ( n + 1 ) A 1 ( C ) / / Hom ( U , R ( n ) A 1 ( C ))[ 1 ] . This implies that the ev aluation at 1 morphism ev 1 : Hom ( U , R A 1 ( C )) / / R A 1 ( C ) induces the zero map Hom D ( A S ) ( K , Hom ( U , R A 1 ( C ))) / / Hom D ( A S ) ( K , R A 1 ( C )) f or an y compact object K of D ( A S ) . Hence the induced map a = Hom ( U , ev 1 ) : Hom ( U , Hom ( U , R A 1 ( C ))) / / Hom ( U , R A 1 ( C )) has the same proper ty: f or any compact object K , the map Hom D ( A S ) ( K , Hom ( U , Hom ( U , R A 1 ( C )))) / / Hom D ( A S ) ( K , Hom ( U , R A 1 ( C ))) is zero. The multiplication map A 1 × A 1 / / A 1 induces a map µ : U ⊗ S U / / U such that the composition of µ ∗ : Hom ( U , R A 1 ( C )) / / Hom ( U ⊗ S U , R A 1 ( C )) = Hom ( U , Hom ( U , R A 1 ( C ))) with a is the identity of Hom ( U , R A 1 ( C )) . As D ( A S ) is compactly g enerated, this implies that Hom ( U , R A 1 ( C )) = 0 in the der iv ed categor y D ( A S ) .  Remar k 5.2.29 Consider a t -ﬂasque resolution functor ( i.e. a ﬁbrant resolution f or the t -local model structure) R t : C ( A S ) / / C ( A S ) , 1 / / R t . As a corollar y of the proposition, the composite functor R A 1 ◦ R t is a resolution functor by t -local and A 1 -local comple xes. Example 5.2.30 Consider an admissible topology t and the P -premotivic A 1 - derived categor y D = D eﬀ A 1 ( Sh t ( P , Λ )) . Suppose that t is bounded for abelian t -sheav es (for e xample, this is the case for the Zar iski and the Nisne vich topologies, see 5.1.29 ). Let C be a comple x of abelian t -shea v es on P / S . If C is A 1 -local, then Hom D ( S ) ( Λ t S ( X ) , C ) = H n t ( X ; C ) (this is tr ue without an y condition on t ). Consider a t -local resolution C t of C in C  Sh t ( P / S , Λ )  . Then we get the f ollowing f or mula: Hom D ( S )  Λ t S ( X ) , C [ n ]  = H n  Γ  X , R A 1 ( C t )   . 198 Construction of ﬁbred categories Corollary 5.2.31 Consider a morphism of abelian P -premotivic categories ϕ : A / / o o B : ψ Suppose there ar e admissible topologies t and t 0 , with t 0 ﬁner than t , such that the f ollowing conditions ar e veriﬁed. (i) A is compatible with t and B is compatible with t 0 . (ii) B and D ( B ) ar e compactly τ -g enerated. (iii) F or any scheme S , the functor ψ S : B S / / A S pr eser v es ﬁltering colimits. Then, ψ S : C ( B S ) / / C ( A S ) preserves A 1 -equiv alences between t 0 -ﬂasque objects. If mor eov er ψ S is exact, the functor ψ S pr eser v es A 1 -equiv alences. Proof W e already kno w that ψ S is a r ight Quillen functor , so that it preser v es local objects and A 1 -ﬁbrant objects. This implies also that ψ S preserves A 1 -equiv alences betw een A 1 -ﬁbrant objects (this is Ken Brown ’ s lemma [ Ho v99 , 1.1.12]). Let D be a t 0 -ﬂasque complex of B S . Then ψ S ( D ) is a t -ﬂasque complex of A S . It f ollo ws from Proposition 5.2.28 that R A 1 ( D ) is A 1 -local and that D / / R A 1 ( D ) is an A 1 - equiv alence. Lemma 5.2.27 implies the map ψ S ( D ) / / R A 1 ( ψ S ( D )) = ψ S ( R A 1 ( D )) is a an A 1 -equiv alence. This implies the ﬁrst asser tion. The last assertion is a direct consequence of the ﬁrst one.  5.2.32 Consider the usual cosimplicial scheme ∆ • deﬁned b y ∆ n = Sp ec ( Z [ t 0 , . . . , t n ]/( t 1 + · · · + t n − 1 ) ) ' A n (see [ MV99 ]). For any scheme S , w e get a cosimplicial object of A S , namely M S ( ∆ • S ) . Giv en any complex C of A S , w e deﬁne its associated Suslin singular complex as (5.2.32.1) C ∗ ( C ) = T ot ⊕ Hom ( M S ( ∆ • S ) , C ) , where Hom ( M S ( ∆ • S ) , C ) is considered as a bicomple x by the Dold-Kan cor respon- dence. The canonical map M S ( ∆ • S ) / / 1 S induces a map C / / C ∗ ( C ) . Lemma 5.2.33 F or any complex C of A S , the map C ∗ ( C ) / / Hom ( M S ( A 1 S ) , C ∗ ( C )) = C ∗ ( Hom ( M S ( A 1 S ) , C )) is a chain homotopy equiv alence. Proof The composite mor phism ( s 0 p × I d ) ∗ : M S ( A 1 × ∆ • S ) / / M S ( A 1 × ∆ • S ) , 5 Fibred der iv ed categor ies 199 where s 0 is the map induced b y the zero section, and p is the map induced b y the obvious projection of A 1 on its base, is chain homotopic to the identity . Indeed, the homotop y relation is given by the f or mula s n = n Õ i = 0 (− 1 ) i . ( 1 ⊗ S ψ i ) : M S ( A 1 × ∆ n + 1 S ) / / M S ( A 1 × ∆ n S ) where 1 is the identity of M S ( A 1 S ) , and ψ i is induced by the map ∆ n + 1 S / / A 1 × ∆ n S which sends the j -th v ertex v j , n + 1 to either 0 × v j , n , if j ≤ i , or to 1 × v j − 1 , n otherwise. This implies the lemma.  Lemma 5.2.34 F or any t -ﬂasque complex C of A S , w e hav e a canonical isomor - phism C ∗ ( C ) ' L lim / / n R Hom ( M S ( ∆ n S ) , C ) in D ( A S ) . This is a variation on the Dold-Kan correspondence. As a direct consequence, we get: Lemma 5.2.35 F or any complex C of A S , the map C / / C ∗ ( C ) is an A 1 - equiv alence. Proposition 5.2.36 If t is bounded in A , then, f or any t -ﬂasque complex C of A S , C ∗ ( C ) is A 1 -local. Proof Using the ﬁrst premotivic adjunction of e xample 5.2.21 and the fact that D ( A ) is compactly g enerated ( 5.1.30 ), w e can reduce the proposition to the case where A S is the categor y of preshea ves of abelian g roups o v er P / S , in which case this is w ell-kno wn.  5.2.d Constructible A 1 -local premotiv es 5.2.37 Consider an abelian P -premotivic categor y A compatible with an admissi- ble topology t . Assume that t is bounded in A (see Deﬁnition 5.1.28 ) and consider a bounded generating famil y N t S f or t -hyperco vers in A S . Let T A 1 S be the famil y of comple x es of C ( A S ) of shape M S ( A 1 X ){ i } / / M S ( X ){ i } f or a P -scheme X ov er S and a twist i ∈ I . Then the functor ( 5.1.31.1 ) ob viously induces the f ollo wing functor (5.2.37.1)  K b  M ( P / S , A )  / N t S ∪ T A 1 S  \ / / D eﬀ A 1 ( A S ) , 200 Constr uction of ﬁbred categor ies where the categor y on the left is the pseudo-abelian category associated to the V erdier quotient of K b  M ( P / S , A )  b y the thick subcategory generated by N t S ∪ T A 1 S . Applying Thomason’ s localization theorem [ Nee01 ], w e g et from Proposition 5.1.32 the f ollo wing result: Proposition 5.2.38 Consider the previous hypothesis and notations and assume that A is ﬁnitely τ -present ed. Then D eﬀ A 1 ( A ) is compactly τ -g ener ated. Moreo ver , the functor ( 5.2.37.1 ) is fully fait hful. Let us denote by D eﬀ A 1 , c ( A ) the subcategor y of D eﬀ A 1 ( A ) made of τ -constr uctible premotiv es in the sense of Deﬁnition 1.4.9 . T aking into account Proposition 1.4.11 , w e deduce from the abov e proposition the f ollowing corollary: Corollary 5.2.39 Under the assumptions of 5.2.38 , f or any pr emotiv e M in D eﬀ A 1 ( A S ) , the f ollowing conditions ar e equiv alent: (i) M is compact; (ii) M is τ -constructible. Mor eov er , the functor ( 5.2.37.1 ) induces an equivalence of categories:  K b  M ( P / S , A )  / N t S ∪ T A 1 S  \ / / D eﬀ A 1 , c ( A S ) . Example 5.2.40 With the notations of 5.1.34 , we get the follo wing equivalences of categories:  K b ( Λ ( Sm / S ) ) /( BG S ∪ T A 1 S )  \ / / D eﬀ A 1 , c ( S , Λ ) .  K b  Λ ( S f t / S )  / C D H S ∪ T A 1 S  \ / / D eﬀ A 1 , c  Sh c dh ( S f t / S , Λ )  . This statement is the analog of the embedding theorem [ VSF00 , chap. 5, 3.2.6]. Proposition 5.2.41 Assume P = S f t is the class of ﬁnite type (resp. separat ed and of ﬁnite type) morphisms. Let A be an abelian g eneralized premo tivic category compatible with an admis- sible topology t and satisfying the property (C) (r esp. (wC)) of P arag raph 5.1.35 . Then the triangulated g eneralized premo tivic category D eﬀ A 1 ( A ) is τ -continuous (r esp. weakly τ -continuous) — see Deﬁnition 4.3.2 . Proof The proof relies on the follo wing lemma: Lemma 5.2.42 U nder t he assump tions of the pr eceding proposition, for any mor - phism of schemes f : T / / S , the functor L f ∗ : D ( A S ) / / D ( A T ) pr eser v es A 1 -local complexes.  5 Fibred der iv ed categor ies 201 When f is a mor phism of ﬁnite type (resp. separated of ﬁnite type), the functor L f ∗ admits L f ] as a left adjoint and the lemma is clear . In the general case, one can wr ite f as a projective limit of a projectiv e sys tem of mor phisms of scheme ( f α : T α / / S ) α ∈ A such that f α is aﬃne of ﬁnite type. Recall from Proposition 5.1.36 , D ( A ) is τ -continuous. Thus, to check that f or an A 1 -local comple x C in D ( A S ) , the complex L f ∗ ( C ) is A 1 -local, we thus are reduced to pro v e that L f ∗ α ( C ) is A 1 -local which f ollow s from the ﬁrst treated case. The lemma is prov en. Giv en the full embedding D eﬀ A 1 ( A ) / / D ( A ) whose image is made of A 1 -local comple x es, the proposition no w directly f ollo ws from the pre vious lemma and the fact D ( A ) is τ -continuous.  Example 5.2.43 T aking into account the second point of Example 5.1.37 , the previous proposition can be applied to the categor y Sh t  S f t , Z  where t = Nis , ´ e t , c dh , qfh , h . Remar k 5.2.44 The previous proposition will be e xtended to the (non generalized) premotivic case in Corollar y 6.1.12 . 5.3 The stable A 1 -deriv ed premotivic category 5.3.a Modules Let A be an abelian P -premotivic categor y with g enerating set of twists τ . A car tesian commutative monoid R of A is a car tesian section of the ﬁbred category A o ver S such that f or any scheme S , R S has a commutativ e monoid structure in A S and f or any mor phism of schemes f : T / / S , the str uctural transition maps φ f : f ∗ ( R S ) / / R T are isomor phisms of monoids. Let us ﬁx a car tesian commutativ e monoid R of A . Consider a base scheme S . W e denote by R S - mo d the categor y of modules in the monoidal categor y A S o v er the monoid R S . For any P -scheme X / S and any twist i ∈ τ , we put R S ( X ){ i } = R S ⊗ S M S ( X ){ i } endo w ed with its canonical R S -module structure. The categor y R S - mo d is a Grothendieck abelian categor y such that the f org etful functor U S : R S - mo d / / A S is exact and conser vativ e. A famil y of generators f or R S - mo d is given b y the mod- ules R S ( X ){ i } f or a P -scheme X / S and a twist i ∈ τ . As A S is commutativ e, R S - mo d has a unique symmetr ic monoidal structure such that the free R S -module functor is symmetric monoidal. W e denote b y ⊗ R this tensor product. Note that R S ( X ) ⊗ R R S ( Y ) = R S ( X × S Y ) . Finally , the categor ies of modules R S - mo d f or m a symmetr ic monoidal P -ﬁbred categor y , such that the f ollo wing proposition holds (see 7.2.10 ). Proposition 5.3.1 Let A be a τ -gener ated abelian P -premotivic category and R be a cartesian commutative monoid of A . 202 Construction of ﬁbred categories Then the categor y R - mo d equipped with the structures introduced abov e is a τ -g enerated abelian P -pr emotivic category. Mor eov er , we have an adjunction of abelian P -pr emotivic categories: (5.3.1.1) R ⊗ (−) : A / / o o R - mo d : U . Remar k 5.3.2 With the h ypothesis of the preceding proposition, f or an y mor phism of schemes f : T / / S , the ex chang e transf ormation f ∗ U S / / U T f ∗ is an isomor- phism b y construction of R - mo d ( 7.2.10 ). Proposition 5.3.3 Let A be a τ -g enerat ed abelian P -pr emotivic categor y compat- ible with an admissible topology t . Consider a cartesian commutative monoid R of A suc h that for any scheme S , tensoring quasi-isomorphisms betw een coﬁbrant complexes by R S giv es quasi-isomorphisms (e.g. R S might be coﬁbrant (as a com- plex concentrated in degr ee zer o), or ﬂat). Then the abelian P -premo tivic categor y R - mo d is compatible with t . Proof In view of Proposition 5.1.26 , w e hav e onl y to sho w that R - mo d satisﬁes cohomological t -descent. Consider a t -h yperco v er p : X / / X in P / S . W e pro v e that the map p ∗ : R S ( X ) / / R S ( X ) is a quasi-isomorphism in C ( R S - mo d ) . The functor U S is conservativ e, and U S ( p ∗ ) is equal to the map: R S ⊗ S M S ( X ) / / R S ⊗ S M S ( X ) . But this is a quasi-isomor phism in C ( A S ) b y assumption on R S .  Remar k 5.3.4 According to Lemma 5.1.27 , f or any simplicial P -scheme X o v er S , an y twist i ∈ τ and any R S -module C , we get canonical isomorphisms: Hom K ( R S - mo d )  R S ( X ) { i } , C  ' Hom K ( A S ) ( M S ( X ) { i } , C ) (5.3.4.1) Hom D ( R S - mo d ) ( R S ( X ) { i } , C ) ' Hom D ( A S ) ( M S ( X ) { i } , C ) . (5.3.4.2) 5.3.b Symmetric sequences Let A be an abelian categor y . Let G be a g roup. An action of G on an object A ∈ A S is a mor phism of groups G / / Aut A ( A ) , g  / / γ A g . W e sa y that A is a G -object of A . A G -equivariant morphism A f / / B of G -objects of A is a mor phism f in A such that γ B g ◦ f = f ◦ γ A g . If E is an y object of A , we put G × E = É g ∈ G E considered as a G -object via the permutation isomor phisms of the summands. If H is a subgroup of G , and E is an H -object, G × E has tw o actions of H : the ﬁrst one, say γ , is obtained via the inclusion H ⊂ G , and the second one denoted b y γ 0 , is obtained using the structural action of H on E . W e deﬁne G × H E as the coequalizer of the famil y of mor phisms ( γ σ − γ 0 σ ) σ ∈ H , and consider it equipped with its induced action of G . 5 Fibred der iv ed categor ies 203 Deﬁnition 5.3.5 Let A be an abelian categor y . A symmetric sequence of A is a sequence ( A n ) n ∈ N such that f or each n ∈ N , A n is a S n -object of A . A morphism of symmetr ic sequences of A is a collection of S n -equiv ariant mor phism ( f n : A n / / B n ) n ∈ N . W e let A S be the categor y of symmetric sequences of A . It is straightf or w ard to check A S is abelian. For any integer n ∈ N , we deﬁne the n -th ev aluation functor as f ollo ws: e v n : A S / / A , A ∗  / / A n . An y object A of A can be considered as the tr ivial symmetr ic sequence ( A , 0 , . . . ) . The functor i 0 : A  / / ( A , 0 , . . . ) is obviousl y left adjoint to e v 0 and w e obtain an adjunction (5.3.5.1) i 0 : A / / o o A S : e v 0 . Remark i 0 is also r ight adjoint to e v 0 . Thus, i 0 preserves ev er y limit and colimit. For any integer n ∈ N and any symmetr ic sequence A ∗ of A , we put ( A ∗ {− n } ) m =  S m × S m − n A m − n if m ≥ n 0 otherwise. (5.3.5.2) This deﬁne an endofunctor on A S , and w e ha ve A ∗ {− n } { − m } = A ∗ {− n − m } (through a canonical isomor phism). Remark ﬁnally that for any integer n ∈ N , the functor i n : A / / A S , A  / / ( i 0 ( A )){ − n } is left adjoint to e v n . Remar k 5.3.6 Let S be the category of ﬁnite sets with bijectiv e maps as morphisms. Then the categor y of symmetr ic sequences is canonically equivalent to the categor y of functors S / / A . This presentation is useful to deﬁne a tensor product on A S . Deﬁnition 5.3.7 Let A be a symmetr ic closed monoidal abelian categor y . Giv en tw o functors A ∗ , B ∗ : S / / A , we put: E ⊗ S F : S  / / A N  / / É N = P t Q E ( P ) ⊗ F ( Q ) . If 1 A is the unit object of the monoidal categor y A , the category A S is then a symmetric closed monoidal categor y with unit object i 0 ( 1 A ) . 5.3.8 Let A be an object of A . Then the n -th tensor pow er A ⊗ n of A is endo w ed with a canonical action of the group S n through the structural permutation isomor phism of the symmetric structure on A . Thus the sequence Sym ( A ) = ( A ⊗ n ) n ∈ N is a symmetric sequence. 204 Construction of ﬁbred categor ies Moreo v er , the isomor phism A ⊗ n ⊗ A ⊗ m / / A ⊗ n + m is S n × S m -equiv ariant. Thus it induces a mor phism µ : Sym ( A ) ⊗ S Sym ( A ) / / Sym ( A ) of symmetr ic sequences. W e also consider the ob vious morphism η : i 0 ( 1 A ) = i 0 ( A ⊗ 0 ) / / Sym ( A ) . One can chec k easily that Sym ( A ) equipped with the multiplication µ and the unit η is a commutativ e monoid in the monoidal categor y A S . Deﬁnition 5.3.9 Let A be an abelian symmetr ic monoidal categor y . The commu- tativ e monoid Sym ( A ) of A S deﬁned abo v e will be called the symmetr ic monoid generated by A . Remar k 5.3.10 One can describe Sym ( A ) by a univ ersal property: given a commuta- tiv e monoid R in A S , to give a mor phism of commutative monoids Sym ( A ) / / R is equiv alent to give a mor phism A / / R 1 in A . 5.3.11 Consider an abelian P -premotivic category A . Consider a base scheme S . According to the previous paragraph, the categor y A S S is an abelian categor y , endo w ed with a symmetr ic tensor product ⊗ S S . For any P -scheme X / S and any integer n ∈ N , using ( 5.3.5.2 ), we put M S ( X , A S ){ − n } = i 0 ( M S ( X , A )) {− n } . It is immediate that the class of symmetr ic sequences of the form M S ( X , A S ){ − n } f or a smooth S -scheme X and an integ er n ≥ 0 is a generating famil y f or the abelian category A S S which is theref ore a Grothendieck abelian categor y . It is clear that for an y P -scheme X and Y o v er S , M S ( X , A S ){ − n } ⊗ S S M S ( Y , A S ){ − n } = M S ( X × S Y , A S ){ − n } . Giv en a mor phism (resp. P -morphism) of schemes f : T / / S and a symmetr ic sequence A ∗ of A S , we put f ∗ S ( A ∗ ) = ( f ∗ A n ) n ∈ N (resp. f S ] ( A ∗ ) = ( f ] A n ) n ∈ N ). This deﬁnes a functor f ∗ S : A S S / / A S T (resp. f S ] : A S T / / A S S ) which is obviousl y r ight e xact. Thus, the fun ctor f ∗ S admits a r ight adjoint which we denote by f S ∗ . When f is in P , w e check easily the functor f S ] is left adjoint to f ∗ S . From cr iterion 1.1.42 and Lemma 1.2.13 , we chec k easily the follo wing proposi- tion: Proposition 5.3.12 Consider the pr evious hypothesis and notations. The association S  / / A S S tog ether with the structures introduced abov e deﬁnes an N × τ -g enerated abelian P -pr emotivic category. Mor eov er , the diﬀerent adjunctions of the form ( 5.3.5.1 ) o v er each ﬁbers ov er a sc heme S deﬁne an adjunction of P -premotivic categories: (5.3.12.1) i 0 : A / / o o A S : e v 0 Indeed, i 0 is trivially compatible with twists. Proposition 5.3.13 Let A be an abelian P -premotivic categor y , and t be an ad- missible topology . If A is compatible with t then A S is compatible with t . 5 Fibred der iv ed categor ies 205 Proof This is based on the follo wing lemma (see [ CD09 , 7.5, 7.6]): Lemma 5.3.14 F or any complex C of A S , any complex E of A S S and any integ er n ≥ 0 , there are canonical isomorphisms: Hom K ( A S S ) ( i 0 ( C ){ − n } , E ) ' Hom K ( A S ) ( C , E n ) (5.3.14.1) Hom D ( A S S ) ( i 0 ( C ){ − n } , E ) ' Hom D ( A S ) ( C , E n ) (5.3.14.2) If A is compatible with t , this implies that E is local (resp. t -ﬂasque) if and only if f or an y n ≥ 0 , E n is local (resp. t -ﬂasque). This concludes.  5.3.c Symmetric T ate spectra 5.3.15 Consider an abelian P -premotivic category A . For any scheme S , the unit point of G m , S deﬁnes a split monomor phism of A - premotiv es 1 S / / M S ( G m , S ) . W e denote by 1 S { 1 } the cokernel of this monomor- phism and call it the suspended T ate S -pr emotiv e with coeﬃcients in A . The collec- tion of these objects f or any scheme S is a car tesian section of A denoted by 1 { 1 } . For any integer n ≥ 0 , we denote b y 1 { n } its n -the tensor pow er . With the notations of 5.3.9 , we deﬁne the symmetric T ate spectr um o v er S as the symmetric sequence 1 S {∗ } = S y m ( 1 S { 1 } ) in A S S . The cor responding collection deﬁnes a cartesian commutativ e monoid of the ﬁbred category A S , called the absolute T ate spectrum . Deﬁnition 5.3.16 Consider an abelian P -premotivic category A . W e denote b y Sp ( A ) the abelian P -premotivic categor y of modules ov er 1 {∗ } in the categor y A S . The objects of Sp ( A ) are called the abelian (symmetr ic) T ate spectra. 80 The categor y Sp ( A ) is ( N × τ ) -generated. Composing the adjunctions ( 5.3.1.1 ) and ( 5.3.12.1 ), w e get an adjunction (5.3.16.1) Σ ∞ : A / / o o Sp ( A ) : Ω ∞ of abelian P -premotivic categories. Let us e xplicit the deﬁnition. An abelian T ate spectr um ( E , σ ) is the data of : 1. f or an y n ∈ N , an object E n of A S endo w ed with an action of S n 2. f or an y n ∈ N , a mor phism σ n : E n { 1 } / / E n + 1 in A S such that the composite map E m { n } σ m { n − 1 } / / E m + 1 { n − 1 } / / . . . σ m + n − 1 / / E m + n 80 As we will almost ne ver consider non symmetr ic spectra, w e will cancel the w ord "symmetr ic" in our terminology . 206 Construction of ﬁbred categor ies is S n × S m -equiv ariant with respect to the canonical action of S n on 1 S { n } and the structural action of S m on E m . By deﬁnition, Ω ∞ ( E ) = E 0 . Thus, the functor Ω ∞ is e xact. Giv en an object A of A S , the abelian T ate spectr um Σ ∞ A is deﬁned suc h that ( Σ ∞ A ) n = A { n } with the action of S n giv en b y its action on 1 S { n } by per mutations of the factors. Be careful we consider the categor y Sp ( A S ) as N -twisted by negativ e twists. For an y abelian T ate spectr um E ∗ , ( E ∗ {− n } ) m = S m × S m − n E m − n f or n ≥ m . 5.3.17 Consider a mor phism ϕ : A / / B of abelian P -premotivic categories. Then as ϕ ( 1 A { 1 } ) = 1 B { 1 } , ϕ can be extended to abelian T ate spectra in such a wa y that the follo wing diagram commutes: A ϕ / / Σ ∞ A   B Σ ∞ B   Sp ( A ) Sp ( ϕ ) / / Sp ( B ) . (Of course the obvious diagram for the cor responding r ight adjoints also commutes.) Deﬁnition 5.3.18 For an y scheme S , a complex of abelian T ate spectra ov er S will be called simply a T ate spectrum o ver S . A T ate spectr um E is a bigraded object. In the notation E m n , the inde x m cor responds to the (cochain) complex str ucture and the index n to the symmetric sequence structure. From propositions 5.3.3 and 5.3.13 , we g et the f ollo wing: Proposition 5.3.19 Let A be an abelian P -premo tivic categor y compatible with an admissible topology t . Then Sp ( A ) is compatible with t . Note also that remark 5.3.4 and Lemma 5.3.14 implies that f or an y simplicial P - scheme X o ver S , any integer n ∈ N , and any T ate spectrum E , we hav e canonical isomorphisms: Hom K ( Sp ( A S )) ( Σ ∞ M S ( X , A ){ − n } , E ) ' Hom K ( A S ) ( Σ ∞ M S ( X , A ) , E n ) (5.3.19.1) Hom D ( Sp ( A S )) ( Σ ∞ M S ( X , A ){ − n } , E ) ' Hom D ( A S ) ( Σ ∞ M S ( X , A ) , E n ) (5.3.19.2) A ccording to the proposition, the category C ( Sp ( A S )) of T ate spectra o v er S has a t - descent model structure. The previous isomor phisms allow to descr ibe this structure as f ollo ws: 1. For any simplicial P -scheme X ov er S , and any integer n ≥ 0 , the T ate spectrum Σ ∞ M S ( X , A ){ − n } is coﬁbrant. 5 Fibred der iv ed categor ies 207 2. A T ate spectr um E ov er S is ﬁbrant if and only if for an y integer n ≥ 0 , the comple x E n o v er A S is local ( i.e. t -ﬂasque). 3. Let f : E / / F be a mor phism of T ate spectra o ver S . Then f is a ﬁbration (resp. q uasi-isomor phism) if and onl y if f or an y integer n ≥ 0 , the morphism f n : E n / / F n of comple xes ov er A S is a ﬁbration (resp. quasi-isomor phism). Note that proper ties (2) and (3) f ollow s from ( 5.3.4.1 ) and ( 5.3.14.1 ). 5.3.20 W e can also introduce the A 1 -localization of this model structure. The cor re- sponding homotopy categor y is the A 1 -derived P -premotivic categor y D eﬀ A 1 ( Sp ( A )) introduced in 5.2.16 . The isomor phism ( 5.3.19.2 ) gives the f ollowing asser tion: From the abo v e, a T ate spectr um E is A 1 -local if and only if f or an y integer n ≥ 0 , E n is A 1 -local. 1. A T ate spectr um E o ver S is A 1 -local if and only if for any integer n ≥ 0 , the comple x E n o v er A S is A 1 -local. 2. Let f : E / / F be a morphism of T ate spectra ov er S . Then f is a A 1 -local ﬁbration (resp. weak A 1 -equiv alence) if and only if f or any integer n ≥ 0 , the morphism f n : E n / / F n of complex es o ver A S is a A 1 -local ﬁbration (resp. w eak A 1 -equiv alence). As a consequence, the isomor phism ( 5.3.19.2 ) induces an isomor phism Hom D eﬀ A 1 ( Sp ( A S )) ( Σ ∞ M S ( X , A ){ − n } , E ) ' Hom D eﬀ A 1 ( A S ) ( Σ ∞ M S ( X , A ) , E n ) . (5.3.20.1) Similarl y , the adjunction ( 5.3.16.1 ) induces an adjunction of tr iangulated P - premotivic categories (5.3.20.2) L Σ ∞ : D eﬀ A 1 ( A ) / / o o D eﬀ A 1 ( Sp ( A )) : R Ω ∞ . 5.3.d Symmetric T ate Ω -spectra 5.3.21 The ﬁnal step is to localize fur ther the categor y D eﬀ A 1 ( Sp ( A )) . The aim is to relate the positiv e twists on D eﬀ A 1 ( A ) obtained by tensor ing with 1 S { 1 } and the negativ e twists on D eﬀ A 1 ( Sp ( A )) induced by the consideration of symmetric sequences. Let X be a P -scheme ov er S . From the deﬁnition of Σ ∞ , there is a canonical morphism of abelian T ate spectra:  Σ ∞  1 S { 1 }   {− 1 } / / Σ ∞ 1 S . T ensoring this map b y Σ ∞ M S ( X , A ) {− n } f or any P -scheme X ov er S and an y integer n ∈ N , we obtain a famil y of mor phisms of T ate spectra concentrated in cohomological degree 0 : 208 Construction of ﬁbred categories  Σ ∞  M S ( X , A ) { 1 }   {− n − 1 } / / Σ ∞ M S ( X , A ) {− n } . W e denote by W Ω this famil y and put W Ω , A 1 = W Ω ∪ W A 1 . Obviousl y , W Ω , A 1 is stable by the operations f ∗ and f ] . Deﬁnition 5.3.22 Let A be an abelian P -premotivic categor y compatible with an admissible topology t . With the notations introduced abov e, we deﬁne the stable A 1 - deriv ed P -premo tivic categor y with coeﬃcients in A as the der iv ed P -premotivic category D A 1 ( A ) : = D ( Sp ( A ))[ W − 1 Ω , A 1 ] deﬁned in Corollar y 5.2.5 . 5.3.23 A ccording to this deﬁnition, we g et the follo wing identiﬁcation: D A 1 ( A ) = D eﬀ A 1 ( Sp ( A ))[ W − 1 Ω ] . Using the left Bousﬁeld localization of the A 1 -local model structure on C ( Sp ( A )) , w e thus obtain a canonical adjunction of tr iangulated P -ﬁbred premotivic categories D eﬀ A 1 ( Sp ( A )) / / o o D eﬀ A 1 ( Sp ( A ))[ W − 1 Ω ] which allow s us to descr ibe D A 1 ( A S ) as the full subcategor y of D eﬀ A 1 ( Sp ( A S )) made of T ate spectra which are W Ω -local in D eﬀ A 1 ( Sp ( A S )) . Recall a T ate spectrum E is a sequence of comple x es ( E n ) n ∈ N o v er A S together with suspension maps in C ( A S ) σ n : 1 S { 1 } ⊗ E n / / E n + 1 . From this, we deduce a canonical mor phism 1 S { 1 } ⊗ L E n / / E n + 1 in D eﬀ A 1 ( A ) whose adjoint mor phism w e denote by (5.3.23.1) u n : E n / / R Hom D eﬀ A 1 ( A S ) ( 1 S { 1 } , E n + 1 ) A ccording to ( 5.3.20.1 ), the condition that E is W Ω -local in D eﬀ A 1 ( Sp ( A )) is equivalent to ask that f or any integer n ≥ 0 , the map ( 5.3.23.1 ) is an isomor phism in D eﬀ A 1 ( Sp ( A )) . Considering the adjunction ( 5.3.20.2 ), we obtain ﬁnally an adjunction of tr iangu- lated P -ﬁbred categor ies: (5.3.23.2) Σ ∞ : D eﬀ A 1 ( A ) / / o o D eﬀ A 1 ( Sp ( A )) / / o o D A 1 ( A ) : Ω ∞ . Note that tautologicall y , the T ate spectr um Σ ∞ ( 1 S { 1 } ) has a tensor inv erse given b y the spectrum ( Σ ∞ 1 S ){ − 1 } in D A 1 ( A S ) . Thus, we hav e obtained from the abelian premotivic category A a triangulated premotivic categor y D A 1 ( A S ) which satisﬁes the proper ties: • the homotop y proper ty (Htp); • the stability property (Stab); 5 Fibred der iv ed categor ies 209 • the t -descent proper ty . As we will see in the follo wings, the construction satisﬁes a univ ersality proper ty that the reader can already guess. Deﬁnition 5.3.24 Consider the assumptions of deﬁnition 5.3.22 . For any scheme S , w e say that a T ate spectr um E ov er S is a T ate Ω -spectrum if the f ollo wing conditions are fulﬁlled: (a) For any integer n ≥ 0 , E n is t -ﬂasque and A 1 -local. (b) For any integer n ≥ 0 , the adjoint of the structural suspension map E n / / Hom C ( A S ) ( 1 S { 1 } , E n + 1 ) is a quasi-isomorphism. In par ticular , a T ate Ω -spectr um is W Ω -local in D eﬀ A 1 ( Sp ( A S )) . In fact, it is also W Ω , A 1 - local in the categor y D ( Sp ( A S )) so that the categor y D A 1 ( A ) is also equiv alent to the full subcategor y of D ( Sp ( A S )) spanned by T ate Ω -spectra. Fibrant objects of the W Ω , A 1 -local model categor y on C ( Sp ( A )) obtained in deﬁnition 5.3.22 are ex actly the T ate Ω -spectra. Proposition 5.3.25 Consider the abov e notations. Let S be a base scheme. 1. If the endofunctor D eﬀ A 1 ( A S ) / / D eﬀ A 1 ( A S ) , C  / / R Hom D eﬀ A 1 ( A S ) ( 1 S { 1 } , C ) is conser v ative, then the functor Ω ∞ S is conser v ative. 2. If the T ate twist E  / / E ( 1 ) is fully faithful in D eﬀ A 1 ( A S ) , then Σ ∞ S is fully faithful. 3. If the T ate twist E  / / E ( 1 ) induces an auto-equivalence of D eﬀ A 1 ( A S ) , then ( Σ ∞ S , Ω ∞ S ) ar e adjoint equivalences of categories. Remar k 5.3.26 Similar statements can be obtained f or the derived categories rather than the A 1 -derived categor ies. W e left their f or mulation to the reader . Proof Consider point (1). W e hav e to prov e that f or an y W Ω -local T ate spectr um E in D eﬀ A 1 ( Sp ( A S )) , if R Ω ∞ ( E ) = 0 , then E = 0 . But R Ω ∞ ( E ) = Ω ∞ ( E ) = E 0 (see 5.3.20 ). Because for an y integer n ≥ 0 , the map ( 5.3.23.1 ) is an A 1 -equiv alence, we deduce that f or an y integ er n ∈ Z , the complex E n is (weakl y) A 1 -acy clic. According to ( 5.3.20.1 ), this implies E = 0 — because D A 1 ( A S ) is N -generated. Consider point (2). W e want to pro v e that f or any complex C o v er A S , the counit map C / / R Ω ∞ L Σ ∞ ( C ) is an isomor phism. It is enough to treat the case where C is coﬁbrant. Considering the left adjoint L Σ ∞ of ( 5.3.20.2 ), w e ﬁrst pro ve that L Σ ∞ ( C ) is W Ω - local. Because C is coﬁbrant, this T ate spectrum is equal in degree n to the comple x C { n } (with its natural action of S n ). Moreo v er , the suspension map is given by the isomorphism (in the monoidal categor y C ( A S ) ) 210 Cons tr uction of ﬁbred categor ies σ n : 1 S { 1 } ⊗ S C { n } / / C { n + 1 } . In par ticular , the cor responding map in D eﬀ A 1 ( A S ) σ 0 n : 1 S { 1 } ⊗ L S C { n } / / C { n + 1 } . is canonically isomor phic to 1 S { 1 } ⊗ L S C { n } 1 ⊗ 1 / / 1 S { 1 } ⊗ L S C { n } . Thus, because the T ate twist is fully f aithful in D eﬀ A 1 ( A S ) , the adjoint map to σ 0 n is an A 1 -equiv alence. In other words, L Σ ∞ ( C ) is W Ω -local. But then, as C is coﬁbrant, C = Ω ∞ Σ ∞ ( C ) = R Ω ∞ L Σ ∞ ( C ) , and this concludes. Point (3) is then a consequence of (1) and (2).  Remar k 5.3.27 1. The construction of the tr iangulated categor y D A 1 ( A ) can also be obtained using the more general constr uction of [ CD09 , §7] — see also [ Ho v01 , 7.11] and [ A y o07b , chap. 4] f or e ven more g eneral accounts. Here, w e e xploit the simpliﬁcation ar ising from the f act that we inv er t a comple x concentrated in degree 0 : this allo wed us to describe D A 1 ( A ) simpl y as a V erdier quotient of the der iv ed categor y of an abelian category . Ho w ev er , we can also consider the category of symmetric spectra in C ( A S ) with respect to one of the complex es 1 S ( 1 )[ 2 ] or 1 S ( 1 ) and this leads to the equivalent categories; see [ Hov01 , 8.3]. 2. Point (3) of Proposition 5.3.25 is a par ticular case of [ Hov01 , 8.1]. 5.3.28 Consider a mor phism of abelian P -premotivic categories ϕ : A / / o o B : ψ such that A (resp. B ) is compatible with a sys tem of topology t (resp. t 0 ). Suppose t 0 is ﬁner than t . According to 5.3.17 , we obtain an adjunction of abelian P -premotivic categories ϕ : C ( Sp ( A )) / / o o C ( Sp ( B )) : ψ . The pair ( ϕ S , ψ S ) is a Quillen adjunction f or the stable model str uctures (appl y again [ CD09 , prop. 3.11]). Thus w e obtain a morphism of tr iangulated P -premotivic categories: L ϕ : D A 1 ( A ) / / o o D A 1 ( B ) : R ψ . Remar k 5.3.29 Under the light of Proposition 5.3.25 , the categor y D A 1 ( A ) might be considered as the univ ersal der iv ed P -premotivic categor y T with a mor phism D ( A ) / / T , and such that T satisﬁes the homotopy and the stability proper ty . This can be made precise in the setting of algebraic derivators or of dg-categor ies (or an y other kind of stable ∞ -categor ies). Proposition 5.3.30 Let t and t 0 be tw o admissible topologies, with t 0 ﬁner than t . Then D A 1 ( Sh t 0 ( P , Λ )) is canonically equiv alent to the the full subcategor y of D A 1 ( Sh t ( P , Λ )) spanned by the objects which satisfy t 0 -descent. 5 Fibred der iv ed categor ies 211 Proof It is suﬃcient to prov e this proposition in the case where t is the coarse topology . W e deduce from [ A y o07b , 4.4.42] that, f or any scheme S in S , w e hav e D A 1 ( Sh t 0 ( P / S , Λ )) = D ( PSh ( P / S , Λ ) ) [ W − 1 ] , with W = W t 0 ∪ W A 1 ∪ W Ω , where W t 0 is the set of maps of shape Σ ∞ M S ( X ) { n } [ i ] / / Σ ∞ M S ( X ){ n } [ i ] , f or an y t 0 -h yperco v er X / / X and an y integers n ≤ 0 and i . The asser tion is then a par ticular case of the description of the homotopy category of a left Bousﬁeld localization.  Example 5.3.31 W e hav e the s table v ersions of the P -premotivic categories intro- duced in e xample 5.2.17 : 1) Consider the admissible topology t = Nis . Follo wing F . Morel, we deﬁne the stable A 1 -deriv ed premo tivic category as (see also the construction of [ A y o07b ]): D A 1 , Λ : = D A 1 ( Sh Nis ( Sm , Λ )) and D A 1 , Λ : = D A 1  Sh Nis  S f t , Λ   , as w ell as the gener alized stable A 1 -deriv ed premo tivic category 81 (5.3.31.1) D A 1 , Λ : = D A 1  Sh Nis  S f t , Λ   . Giv en a scheme S , we shall also write: (5.3.31.2) D A 1 ( S , Λ ) : = D A 1 , Λ ( S ) and D A 1 ( S , Λ ) : = D A 1 , Λ ( S ) . In the case when t = ´ e t , we get the tr iangulated premotivic categor ies of é tale pr emotiv es : D A 1 ( Sh ´ e t ( Sm , Λ )) and D A 1  Sh ´ e t  S f t , Λ   . In each of these cases, w e denote b y Σ ∞ Λ t S ( X ) the premotiv e associated with a smooth S -scheme X . From the adjunction ( 5.1.24.2 ), we get an adjunction of tr iangulated premotivic categories: a ´ e t : D A 1 , Λ / / o o D A 1 ( Sh ´ e t ( Sm , Λ )) : R O ´ e t . 2) Assume P = S f t : Consider the S f t -admissible topology t = h (resp. t = qfh ). In [ V oe96 ], V o- ev odsky has introduced the categor y of eﬀectiv e h -motiv es (resp. qfh -motiv es). A ccording to the theor y presented abo v e, one can e xtend this deﬁnition to the stable 81 W e will see in Example 6.1.10 that the generalized v ersion contains the usual one as a full subcategory . 212 Construction of ﬁbred categories setting: one deﬁnes the categor y of stable h -motiv es (resp. qfh -motiv es ) ov er S with coeﬃcients in Λ as: DM h ( S , Λ ) : = D A 1  Sh h  S f t / S , Λ   . resp. DM qfh ( S , Λ ) : = D A 1  Sh qfh  S f t / S , Λ   . In other words, this is the stable A 1 -derived categor y of h -sheav es (resp. qfh -shea v es) of Λ -modules. Moreov er , we get the g eneralized triangulated premo tivic categor y of h -motiv es (resp. qfh -motives) with coeﬃcients in Λ o v er S : DM h , Λ : = D A 1  Sh h  S f t , Λ   . resp. DM qfh , Λ : = D A 1  Sh qfh  S f t , Λ   . For an S -scheme of ﬁnite type X , w e will denote by Σ ∞ Λ h S ( X ) (resp Σ ∞ Λ qfh S ( X ) ) the corresponding premotiv e associated with X in DM t ( S , Λ ) . Note that the h - sheaﬁﬁcation functor induces a premotivic adjunction (see Paragraph 5.3.28 ): (5.3.31.3) DM qfh , Λ / / o o DM h , Λ . These generalized premotivic categor ies are too big to be reasonable (in particular f or the localization proper ty — see Remark 2.3.4 ). Theref ore, we introduce the tr ian- gulated category DM t ( S , Λ ) as the localizing subcategory of DM t ( S , Λ ) generated b y objects of shape Σ ∞ Λ t S ( X )( p )[ q ] f or an y smooth S -scheme of ﬁnite type X and an y integers p and q . The ﬁbred categor y DM h , Λ (resp. DM qfh , Λ ) deﬁned abo v e is premotivic. W e call it the pr emotivic category of h -motiv es (resp. qfh -motiv es) . The famil y of inclusions (5.3.31.4) DM t ( S , Λ ) / / DM t ( S , Λ ) inde x ed by a scheme S deﬁnes a premotivic morphism (the e xistence of r ight adjoints is ensured by the Brown representability theorem). Remar k 5.3.32 When Λ = Q , we will show that the categories DM h , Q and DM qfh , Q are equiv alent and satisfy the axioms of a motivic categor y . In fact, the y are equiv alent to the categor y of Beilinson motiv es. See Theorem 16.1.2 for all these results. Proposition 5.3.33 Consider the notations of the second point in the abov e example. Then the pr emotivic category DM t , Λ satisﬁes t -descent. Proof This is true f or DM t , Λ b y construction, which implies f ormally the asser tion f or DM t , Λ .  Remar k 5.3.34 According to Proposition 5.2.10 and Remark 5.3.29 , for any admis- sible topology t , D A 1 ( Sh t ( P , Z ) ) is the univ ersal deriv ed P -premotivic category satisfying t -descent as well as the homotopy and stability proper ties. 5 Fibred der iv ed categor ies 213 A crucial ex ample f or us: the stable A 1 -derived premotivic categor y D A 1 is the univ ersal der iv ed premotivic categor y satisfying the proper ties of homotopy , of stability and of Nisnevic h descent. 5.3.35 W e assume P = Sm . Let Sh • ( Sm ) be the categor y of pointed Nisnevich shea v es of sets. Consider the pointed v ersion of the adjunction of P -premotivic categor ies N : ∆ o p Sh • ( Sm ) / / o o C ( Sh Nis ( Sm , Z ) ) : K constructed in 5.2.25 . If we consider on the left-hand side the A 1 -model categor y deﬁned by Blan- der [ Bla03 ], ( N S , K S ) is a Quillen adjunction for any scheme S . W e consider ( G m , 1 ) as a constant pointed simplicial sheaf. The constr uction of symmetric G m -spectra respectivel y to the model category ∆ o p Sh • ( Sm ) can no w be carr ied out follo wing [ Jar00 ] or [ A y o07b ] and yields a symmetr ic monoidal model category whose homotopy categor y is the stable homotopy categor y of Morel and V oe v odsky SH ( S ) . Using the functoriality statements [ Hov01 , th. 8.3 and 8.4], w e ﬁnally obtain a P -premotivic adjunction (5.3.35.1) N : SH / / o o D A 1 : K . The functor K is the analog of the Eilenberg-Mac Lane functor in alg ebraic topology; in f act, this adjunction is actuall y induced by the Eilenberg-MacLane functor (see [ A y o07b , chap. 4]). In particular, as the rational model categor y of topological (symmetr ic) S 1 -spectra is Quillen equiv alent to the model category of comple x es of Q -v ector spaces, w e ha v e a natural equiv alence of premotivic categor ies (5.3.35.2) SH Q / / o o D A 1 , Q , (where SH Q ( S ) denotes the V erdier quotient of SH ( S ) by the localizing subcategory generated by compact torsion objects). 5.3.36 W e can extend the considerations of Ex ample 5.1.25 and P aragraph 5.2.22 on changing coeﬃcients in categories of shea ves. Let t be an admissible topology and ϕ : Λ / / Λ 0 be an e xtension of rings. Using the P -premotivic adjunction ( 5.1.25.1 ) and according to Paragraph 5.3.28 , w e g et an adjunction of tr iangulated P -premotivic categor ies: L ϕ ∗ : D A 1  Sh t ( P , Λ )  / / o o D A 1  Sh t ( P , Λ 0 )  : R ϕ ∗ . Giv en two T ate spectra C and D of t -shea v es of Λ -modules o v er P S , we get a canonical mor phism of Λ 0 -modules: (5.3.36.1) Hom D A 1 ( Sh t ( P S , Λ ) )  C , D  ⊗ Λ Λ 0 / / Hom D A 1 ( Sh t ( P S , Λ 0 ) )  L ϕ ∗ ( C ) , L ϕ ∗ ( D )  Then the stable v ersion of Proposition 5.2.23 holds (the proof is the same): 214 Cons truction of ﬁbred categor ies Proposition 5.3.37 Consider the abov e assumptions. Then the map ( 5.3.36.1 ) is an isomorphism in the two following cases: 1. If Λ 0 is a free Λ -module and C is compact; 2. If Λ 0 is a free Λ -module of ﬁnite rank. 5.3.e Constructible premotivic spectra Lemma 5.3.38 Let A be an abelian P -premo tivic cat egor y compatible with a topology t and suc h that the categor y A 1 -deriv ed category D eﬀ A 1 ( A ) satisﬁes Nis- nevic h descent. Then, for any scheme S , the non-trivial cyclic per mutation ( 123 ) of order 3 acts as the identity on the pr emotiv e 1 S { 1 } ⊗ 3 in D eﬀ A 1 ( A S ) . Proof Using e xample 5.2.21 , it is suﬃcient to pro ve this in D eﬀ A 1 , Λ ( S ) , which is w ell-kno wn; see for ex ample [ A y o07b , 4.5.65].  Proposition 5.3.39 Consider the hypothesis of the pr evious lemma and assume that the triangulated premo tivic categor y D eﬀ A 1 ( A ) is compactly τ -g enerat ed. Then, f or any sc heme S , any couple of integ ers ( i , a ) , any compact object C of D eﬀ A 1 ( A S ) and any T ate spectrum E in A S , w e have a canonical isomorphism Hom D A 1 ( A S ) ( L Σ ∞ ( C ){ a } , E [ i ]) ' lim / / r > > 0 Hom D eﬀ A 1 ( A S ) ( C { a + r } , E r [ i ]) . Proof Giv en the previous lemma, this is a direct consequence of [ A y o07b , theorems 4.3.61 and 4.3.79].  Corollary 5.3.40 Under the assumptions of the pr eceding proposition, the triangu- lated categor y D A 1 ( A S ) is compactly ( Z × τ ) -gener ated where the factor Z corre- sponds to the T ate twist. Mor e precisely , if D eﬀ A 1 , c ( A S ) denotes the categor y of compact objects in D eﬀ A 1 ( A S ) , then the category of compact objects in D A 1 ( A S ) is canonically equiv alent to the pseudo-abelian completion of the category obtained as the 2 -colimit of the f ollowing diagr am: D eﬀ A 1 , c ( A S ) ⊗ 1 S { 1 } / / D eﬀ A 1 , c ( A S ) / / · · · / / D eﬀ A 1 , c ( A S ) ⊗ 1 S { 1 } / / D eﬀ A 1 , c ( A S ) / / · · · 5.3.41 Let A be an abelian P -premotivic categor y compatible with an admissible topology t . Assume that: • The topology t is bounded in A (Deﬁnition 5.1.28 ). • The abelian P -premotivic category A is ﬁnitely τ -presented. W e will denote by N t S a bounded generating famil y f or t -h yperco v ers in A S . 5 Fibred der iv ed categor ies 215 Recall from Proposition 5.2.38 that the categor y of compact objects of the tri- angulated categor y D eﬀ A 1 ( A S ) is canonically equivalent to the triangulated monoidal category:  K b  Z S ( Sm / S ; A )  /( N t S ∪ T A 1 S )  \ Let us denote by D A 1 , g m ( A S ) the categor y obtained from the monoidal categor y on the left-hand side of the abo ve functor by f ormally in v er ting the T ate twist Z A S ( 1 ) . Because D A 1 ( A ) satisﬁes the stability proper ty b y construction, we readily obtains a canonical monoidal functor (5.3.41.1) D A 1 , g m ( A S ) / / D A 1 ( A S ) . Then applying Proposition 5.2.38 , the abo v e corollar y and Proposition 1.4.11 , we deduce: Corollary 5.3.42 Consider the abov e hypothesis and notations. Then the triangulated premo tivic category D A 1 ( A ) is compactly ( Z × τ ) - g enerated. F or any premo tive M in D A 1 ( A S ) the follo wing conditions are equiv alent : (i) M is compact; (ii) M is ( Z × τ ) -constructible. Mor eov er , t he functor ( 5.3.41.1 ) is fully fait hful and has for essential imag e the compact ( i.e. τ -constructible) objects of D A 1 ( A S ) . Example 5.3.43 From the considerations of Example 5.2.40 , we get that, f or any scheme S , the category of compact objects of D A 1 ( S , Λ ) (resp., of its c dh -local counterpar t D A 1  Sh c dh ( S f t / S , Λ )  ) is obtained from the monoidal tr iangulated category K b ( Λ ( Sm / S ) ) (resp. K b  Λ ( S f t / S )  ) b y the f ollowing steps: • one mods out b y the triangulated subcategor ies T A 1 S and B G S (resp. C D H S ) corresponding to the A 1 -homotop y property and the Bro wn-Gersten tr iangles (resp. cdh-triangles), • one takes the pseudo-abelian en velope, • one f ormally in v er ts the T ate twist. Proposition 5.3.44 Assume P = S f t is the class of ﬁnite type (resp. separated and of ﬁnite type) morphisms. Let A be an abelian g eneralized premo tivic category compatible with an admis- sible topology t suc h that : • A satisﬁes property (C) (r esp. (wC)) of P arag raph 5.1.35 . • The A 1 -deriv ed categor y D eﬀ A 1 ( A ) is compactly τ -gener ated and satisﬁes Nis- nevic h descent. 216 Construction of ﬁbred categories Then the stable A 1 -deriv ed premo tivic categor y D A 1 ( A ) is ( Z × τ ) -continuous (resp. w eakly ( Z × τ ) -continuous) — see Deﬁnition 4.3.2 . Proof This is an immediate corollar y of Proposition 5.2.41 combined with Proposi- tion 5.3.39 .  Example 5.3.45 A ccording to the previous proposition and the second point of Ex- ample 5.1.37 , the generalized tr iangulated premotivic category D A 1 , Λ is continuous. W e also ref er the reader to Corollary 6.1.12 f or an extension of this result to the non generalized case. 6 Localization and the univ ersal derived exam ple 6.0.1 In this section, S is an adequate categor y of S -schemes as in 2.0.1 . In sections 6.2 and 6.3 , we assume in addition that the schemes in S are ﬁnite dimensional. W e will apply the deﬁnitions of the preceding section to the admissible class made of mor phisms of ﬁnite type (resp. smooth mor phisms of ﬁnite type) in S , denoted b y S f t (resp. Sm ). Recall the general con v ention of section 1.4 : • pr emotivic means Sm -premotivic; • g eneralized pr emotivic means S f t -premotivic. 6.1 Generalized deriv ed premotivic categories Example 6.1.1 Let t be a S f t -admissible topology . F or a scheme S , we denote b y Sh t  S f t / S , Λ  the categor y of sheav es of abelian groups on S f t / S f or the topology t S . For an S -scheme of ﬁnite type X , we let Λ t S ( X ) be the free t-sheaf of Λ -modules represented by X . Recall Sh t  S f t , Λ  is a generalized abelian premotivic category (see 5.1.4 ). Let ρ : Sm / S / / S f t / S be the obvious inclusion functor and let us denote b y t S the initial topology on Sm / S such that ρ is continuous. Then it induces ( cf. [ A GV73 , IV , 4.10]) a sequence of adjoint functors Sh t ( Sm / S , Λ ) ρ ] - - ρ ∗ 1 1 Sh t  S f t / S , Λ  ρ ∗ o o and we chec ked easily that this induces an enlarg ement of abelian premotivic cate- gories: (6.1.1.1) ρ ] : Sh t ( Sm , Λ ) / / o o Sh t  S f t , Λ  : ρ ∗ . 6 Localization and the universal der iv ed ex ample 217 Remar k 6.1.2 Note that f or an y scheme S , the abelian category Sh t ( Sm / S , Λ ) can be described as the Gabr iel quotient of the abelian categor y Sh t  S f t / S , Λ  with respect to the sheav es F o v er S f t / S such that ρ ∗ ( F ) = 0 . An e xample of suc h a sheaf in the case where t = Nis and dim ( S ) > 0 is the Nisnevic h sheaf Λ S ( Z ) on S f t / S represented b y a nowhere dense closed subscheme Z of S is zero when restricted to Sm / S . 6.1.3 Consider an abelian premotivic categor y A compatible with an admissible topology t on Sm and a generalized abelian premotivic categor y A compatible with an admissible topology t 0 on S . W e denote by M (resp. M ) the geometric sections of A (resp. A ). W e assume that t 0 restricted to Sm is ﬁner that t , and consider an adjunction of abelian premotivic categor ies: ρ ] : A / / o o A : ρ ∗ . Let S be a scheme in S . The functors ρ ] and ρ ∗ induce a der iv ed adjunction (see 5.2.19 ): L ρ ] : D eﬀ A 1 ( A S ) / / o o D eﬀ A 1 ( A S ) : R ρ ∗ (where A is considered as an Sm -ﬁbred categor y). Proposition 6.1.4 Consider the previous hypothesis, and ﬁx a scheme S . Assume furthermore that we have the following properties. (i) The functor ρ ] : A S / / A S is fully fait hful. (ii) The functor ρ ∗ : A S / / A S commutes with small colimits. Then, the f ollowing conditions hold : (a) The induced functor ρ ∗ : C ( A S ) / / C ( A S ) pr eser v es A 1 -equiv alences. (b) The A 1 -deriv ed functor L ρ ] : D eﬀ A 1 ( A S ) / / D eﬀ A 1 ( A S ) is fully faithful. Proof Point (a) f ollo ws from Proposition 5.2.24 . T o pro v e (b), w e ha v e to pro v e that the unit map M / / ρ ∗ L ρ ] ( M ) is an isomor phism f or any object M of D eﬀ A 1 ( A S ) . For this pur pose, w e ma y assume that M is coﬁbrant, so that we hav e M ' ρ ∗ ρ ] ( M ) ' ρ ∗ L ρ ] ( M ) (where the ﬁrst isomorphism holds already in C ( A S ) ).  Corollary 6.1.5 Consider the hypothesis of the previous proposition. Then the family of adjunctions L ρ ] : D eﬀ A 1 ( A S ) / / D eﬀ A 1 ( A S ) : R ρ ∗ indexed by a scheme S induces an enlarg ement of triangulated pr emotivic categories L ρ ] : D eﬀ A 1 ( A ) / / o o D eﬀ A 1 ( A ) : R ρ ∗ . 218 Construction of ﬁbred categor ies Example 6.1.6 Considering the situation of 6.1.1 , we will be particularly interested in the case of the Nisne vich topology . W e denote by D eﬀ A 1 , Λ the generalized A 1 -derived premotivic categor y associated with Sh  S f t , Λ  (see also Example 5.3.31 ). The preceding corollary giv es a canonical enlargement: (6.1.6.1) D eﬀ A 1 , Λ / / o o D eﬀ A 1 , Λ 6.1.7 Consider again the h ypothesis of 6.1.3 . W e denote simply b y M (resp. M ) the geometric sections of the premotivic tr iangulated categor y D A 1 ( A ) (resp. D A 1 ( A ) ). Recall from 5.3.15 that we hav e deﬁned 1 S { 1 } (resp. 1 S { 1 } ) as the cokernel of the canonical map 1 S / / M S ( G m , S ) (resp. 1 S / / M S ( G m , S ) ). Thus, it is obvious that w e get a canonical identiﬁcation ρ ] ( 1 S { 1 } ) = 1 S { 1 } . Theref ore, the enlargement ρ ] can be e xtended canonically to an enlarg ement ρ ] : Sp ( A ) / / o o Sp ( A ) : ρ ∗ of abelian premotivic categor ies in such a wa y that f or any scheme S , the follo wing diagram commutes: A S ρ ] / / Σ ∞ A   A S Σ ∞ A   Sp ( A S ) ρ ] / / Sp ( A S ) . A ccording to Proposition 5.3.13 , Sp ( A ) (resp. Sp ( A ) ) is compatible with t (resp. t 0 ), and we obtain an adjoint pair of functors ( 5.3.28 ): L ρ ] : D A 1 ( A S ) / / o o D A 1 ( A S ) : R ρ ∗ . From the preceding commutative square, we get the identiﬁcation: (6.1.7.1) L ρ ] ◦ Σ ∞ A = Σ ∞ A ◦ L ρ ] As in the non-eﬀective case, we g et the f ollo wing result: Proposition 6.1.8 Keep the assumptions of Proposition 6.1.4 , and suppose fur ther - mor e that both D eﬀ A 1 ( A ) and D eﬀ A 1 ( A ) are compactly τ -gener ated. Then the derived functor L ρ ] : D A 1 ( A S ) / / D A 1 ( A S ) is fully faithful. Proof W e hav e to prov e that f or an y T ate spectrum E of D A 1 ( A S ) , the adjunction morphism E / / L ρ ∗ R ρ ] ( E ) is an isomor phism. According to Proposition 1.3.19 , the functor L ρ ∗ admits a right adjoint. Thus, applying Lemma 1.1.43 , it is suﬃcient to consider the case where E = M S ( X ){ i } [ n ] for a smooth S -scheme X , and a couple ( n , i ) ∈ Z × τ . Moreo v er , it is suﬃcient to prov e that f or another smooth S -scheme Y and an integer j ∈ Z , the induced mor phism 6 Localization and the universal der iv ed ex ample 219 Hom ( Σ ∞ M S ( Y ) { j } , Σ ∞ M S ( X ){ i } [ n ]) / / Hom ( Σ ∞ M S ( Y ) { j } , Σ ∞ M S ( X ){ i } [ n ]) is an isomorphism. Using the identiﬁcation ( 6.1.7.1 ), propositions 5.3.39 and 6.1.4 allo ws us to conclude.  Corollary 6.1.9 If the assumptions of Proposition 6.1.8 hold for any scheme S in S , then w e obtain an enlarg ement of triangulated premo tivic categories L ρ ] : D A 1 ( A ) / / o o D A 1 ( A ) : R ρ ∗ . Example 6.1.10 Considering again the situation of 6.1.1 , in the case of the Nisnevic h topology . W e denote b y D A 1 , Λ the generalized stable A 1 -derived premotivic category associated with Sh  S f t , Λ  . The preceding corollary gives a canonical enlarg ement: (6.1.10.1) L ρ ] : D A 1 , Λ / / o o D A 1 , Λ : R ρ ∗ which is compatible with the enlarg ement ( 6.1.6.1 ) in the sense that the f ollo wing diagram is essentially commutativ e: D eﬀ A 1 , Λ / / Σ ∞   D eﬀ A 1 , Λ Σ ∞   D A 1 , Λ / / D A 1 , Λ Corollary 6.1.11 Consider a Gro thendiec k topology t on our categor y of schemes S . Let S be a sc heme in S , and M an object of D A 1 , Λ ( S ) . Then M satisﬁes t -descent in D A 1 , Λ ( S ) if and only if L ρ ] ( M ) satisﬁes t -descent in D A 1 , Λ ( S ) . Proof Let f : X / / S be a diag ram of S -schemes of ﬁnite type. Deﬁne H q ( X , M ( p )) = Hom D A 1 , Λ ( S ) ( Λ X , L f ∗ ( M )( p )[ q ]) H q ( X , M ( p )) = Hom D A 1 , Λ ( S ) ( Λ X , L f ∗ L ρ ] ( M )( p )[ q ]) f or an y integers p and q . The full faithfulness of L ρ ] ensures that the compar ison map H q ( X , M ( p )) / / H q ( X , M ( p )) is alwa ys bijectiv e. This proposition f ollo ws then from the fact that M (resp. L ρ ] ( M ) ) satisﬁes t -descent if and only if, f or an y integers p and q , f or any S -scheme of ﬁnite type X , and any t -hyperco ver X / / X , the induced map H q ( X , M ( p )) / / H q ( X , M ( p )) (resp. H q ( X , M ( p )) / / H q ( X , M ( p )) ) is bi jectiv e.  W e end-up this section with another interesting application of the preceding results. 220 Construction of ﬁbred categories Corollary 6.1.12 Consider the hypothesis and assumptions of Proposition 6.1.4 . W e suppose furthermore that the g eneralized abelian premo tivic categor y A satisﬁes condition (C) of P ar agr aph 5.1.35 . 1. Then the triangulated premotivic categor y D eﬀ A 1 ( A ) is τ -continuous. 2. Assume furthermore that D eﬀ A 1 ( A ) and D eﬀ A 1 ( A ) are compactly τ -g enerated. Then the triangulated premotivic categor y D A 1 ( A ) is τ -continuous. Proof A ccording to Proposition 5.2.41 , the categor y D eﬀ A 1 ( A ) is τ -continuous. Ac- cording to Corollary 6.1.5 , the functor L ρ ] : D eﬀ A 1 ( A ) / / D eﬀ A 1 ( A ) : R ρ ∗ is fully faithful and commutes with L f ∗ . Thus Point (1) follo ws. In the assumption of Point (2), we deduce from Proposition 5.3.44 that D A 1 ( A ) is ( Z × τ ) -continuous. Thus it is suﬃcient to apply Corollar y 6.1.9 as in the eﬀectiv e case to get the asser tion of Point (2).  Example 6.1.13 A ccording to the second point of Example 5.1.37 , w e can apply this corollary to the enlarg ement Sh Nis ( Sm , Λ ) / / Sh Nis  S f t , Λ  . Thus, we deduce that the tr iangulated premotivic categories D eﬀ A 1 , Λ and D A 1 , Λ both are continuous. 6.2 The fundamental example Recall the follo wing theorem of A youb [ A yo07b ]: Theorem 6.2.1 The triangulated premo tivic categories D eﬀ A 1 , Λ and D A 1 , Λ satisfy the localization property . Corollary 6.2.2 1. The premo tivic categor y D A 1 , Λ is a motivic category. 2. It is compactly g enerat ed by the T ate twist. 3. Suppose that T is a deriv ed premo tivic category (see 5.2.9 ) which is a mo- tivic category. Then ther e exists a canonical morphism of derived pr emotivic categories: D A 1 , Z / / T . Proof The ﬁrst asser tion follo ws from the previous theorem and R emark 2.4.47 . The second one f ollow s from Corollar y 5.3.42 . The last one f ollow s from Proposition 3.3.5 and Example 5.3.34 .  Remar k 6.2.3 Thus, Theorem 2.4.50 can be applied to D A 1 , Λ . In par ticular , for an y separated morphism of ﬁnite type f : T / / S , there e xists a pair of adjoint functors f ! : D A 1 , Λ ( T ) / / o o D A 1 , Λ ( S ) : f ! 6 Localization and the universal der iv ed ex ample 221 as in the theorem loc. cit. so that we hav e remo v ed the quasi-projectiv e assumption in [ A y o07a ]. 6.2.4 Because the cdh topology is ﬁner than the Nisnevic h topology , we get an adjunction of generalized premotivic categor ies: a ∗ c dh : D A 1 , Λ / / o o D A 1  Sh c dh  S f t , Λ   : R a c dh , ∗ . Corollary 6.2.5 F or any scheme S , the composite functor D A 1 ( S , Λ ) / / D A 1 ( S , Λ ) a c dh / / D A 1  Sh c dh  S f t / S , Λ   is fully fait hful. Mor eov er , it induces an enlarg ement of premo tivic categories: (6.2.5.1) D A 1 , Λ / / o o D A 1  Sh c dh  S f t , Λ   Remar k 6.2.6 This corollary is a generalization in our derived setting of the main theorem of [ V oe10c ]. Note that if dim ( S ) > 0 , there is no hope that the abov e composite functor is essentially surjective because as soon as Z is a nowhere dense closed subscheme of S , the premotive M c dh S ( Z , Λ ) does not belong to its image ( cf. remark 6.1.2 ). Proof A ccording to Corollar y 6.2.2 and Proposition 3.3.10 , any T ate spectr um E of D A 1 ( S , Λ ) satisﬁes c dh -descent in the derived premotivic categor y D A 1 , Λ , and this implies the ﬁrst asser tion by 5.3.30 and 6.1.11 . The second one then f ollo ws from the fact the for getful functor D A 1  Sh c dh  S f t / S , Λ   / / D A 1 ( S , Λ ) . commutes with direct sums (its left adjoint preserves compact objects).  6.3 Near ly Nisnevic h shea ves 6.3.1 In all this section, we ﬁx an abelian premotivic categor y A and we consider the canonical premotivic adjunction ( 5.1.2.1 ) associated with A . W e assume A satisﬁes the f ollo wing proper ties. (i) A is compatible with Nisnevic h topology , so that w e hav e from ( 5.1.2.1 ) a premotivic adjunction: (6.3.1.1) γ ∗ : Sh Nis ( Sm , Z ) / / o o A : γ ∗ . (ii) A is ﬁnitely presented ( i.e. the functors Hom A S ( M S ( X ) , −) preser v e ﬁltered colimits and f orm a conser v ativ e famil y , Def. 1.3.11 ). 222 Construction of ﬁbred categor ies (iii) For any scheme S , and for any open immersion U / / X of smooth S -schemes, the map M S ( U ) / / M S ( X ) is a monomor phism. (iv) For any scheme S , the functor γ ∗ : A S / / Sh Nis ( Sm / S , Z ) is e xact. Note that the functor γ ∗ : A S / / Sh Nis ( Sm / S , Z ) is e xact and conser v ative. As it also preserves ﬁltered colimits, this functor preser v es in fact small colimits. Observe also that, according to assumptions (i)-(iv), the abelian premotivic cate- gory of T ate spectra Sp ( A ) is compatible with Nisnevic h topology and N -generated. Moreo v er , w e get a canonical premotivic adjunction (6.3.1.2) γ ∗ : Sp ( Sh Nis ( Sm , Z ) ) / / o o Sp ( A ) : γ ∗ such that γ ∗ is conservativ e and preser v es small colimits. In the follo wing, we sho w ho w one can deduce proper ties of the premotivic triangulated categor ies D eﬀ A 1 ( A ) and D A 1 ( A ) from the good proper ties of D eﬀ A 1 , Z and D A 1 , Z . 6.3.a Support property (eﬀectiv e case) Proposition 6.3.2 F or any scheme S , the functor γ ∗ : C ( A S ) / / C ( Sh Nis ( Sm / S , Z ) ) pr eser v es and detects A 1 -equiv alences. Proof It f ollow s immediately from Corollar y 5.2.31 that γ ∗ preserves A 1 -equiv alences. The fact it detects them can be rephrased by saying that the induced functor γ ∗ : D eﬀ A 1 ( A S ) / / D eﬀ A 1 , Z ( S ) is conser vativ e. This is obviousl y tr ue once we noticed that its left adjoint is essen- tially sur jectiv e on generators.  Corollary 6.3.3 The right derived functor R γ ∗ = γ ∗ : D eﬀ A 1 ( A S ) / / D eﬀ A 1 , Z ( S ) is conser v ative. Proposition 6.3.4 Let f : S 0 / / S be a ﬁnite mor phism of schemes. Then the in- duced functor f ∗ : C ( A S 0 ) / / C ( A S ) pr eser v es colimits and A 1 -equiv alences. Proof W e ﬁrst prov e f ∗ preserves colimits. W e kno w the functors γ ∗ preserve colimits and are conser vativ e. As we hav e the identiﬁcation γ ∗ f ∗ = f ∗ γ ∗ , it is suﬃcient to pro v e the proper ty f or A = Sh Nis ( Sm , Z ) . Let X be a smooth S -scheme. It is suﬃcient to pro v e that, f or any point x of X , if X h x denotes the henselization of X at x , the functor 6 Localization and the universal der iv ed ex ample 223 Sh Nis ( Sm / S 0 , Z ) / / A b , F  / / f ∗ ( F )( X h x ) = F ( S 0 × S X h x ) commutes to colimits. Moreov er , the scheme S 0 × S X h x is ﬁnite ov er X h x , so that we ha v e S 0 × S X h x = q i Y i , where the Y i ’ s are a ﬁnite famil y of henselian local schemes o v er S 0 × S X h x . Hence, we hav e to check that the functor F  / / É i F ( Y i ) preser v es colimits. As colimits commute to sums, it is thus suﬃcient to pro ve that the functors F  / / F ( Y i ) commute to colimits. This f ollo ws from the fact that the local henselian schemes Y i are points of the topos of sheav es o v er the small Nisnevich site of X . W e are left to pro v e that the functor f ∗ : C ( A S 0 ) / / C ( A S ) respects A 1 - equiv alences. For this, w e shall study the behavior of f ∗ with respect to the A 1 - resolution functor constructed in 5.2.26 . Note that f ∗ commutes to limits because it has a left adjoint. In par ticular , w e kno w that f ∗ is exact. Moreo v er , one chec ks easily that f ∗ R ( n ) A 1 = f ∗ R ( n ) A 1 . As f ∗ commutes to colimits, this giv es the formula f ∗ R A 1 = R A 1 f ∗ . Let C be a comple x of Nisnevic h sheav es of abelian groups on Sm / S 0 . Choose a quasi-isomorphism C / / C 0 with C 0 a Nis -ﬂasque complex. Ap- plying Proposition 5.2.28 , w e know that R A 1 ( C 0 ) is A 1 -ﬁbrant and that w e get a canonical A 1 -equiv alence f ∗ ( C ) / / f ∗ ( C 0 ) / / f ∗ ( R A 1 ( C 0 )) = R A 1 ( f ∗ ( C 0 )) . Hence, w e are reduced to prov e that f ∗ preserves A 1 -equiv alences betw een A 1 - ﬁbrant objects. But such A 1 -equiv alences are quasi-isomorphisms, so that w e can conclude using the ex actness of f ∗ .  Proposition 6.3.5 F or any open immer sion of sc hemes j : U / / S , the exchang e transf ormation j ] γ ∗ / / γ ∗ j ] is an isomorphism of functors. Proof Let X be a scheme, and F a Nisne vich sheaf of abelian groups on Sm / X . Deﬁne the category C F as f ollow s. The objects are the couples ( Y , s ) , where Y is a smooth scheme o v er X , and s is a section of F o v er Y . The ar ro ws ( Y , s ) / / ( Y 0 , s 0 ) are the mor phisms f ∈ Hom Sh Nis ( Sm / X , Z ) ( Z X ( Y ) , Z X ( Y 0 )) such that f ∗ ( s 0 ) = s . W e ha v e a canonical functor ϕ F : C F / / Sh Nis ( Sm / X , Z ) deﬁned b y ϕ F ( Y , s ) = Z X ( Y ) , and one easily checks that the canonical map lim / / C F ϕ F = lim / / ( Y , s ) ∈ C F Z X ( Y ) / / F is an isomorphism in Sh Nis ( Sm / X , Z ) (this is essentiall y a ref or mulation of the Y oneda lemma). Consider no w an object F in the categor y A U . W e get two categories C γ ∗ ( F ) and C γ ∗ ( j ] ( F )) . There is a functor i : C γ ∗ ( F ) / / C γ ∗ ( j ] ( F )) 224 Constr uction of ﬁbred categor ies which is deﬁned by the f ormula i ( Y , s ) = ( Y , j ] ( s )) . T o e xplain our notations, let us sa y that we see s as a morphism from M S ( U , A ) to F , so that j ] ( s ) is a mor phism from M S ( Y , A ) = j ] M S ( U , A ) to j ] ( F ) . This functor i has r ight adjoint i 0 : C γ ∗ ( j ] ( F )) / / C γ ∗ ( F ) deﬁned by i 0 ( Y , s ) = ( Y U , s U ) , where Y U = Y × S U , and s U is the section of γ ∗ ( F ) o v er Y U that cor responds to the section j ∗ ( s ) of j ∗ j ] γ ∗ ( F ) o v er Y U under the canonical isomorphism γ ∗ ( F ) ' j ∗ j ] γ ∗ ( F ) (here, we use strongly the fact the functor j ] is fully faithful). The e xistence of a r ight adjoint implies i is coﬁnal. This latter proper ty is suﬃcient f or the canonical mor phism lim / / C γ ∗ ( F ) ϕ γ ∗ ( j ] ( F )) ◦ i / / lim / / C γ ∗ ( j ] ( F )) ϕ γ ∗ ( j ] ( F )) = γ ∗ ( j ] ( F )) to be an isomor phism. But the functor ϕ γ ∗ ( j ] ( F )) ◦ i is e xactl y the composition of the functor j ] with ϕ γ ∗ ( F ) . As the functor j ] commutes with colimits, we hav e lim / / C γ ∗ ( F ) ϕ γ ∗ ( j ] ( F )) ◦ i = lim / / C γ ∗ ( F ) j ] ϕ γ ∗ ( F ) ' j ] lim / / C γ ∗ ( F ) ϕ γ ∗ ( F ) ' j ] ( γ ∗ ( F )) . Hence we obtain a canonical isomorphism j ] ( γ ∗ ( F )) ' γ ∗ ( j ] ( F )) . It is easily seen that the cor responding map γ ∗ ( F ) / / j ∗ ( γ ∗ ( j ] ( F ))) = γ ∗ ( j ∗ j ] ( F )) is the imag e by γ ∗ of the unit map F / / j ∗ j ] ( F ) . This show s the isomor phism we hav e constructed is the e x chang e mor phism.  Corollary 6.3.6 F or any open immer sion of schemes j : U / / S , the functor j ] : A U / / A S is exact. Moreo ver , the induced functor j ] : C ( A U ) / / C ( A S ) pr eser v es A 1 -equiv alences. Proof Using the fact γ ∗ is exact and conser vativ e, and propositions 6.3.2 and 6.3.5 , it is suﬃcient to pro v e this corollary when A = Sh Nis ( Sm , Z ) . It is straightf or w ard to pro ve exactness using Nisnevic h points. The f act j ] preserves A 1 -equiv alences f ollow s from the e xactness proper ty and from the ob vious f act it preserv es strong A 1 -equiv alences.  Corollary 6.3.7 Let j : U / / S be an open immersion of sc hemes. F or any object M of D eﬀ A 1 ( A U ) the exc hang e morphism (6.3.7.1) L j ] ( R γ ∗ ( M )) / / R γ ∗ ( L j ] ( M )) is an isomorphism in D eﬀ A 1 ( S , Z ) . 6 Localization and the universal der iv ed ex ample 225 6.3.b Support property (stable case) 6.3.8 Recall from 5.3.17 that the premotivic adjunction ( γ ∗ , γ ∗ ) induces a canonical adjunction of abelian premotivic categor ies that we denote by : ˜ γ ∗ : Sp ( Sh Nis ( Sm , Z ) ) / / o o Sp ( A S ) : ˜ γ ∗ Proposition 6.3.9 F or any scheme S , the functor induced functor ˜ γ ∗ : C  Sp ( A S )  / / o o C  Sp ( Sh Nis ( Sm / S , Z ) )  pr eser v es and detects stable A 1 -equiv alences. Proof Using the equivalence between symmetric T ate spectra and non symmetr ic T ate spectra, w e are reduced to pro v e this f or comple xes of non symmetr ic T ate spectra. Consider a non symmetric T ate spectr um ( E n ) n ∈ N with suspension maps σ n : E n { 1 } / / E n + 1 . The non symmetric T ate spectrum ˜ γ ∗ ( E ) is equal to γ ∗ ( E n ) in degree n ∈ Z , and the suspension map is given by the composite: 1 S { 1 } ⊗ S γ ∗ ( E n ) / / γ ∗ ( γ ∗ ( 1 S { 1 } ) ⊗ S E n ) = γ ∗ ( E n { 1 } ) γ ∗ ( σ n ) / / E n + 1 . Thus, propositions 6.3.2 and 5.3.40 allow s us to conclude.  Corollary 6.3.10 The right derived functor R γ ∗ = γ ∗ : D A 1 ( A S ) / / D A 1 , Z ( S ) is conser v ative. Proposition 6.3.11 Let j : U / / X be an open immersion of schemes. F or any object M of D A 1 ( A U ) , the exc hang e morphism L j ] ( R γ ∗ ( M )) / / R γ ∗ ( L j ] ( M )) is an isomorphism in D A 1 , Z ( X ) . Proof From Corollary 6.3.6 and the P -base chang e f or mula f or the open immersion j , one deduces easily that j ] preserves stable A 1 -equiv alences of (non symmetric) T ate spectra. Moreo v er , Proposition 6.3.5 sho ws that j ] γ ∗ = γ ∗ j ] at the le v el of T ate spectra. This concludes.  Corollary 6.3.12 The triangulated pr emotivic categor y D A 1 ( A ) satisﬁes the sup- port property . Proof A ccording to corollar y 6.3.10 , the functor R γ ∗ is conser vativ e. Thus, by vir tue of the preceding proposition, to pro v e the suppor t proper ty in the case of D A 1 ( A ) it is suﬃcient to prov e it in the case where A = Sh Nis ( Sm , Z ) . This follo ws from theorems 6.2.1 and 2.4.50 .  226 Construction of ﬁbred categor ies 6.3.c Localization f or smooth schemes Lemma 6.3.13 Let i : Z / / S be a closed immer sion whic h admits a smoo th r e- traction p : S / / Z . Then the exchang e transf or mation L γ ∗ R i ∗ / / R i ∗ L γ ∗ is an isomorphism in D eﬀ A 1 ( A S ) (r esp. D A 1 ( A S ) ). Proof W e ﬁrst remark that f or any object C of C ( A Z ) (resp. C ( Sp ( A Z )) ) the canonical sequence j ] ( p j ) ∗ ( C ) / / p ∗ ( C ) / / i ∗ ( C ) is a coﬁber sequence in D eﬀ A 1 ( A S ) (resp. D A 1 ( A ) S ) ). Indeed, w e can chec k this after applying the ex act conser v ativ e functor γ ∗ . The sequence w e obtain is canonically isomorphic through e x chang e transf or mations to j ] j ∗ p ∗ ( γ ∗ C ) / / p ∗ ( γ ∗ C ) / / i ∗ i ∗ p ∗ ( γ ∗ C ) using Corollary 6.3.7 , the commutation of γ ∗ with j ∗ , p ∗ and i ∗ (recall it is the right adjoint of a premotivic adjunction) and the relation pi = 1 . But this last sequence is a coﬁber sequence in D eﬀ A 1 , Z ( S ) (resp. D A 1 , Z ( S ) ) because it satisﬁes the localization property (see 6.2.1 ). Using ex chang e transf or mations, w e obtain a mor phism of distinguished tr iangles in DM eﬀ Z ( S ) γ ∗ j ] j ∗ p ∗ ( C ) / / γ ∗ p ∗ ( C ) / / γ ∗ i ∗ ( C ) / / E x ( γ ∗ , i ∗ )   γ ∗ j ] j ∗ p ∗ ( C )[ 1 ] j ] j ∗ p ∗ ( γ ∗ C ) / / p ∗ ( γ ∗ C ) / / i ∗ ( γ ∗ C ) / / j ] j ∗ p ∗ ( γ ∗ C )[ 1 ] The ﬁrst two vertical ar ro ws are isomor phisms as γ ∗ is the left adjoint of a premotivic adjunction; thus the mor phism E x ( γ ∗ , i ∗ ) is also an isomor phism.  Proposition 6.3.14 Let i : Z / / S be a closed immersion. If i admits a smoot h r etr action, then D eﬀ A 1 ( A ) satisﬁes (Loc i ). Proof This f ollo ws from Proposition 2.3.19 and the preceding lemma.  Corollary 6.3.15 Let S be a scheme. Then the pr emotivic categor y D eﬀ A 1 ( A ) (resp. D A 1 ( A ) ) satisﬁes localization with respect to any closed immersion betw een smooth S -sc hemes. Proof Let i : Z / / X be closed immersion betw een smooth S -schemes. W e want to pro v e that D eﬀ A 1 ( A ) (resp. D A 1 ( A ) ) satisﬁes localization with respect to i . According to 2.3.18 , it is suﬃcient to prov e that f or any smooth S -scheme S , the canonical map M S ( X / X − X Z ) / / i ∗ M Z ( X Z ) 7 Basic homotopy commutativ e algebra 227 is an isomorphism where we use the notation of loc. cit. and M ( . , A ) denotes the geometric sections of D eﬀ A 1 ( A ) (resp. D A 1 ( A ) ). But the premotivic triangulated category D A 1 ( A ) (resp. D eﬀ A 1 ( A ) ) satisﬁes the Nisnevic h separation proper ty and the Sm -base change proper ty . Thus, w e can argue locally in S f or the Nisnevich topology . Thus, the statement is reduced to the preceding proposition as i admits locally f or the Nisnevic h topology a smooth retraction (see f or e xample [ Dég07 , 4.5.11]).  7 Basic homotop y commutative algebra 7.1 Rings Deﬁnition 7.1.1 A symmetric monoidal model category V satisﬁes the monoid axiom if, for any tr ivial coﬁbration A / / B and any object X , the smallest class of maps of V which contains the map X ⊗ A / / X ⊗ B and is stable by pushouts and transﬁnite compositions is contained in the class of weak equivalences. 7.1.2 Let V be a symmetr ic monoidal categor y . W e denote by Mon ( V ) the category of monoids in V . If V has small colimits, the f org etful functor U : Mon ( V ) / / V has a left adjoint F : V / / Mon ( V ) . Theorem 7.1.3 Let V a symmetric monoidal combinatorial model categor y which satisﬁes the monoid axiom. The categor y of monoids Mon ( V ) is endow ed with the structure of a combinatorial model categor y whose weak equivalences (r esp. ﬁbrations) are the morphisms of commutativ e monoids which are weak equivalences (r esp. ﬁbrations) in V . In particular , the for g etful functor U : Mon ( V ) / / V is a right Quillen functor . Moreo ver , if the unit object of V is coﬁbrant, then any coﬁbrant object of Mon ( V ) is coﬁbrant as an object of V . Proof This is very a par ticular case of the third assertion of [ SS00 , Theorem 4.1] (the fact that Mon ( V ) is combinator ial whenev er V is so comes for instance from [ Bek00 , Proposition 2.3]).  Deﬁnition 7.1.4 A symmetr ic monoidal model category V is str ong ly Q -linear if the underlying categor y of V is additiv e and Q -linear (i.e. all the objects of V are uniquel y divisible). Remar k 7.1.5 If V is a strongl y Q -linear stable model categor y , then it is Q -linear in the sense of 3.2.14 . 228 Construction of ﬁbred categor ies Lemma 7.1.6 Let V be a str ong ly Q -linear model category, G a ﬁnite gr oup, and u : E / / F an equiv ariant mor phism of r epresentations of G in V . Then, if u is a coﬁbration in V , so is the induced map E G / / F G (wher e the subscript G denotes the coinv ariants under the action of the gr oup G ). Proof The map E G / / F G is easily seen to be a direct f actor (retract) of the coﬁbra- tion E / / F .  7.1.7 If V is a symmetric monoidal category , w e denote by Comm ( V ) the category of commutativ e monoids in V . If V has small colimits, the f org etful functor U : Comm ( V ) / / V has a left adjoint F : V / / Comm ( V ) . Theorem 7.1.8 Let V a symmetric monoidal combinatorial model category. Assume that V is lef t proper and tractable, satisﬁes the monoid axiom, and is strong ly Q - linear . Then the category of commutativ e monoids Comm ( V ) is endow ed with the structure of a combinatorial model categor y whose weak equivalences (r esp. ﬁbrations) are the morphisms of commutativ e monoids which are weak equivalences (r esp. ﬁbrations) in V . In particular , the for ge tful functor U : Comm ( V ) / / V is a right Quillen functor . If mor eov er the unit object of V is coﬁbr ant, then any coﬁbrant object of Comm ( V ) is coﬁbrant as an object of V . Proof W e will obser v e ﬁrst that V is freely pow ered in the sense of [ Lur17 , Deﬁnition 4.5.4.2]. Theref ore, the e xistence of this model category str ucture will f ollow from a general result of Lur ie [ Lur17 , Proposition 4.5.4.6]. For this, it is suﬃcient to chec k that a G -equiv ar iant map f : A / / B in V which is a tr ivial coﬁbration when we f orget the G -action has the left lifting proper ty with respect to any G -equivariant map p : X / / Y which is a ﬁbration in V (after f org etting the G -action). In other w ords, we hav e to check that the map induced by f and p in V Hom V ( B , X ) / / Hom V ( A , X ) × Hom V ( A , X ) Hom V ( Y , B ) will induce a surjectiv e map after w e apply the G -inv ar iants functor (w e let the reader construct a natural G -action on Hom V ( B , X ) , the G -in variants of which gives the Q -vector space of G -equiv ariant maps from B to X ). Since G is a ﬁnite group, the G -inv ar iant subspace functor is e xact, hence this is obvious. This prov es the ﬁrst assertion. The second asser tion of the theorem is tr ue by deﬁnition. The last asser tion is prov ed b y a careful analy sis of pushouts by free maps in Comm ( V ) as f ollow s. For two coﬁbrations u : A / / B and v : C / / D in V , wr ite u ∧ v f or the map u ∧ v : A ⊗ D q A ⊗ C B ⊗ C / / B ⊗ D (which is a coﬁbration by deﬁnition of monoidal model categor ies). By iterating this construction, we get, f or a coﬁbration u : A / / B in V , a coﬁbration 7 Basic homotopy commutativ e algebra 229 ∧ n ( u ) = u ∧ · · · ∧ u | {z } n times :  n ( u ) / / B ⊗ n . Note that the symmetric g roup S n acts naturally on B ⊗ n and  n ( u ) . W e deﬁne Sym n ( B ) = ( B ⊗ n ) S n and Sym n ( B , A ) =  n ( u ) S n . By vir tue of Lemma 7.1.6 , we get a coﬁbration of V : σ n ( u ) : Sym n ( B , A ) / / Sym n ( B ) . Consider now the free map F ( u ) : F ( A ) / / F ( B ) can be ﬁltered b y F ( A ) -modules as follo ws. Deﬁne D 0 = F ( A ) . As A = Sym 1 ( B , A ) , we hav e a natural morphism F ( A ) ⊗ Sym 1 ( B , A ) / / F ( A ) . The objects D n are then deﬁned by induction with the pushouts belo w . F ( A ) ⊗ Sym n ( B , A ) 1 F ( A )⊗ σ n ( u ) / /   F ( A ) ⊗ Sym n ( B )   D n − 1 / / D n W e get natural maps D n / / F ( B ) which induce an isomor phism lim / / n ≥ 0 D n ' F ( B ) in such a w ay that the mor phism F ( u ) correspond to the canonical map F ( A ) = D 0 / / lim / / n ≥ 0 D n . Hence, if F ( A ) is coﬁbrant, all the maps D n − 1 / / D n are coﬁbrations, so that the map F ( A ) / / F ( B ) is a coﬁbration in V . In the par ticular case where A is the initial object of V , we see that f or any coﬁbrant object B of V , the free commutativ e monoid F ( B ) is coﬁbrant as an object of V (because the initial object of Comm ( V ) is the unit object of V ). This also implies that, if u is a coﬁbration between coﬁbrant objects, the map F ( u ) is a coﬁbration in V . This descr iption of F ( u ) also allo ws to compute the pushouts of F ( u ) in Comm ( V ) in V as f ollow s. Consider a pushout F ( A ) F ( u ) / /   F ( B )   R v / / S 230 Construction of ﬁbred categories in Comm ( V ) . For n ≥ 0 , deﬁne R n b y the pushouts of V : F ( A ) / /   D n   R / / R n W e then ha v e an isomor phism lim / / n ≥ 0 R n ' S . In par ticular , if u is a coﬁbration betw een coﬁbrant objects, the morphism of com- mutativ e monoids v : R / / S is then a coﬁbration in V . As the f org etful functor U preser v es ﬁltered colimits, w e conclude easil y from there (with the small object argument [ Ho v99 , Theorem 2.1.14]) that any coﬁbration of Comm ( V ) is a coﬁbra- tion of V . Using again that the unit object of V is coﬁbrant in V (i.e. that the initial object of Comm ( V ) is coﬁbrant in V ) this pro v es the last asser tion of the theorem.  Corollary 7.1.9 Let V a symmetric monoidal combinatorial model categor y . As- sume that V is lef t pr oper and tr actable, satisﬁes the monoid axiom, and is strong ly Q -linear . Consider a small set H of maps of V , and denote by L H V the lef t Bous- ﬁeld localization of V by H ; see [ Bar10 , Theor em 4.7]. Deﬁne the class of H - equiv alences in Ho ( V ) to be the class of maps which become invertible in Ho ( L H V ) . If H -equivalences are stable by (deriv ed) tensor product in Ho ( V ) , then L H V is a symmetric monoidal combinatorial model category (which is ag ain lef t proper and tractable, satisﬁes the monoid axiom, and is str ong ly Q -linear). In particular , under these assumptions, there exists a morphism of commutative monoids 1 / / R in V whic h is a weak equiv alence of L H V , with R a coﬁbrant and ﬁbrant object of L H V . Proof The ﬁrst asser tion is a triviality . The last asser tion f ollo ws immediately : the map 1 / / R is simply obtained as a ﬁbrant replacement of 1 in the model category Comm ( L H V ) obtained from Theorem 7.1.8 applied to L H V .  7.1.10 Consider now a category S , as well as a closed symmetr ic monoidal biﬁbred category M o v er S . W e shall also assume that the ﬁbers of M admit limits and colimits. Then the categor ies Mon ( M ( X )) (resp. Comm ( M ( X )) ) deﬁne a biﬁbred cate- gory ov er S as follo ws. Given a mor phism f : X / / Y , the functor f ∗ : M ( Y ) / / M ( X ) is symmetric monoidal, so that it preser v es monoids (resp. commutative monoids) as w ell as mor phisms between them. It thus induces a functor (7.1.10.1) f ∗ : Mon ( M ( Y )) / / Mon ( M ( X )) ( resp. f ∗ : Comm ( M ( Y )) / / Comm ( M ( X )) ) . 7 Basic homotopy commutativ e algebra 231 As f ∗ : M ( Y ) / / M ( X ) is symmetr ic monoidal, its right adjoint f ∗ is lax monoidal: there is a natural mor phism (7.1.10.2) 1 Y / / f ∗ ( 1 X ) = f ∗ f ∗ ( 1 Y ) , and, f or an y objects A and B of M ( X ) , there is a natural mor phism (7.1.10.3) f ∗ ( A ) ⊗ Y f ∗ ( B ) / / f ∗ ( A ⊗ X B ) which cor responds by adjunction to the map f ∗ ( f ∗ ( A ) ⊗ Y f ∗ ( B )) ' f ∗ f ∗ ( A ) ⊗ f ∗ f ∗ ( B ) / / A ⊗ B . Hence the functor f ∗ preserves also monoids (resp. commutative monoids) as w ell as mor phisms betw een them, so that we g et a functor (7.1.10.4) f ∗ : Mon ( M ( X )) / / Mon ( M ( Y )) ( resp. f ∗ : Comm ( M ( X )) / / Comm ( M ( Y )) ) . By construction, the functor f ∗ of ( 7.1.10.1 ) is a left adjoint ot the functor f ∗ of ( 7.1.10.4 ). These constr uctions e xtend to morphisms of S -diagrams in a similar wa y . Proposition 7.1.11 Let M be a symmetric monoidal combinatorial ﬁbred model category ov er S . Assume that, f or any object X of S , the model category M ( X ) satisﬁes the monoid axiom (resp. is lef t pr oper and tractable, satisﬁes the monoid axiom, and is strong ly Q -linear). (a) F or any object X of S , the category Mon ( M )( X ) (resp. Comm ( M )( X ) ) of monoids (resp. of commutative monoids) in M ( X ) is a combinatorial model cat- egor y structur e whose w eak equivalences (resp. ﬁbrations) are the mor phisms of commutativ e monoids which are weak equivalences (resp. ﬁbrations) in M ( X ) . This turns Mon ( M ) (resp. Comm ( M ) ) into a combinatorial ﬁbr ed model cat- egor y ov er S . (b) F or any mor phism of S -diagr ams ϕ : ( X , I ) / / ( Y , J ) , the adjunction ϕ ∗ : Mon ( M )( Y , J ) / / o o Mon ( M )( X , I ) : ϕ ∗ ( r esp. ϕ ∗ : Comm ( M )( Y , J ) / / o o Comm ( M )( X , I ) : ϕ ∗ ) is a Quillen adjunction (where the categories of monoids Mon ( M )( X , I ) (resp. of commutative monoids Comm ( M )( X , I ) ) are endow ed with the injectiv e model category structure obtained from Proposition 3.1.7 applied to Mon ( M ) (r esp. to Comm ( M ) ). (d) If moreo v er , for any object X of S , the unit 1 X is coﬁbrant in M ( X ) , then, for morphism of S -diagr ams ϕ : ( X , I ) / / ( Y , J ) , the squar e 232 Construction of ﬁbred categor ies Ho ( Mon ( M ))( Y , J ) L ϕ ∗ / / U   Ho ( Mon ( M ))( X , I ) U   Ho ( M )( Y , J ) L ϕ ∗ / / Ho ( M )( X , I ) (7.1.11.1) is essentially commutative. Similarly , in the respectiv e case, the squar e Ho ( Comm ( M ))( Y , J ) L ϕ ∗ / / U   Ho ( Comm ( M ))( X , I ) U   Ho ( M )( Y , J ) L ϕ ∗ / / Ho ( M )( X , I ) (7.1.11.2) is essentially commutative. Proof Asser tion (a) is an immediate consequence of Theorem 7.1.3 (resp. of The- orem 7.1.8 ), and assertion (b) is a par ticular case of Proposition 3.1.11 (be ware that the injectiv e model categor y structure on Comm ( M )( X , I ) does not necessarily coincide with the model categor y str ucture giv en by Theorem 7.1.3 (resp. of Theo- rem 7.1.8 ) applied to the injectiv e model structure on M ( X , I ) ). For asser tion (d), w e see b y the second asser tion of Proposition 3.1.6 that it is suﬃcient to prov e it when ϕ : X / / Y is simply a mor phism of S . In this case, by construction of the total left der iv ed functor of a left Quillen functor , this follo ws from the fact that ϕ ∗ commutes with the f org etful functor and from the fact that, by vir tue of the last assertion of Theorem 7.1.3 (resp. of Theorem 7.1.8 ), the for getful functor U preser v es w eak equivalences and coﬁbrant objects.  Remar k 7.1.12 The main application of the preceding corollary will come from as- sertion (d): it say s that, given a monoid (resp. a commutative monoid) R in M ( Y ) and a mor phism f : X / / Y , the image of R by the functor L f ∗ : Ho ( M )( Y ) / / Ho ( M )( X ) is canonically endow ed with a structure of monoid (resp. of commutativ e monoid) in the stronges t sense possible. Under the assumptions of assertion (c) of Proposition 7.1.11 , we shall often make the abuse of sa ying that L f ∗ ( R ) is a monoid (resp. a commutativ e monoid) in M ( X ) without refereeing explicitl y to the model categor y structure on Mon ( M )( X ) (resp. on Comm ( M )( X ) ). Similarl y , f or an y monoid (resp. commutativ e monoid) R in M ( X ) , R f ∗ ( R ) will be canonically endo w ed with a str ucture of a monoid (resp. a commutativ e monoid) in M ( Y ) . In par ticular , f or an y monoid (resp. commutative monoid) R in M ( Y ) , the adjunction map R / / R f ∗ L f ∗ ( R ) is a mor phism of monoids (i.e. is a map in the homotopy categor y Ho ( Mon ( M ))( X ) (resp. Ho ( Comm ( M ))( X ) )), and, f or any monoid (resp. commutativ e monoid) R in M ( X ) , the adjunction map 7 Basic homotopy commutativ e algebra 233 L f ∗ R f ∗ ( R ) / / R is a mor phism of monoids (i.e. is a map in the homotopy categor y Ho ( Mon ( M ))( Y ) (resp. Ho ( Comm ( M ))( Y ) )). Remar k 7.1.13 In order to get a good homotopy theor y of commutativ e monoids wihout the strongl y Q -linear assumption, we should replace commutative monoids b y E ∞ -algebras (i.e. objects endow ed with a s tr ucture of commutativ e monoid up to a bunch of coherent homotopies). More generall y , we should pro v e the analog of Theorem 7.1.3 and of Theorem 7.1.8 by replacing Mon ( V ) by the categor y of algebras of some ‘well-beha v ed’ operad, and then get as a consequence the analog of Proposition 7.1.11 . All this is a consequence of the general constructions and results of [ Spi01 , BM03 , BM09 ]. Ho w ev er , in the case we are interested in the homotop y theor y of commutativ e monoids in some categor y of spectra V , it seems that some v ersion of Shipley’ s posi- tiv e stable model structure ( cf. [ Shi04 , Proposition 3.1]) w ould provide a good model category for commutativ e monoids, which, b y Lur ie ’s strictiﬁcation theorem [ Lur17 , Theorem 4.5.4.7], w ould be equivalent to the homotopy theor y of E ∞ -algebras in V . This kind of technics is now av ailable in the context of stable homotopy theor y of schemes, whic h pro vides a good setting to speak of motivic commutative r ing spectra; see [ Hor13 , GG16 , GG18 , PS18 ]. Theref ore, Theorem 7.1.8 and Proposition 7.1.11 are in f act true in SH for g enuine commutativ e monoids without an y Q -linearity assumption. 7.2 Modules 7.2.1 Giv en a monoid R in a symmetr ic monoidal category V , we shall write R - mo d ( V ) for the categor y of (left) R -modules. The f orgetful functor U : R - mo d ( V ) / / V is a left adjoint to the free R -module functor R ⊗ (−) : V / / R - mo d ( V ) . If V has enough small colimits, and if R is a commutative monoid, the categor y R - mo d ( V ) is endo w ed with a unique symmetric monoidal structure such that the functor R ⊗ (−) is naturally symmetric monoidal. W e shall denote by ⊗ R the tensor product of R - mo d ( V ) . Theorem 7.2.2 Let V be a combinatorial symmetric model categor y whic h satisﬁes the monoid axiom. (i) F or any monoid R in V , the category of right (resp. lef t) R -modules is a combinatorial model category with weak equivalences (resp. ﬁbrations) the morphisms of R -modules whic h are weak equivalences (r esp. ﬁbrations) in V . 234 Construction of ﬁbred categories (ii) F or any commutative monoid R in V , the model category of R -modules giv en by (i) is a combinatorial symmetric monoidal model category which satisﬁes the monoid axiom. Proof Asser tions (i) and (ii) are par ticular cases of the ﬁrst tw o assertions of [ SS00 , Theorem 4.1].  Deﬁnition 7.2.3 A symmetric monoidal model category V is perfect if it has the f ollowing proper ties. (a) V is combinatorial and tractable ( 3.1.27 ); (b) V satisﬁes the monoid axiom; (c) For any weak equiv alence of monoids R / / S , the functor M  / / S ⊗ R M is a left Quillen equivalence from the categor y of left R -modules to the categor y of left S -modules. (d) w eak equivalences are stable by small sums in V . Remar k 7.2.4 If V is a perfect symmetr ic monoidal model category , then, for any commutativ e monoid R , the symmetr ic monoidal model category of R -modules in V given by Theorem 7.2.2 (ii) is also perfect: condition (c) is quite obvious, and condition (d) comes from the f act that the f orgetful functor U : R - mo d / / V commutes with small sums, while it preser v es and detects weak equiv alences. Note that condition (d) implies that the functor U : Ho ( R - mo d ) / / Ho ( V ) preserves small sums. Remar k 7.2.5 If V is a stable symmetr ic monoidal model categor y which satisﬁes the monoid axiom, then f or an y monoid R of V , the model category of (left) R -modules giv en by Theorem 7.2.2 is stable as well: the suspension functor of Ho ( R - mo d ) is given by the derived tensor product b y the R -bimodule R [ 1 ] , which is clear ly in v er tible with in verse R [− 1 ] . In this w ork, a basic example of per f ect model categor ies are those coming from stable A 1 -derived premotivic categories (cf Def. 5.3.22 ): Proposition 7.2.6 Let t be an admissible topology. Then, f or any sc heme S in S , the symme tric monoidal model structur e on C ( Sp ( Sh t ( P / S , Z ) )) underlying the triangulated categor y D A 1 ( Sh t ( P / S , Z )) is per f ect. Proof The g enerating famil y of Sh t ( P / S , Z ) is ﬂat in the sense of [ CD09 , 3.1], so that, b y vir tue of [ CD09 , prop. 7.22 and cor . 7.24], the assumptions of Proposition 7.2.9 are fulﬁlled.  Proposition 7.2.7 Let V be a stable per fect symmetric monoidal model categor y . Assume furthermore that Ho ( V ) admits a small family G of compact g enerat ors (as a triangulated categor y). F or any monoid R in V , the triangulated categor y Ho ( R - mo d ( V )) admits the set { R ⊗ L E | E ∈ G } as a family of compact gener ators. 7 Basic homotopy commutativ e algebra 235 Proof W e ha v e a der iv ed adjunction R ⊗ L (−) : Ho ( V ) / / o o Ho ( R - mo d ( V )) : U . As the functor U preser v es small sums the functor R ⊗ L (−) preser v es compact objects. But U is also conser vativ e, so that { R ⊗ L E | E ∈ G } is a famil y of compact generators of Ho ( R - mo d ( V )) .  Remar k 7.2.8 If V is a combinatorial symmetric model category which satisﬁes the monoid axiom, then there are tw o w ay s to der iv e the tensor product. The ﬁrst one consists in der iving the left Quillen bifunctor (−) ⊗ (−) , which gives the usual derived tensor product (−) ⊗ L (−) : Ho ( V ) × Ho ( V ) / / Ho ( V ) . Remember that, by construction, A ⊗ L B = A 0 ⊗ B 0 , where A 0 and B 0 are coﬁbrant replacements of A and B respectiv ely . On the other hand, the monoid axiom gives that, for an y object A of V , the functor A ⊗ (−) preser v es weak equivalences between coﬁbrant objects, which implies that it has also a total left derived functor A ⊗ L (−) : Ho ( V ) / / Ho ( V ) . Despite the fact we ha ve adopted very similar (not to say identical) notations for these tw o der iv ed functor , there is no reason they would coincide in general: b y construction, the second one is deﬁned by A ⊗ L B = A ⊗ B 0 , where B 0 is some coﬁbrant replacement of B . Ho w ev er , they coincide quite often in practice (e.g. f or simplicial sets, f or the good reason that all of them are coﬁbrant, or f or symmetr ic S 1 -spectra, or f or complex es of quasi-coherent O X -modules ov er a quasi-compact and quasi-separated scheme X ). Proposition 7.2.9 Let V be a stable combinatorial symmetric monoidal model cat- egor y whic h satisﬁes the monoid axiom. Assume furthermore that, for any coﬁbrant object A of V , the functor A ⊗ (−) preserve weak equiv alences (in ot her w ords, that the two w ays to derive the tensor product explained in Remar k 7.2.8 coincide), and that weak equivalences are stable by small sums in V . Then the symmetric monoidal model category V is per f ect. Proof W e jus t hav e to check condition (c) of Deﬁnition 7.2.3 . Consider a w eak equiv alence of monoids R / / S . W e then get a der iv ed adjunction S ⊗ L R (−) : Ho ( R - mo d ( V )) / / o o Ho ( S - mo d ( V )) : U , where S ⊗ L R (−) is the left der iv ed functor of the functor M  / / S ⊗ R M . W e hav e to pro v e that, f or any left R -module M , the map M / / S ⊗ L R M is an isomor phism in Ho ( V ) . As this is a mor phism of tr iangulated functors which commutes with sums, and as Ho ( R - mo d ( V )) is w ell generated in the sense of 236 Construction of ﬁbred categor ies Neeman [ Nee01 ] (as the localization of a stable combinator ial model category), it is suﬃcient to check this when M r uns o v er a small famil y of g enerators of Ho ( R - mo d ( V )) . Let us chose is a small famil y of generators G of Ho ( V ) . As the f org etful functor from Ho ( R - mo d ( V )) to Ho ( V ) is conser vativ e, we see that { R ⊗ L E | E ∈ G } is a small generating f amily of Ho ( R - mo d ( V )) . W e are thus reduced to pro v e that the map R ⊗ L E / / S ⊗ L R ( R ⊗ L E ) ' S ⊗ L E is an isomor phism f or any object E in G . F or this, w e can assume that E is coﬁ- brant, and this f ollo ws then from the fact that the functor (−) ⊗ E preser v es weak equiv alences by assumption.  7.2.10 Let S be a category endo w ed with an admissible class of mor phisms P , and M a cocomplete symmetr ic monoidal P -ﬁbred category . Consider a monoid R in the symmetr ic monoidal categor y M ( 1 S , S ) (i.e. a section of the ﬁbred category Mon ( M ) ov er S ). In other words, R consists of the data of a monoid R X f or each object X of S , and of a morphism of monoids a f : f ∗ ( R Y ) / / R X f or each map f : X / / Y in S , subject to coherence relations; see 3.1.2 . For an object X of S , we shall wr ite R - mo d ( X ) for the categor y of (left) R X - modules in M ( X ) , i.e. R - mo d ( X ) = R X - mo d ( M ( X )) . This deﬁnes a ﬁbred categor y R - mo d o v er S as f ollo ws. For a mor phism f : X / / Y , the inv erse image functor (7.2.10.1) f ∗ : R - mo d ( Y ) / / R - mo d ( X ) is deﬁned by (7.2.10.2) M  / / R X ⊗ f ∗ ( R Y ) f ∗ ( M ) (where, on the r ight-hand side, f ∗ stands f or the inv erse image functor in M ). The functor ( 7.2.10.1 ) has a r ight adjoint (7.2.10.3) f ∗ : R - mo d ( X ) / / R - mo d ( Y ) which is simply the functor induced by f ∗ : M ( X ) / / M ( Y ) (as the latter sends R X -modules to f ∗ ( R X ) -modules, which are themselv es R Y -modules via the map a f ). If the map f is a P -mor phism, then, for an y R X -module M , the object f ] ( M ) has a natural structure of R Y -module: using the map a f , M has a natural str ucture of f ∗ ( R Y ) -module f ∗ ( R Y ) ⊗ X M / / M , and applying f ] , w e get b y the P -projection formula ( 1.1.26 ) a mor phism R Y ⊗ f ] ( M ) ' f ] ( f ∗ ( R Y ) ⊗ M ) / / f ] ( M ) 7 Basic homotopy commutativ e algebra 237 which deﬁnes a natural R Y -module structure on f ] ( M ) . For a P -morphism f : X / / Y , we deﬁne a functor (7.2.10.4) f ] : R - mo d ( X ) / / R - mo d ( Y ) as the functor induced b y f ] : M ( X ) / / M ( Y ) . Note that the functor ( 7.2.10.4 ) is a left adjoint to the functor ( 7.2.10.1 ) whene v er the map a f : f ∗ ( R Y ) / / R X is an isomorphism in M ( X ) . W e shall sa y that R is a cartesian monoid in M o ver S if R is a monoid of M ( 1 C , C ) such that all the structural maps f ∗ ( R Y ) / / R X are isomor phisms (i.e. if R is a car tesian section of the ﬁbred categor y Mon ( M ) ov er S ) If R is a car tesian monoid in M ov er S , then R - mo d is a P -ﬁbred category o v er S : to see this, it remains to prov e that, for any pullback square of S X 0 g / / f 0   X f   Y 0 h / / Y in which f is a P -mor phism, and for any R X -module M , the base change map f 0 ] g ∗ ( M ) / / h ∗ f ] ( M ) is an isomor phism, which follo ws immediately from the analogous f or mula f or M . Similarl y , w e see that whenev er R is a commutativ e monoid of M ( 1 S , S ) (i.e. R X is a commutative monoid in M ( X ) for all X in S ), then R - mo d is a symmetric monoidal P -ﬁbred categor y . Proposition 7.2.11 Let M be a combinatorial symmetric monoidal P -ﬁbred model category ov er S which satisﬁes the monoid axiom, and R a monoid in M ( 1 S , S ) (r esp. a car tesian monoid in M ov er S ). Then 7.2.2 (i) applied termwise tur ns R - mo d into a combinatorial ﬁbr ed model category (resp. a combinatorial P -ﬁbr ed model category). If mor eov er R is commutativ e, then R - mo d is a combinatorial symmetric monoidal ﬁbred model category (r esp. a combinatorial symmetric monoidal P - ﬁbr ed model categor y). Proof Choose, f or each object X of S , tw o small sets of maps I X and J X which generate the class of coﬁbrations and the class of tr ivial coﬁbrations in M ( X ) re- spectiv ely . Then R X ⊗ X I X and R X ⊗ X J X generate the class of coﬁbrations and the class of tr ivial coﬁbrations in R - mo d ( X ) respectivel y . For a map f : X / / Y in S , w e see from f or mula ( 7.2.10.2 ) that the functor ( 7.2.10.1 ) sends these generating coﬁbrations and trivial coﬁbrations to coﬁbrations and tr ivial coﬁbrations respec- tiv ely , from which we deduce that the functor ( 7.2.10.1 ) is a left Quillen functor . In the respective case, if f is a P -morphism, then we deduce similarly from the projec- tion f or mula ( 1.1.26 ) in M that the functor ( 7.2.10.4 ) sends generating coﬁbrations 238 Construction of ﬁbred categories and trivial coﬁbrations to coﬁbrations and tr ivial coﬁbrations respectivel y . The last assertion f ollo ws easily by appl ying 7.2.2 (ii) ter mwise.  Deﬁnition 7.2.12 Let M be a symmetr ic monoidal P -ﬁbred model category o v er S . A homotopy cartesian monoid R in M will be a homotopy car tesian section of Mon ( M ) . Proposition 7.2.13 Let M be a per fect symmetric monoidal P -ﬁbr ed model cate- gor y ov er S , and consider a homotopy cartesian monoid R in M ov er S . Then Ho ( R - mo d ) is a P -ﬁbred category ov er S , and R ⊗ L (−) : Ho ( M ) / / Ho ( R - mo d ) is a mor phism of P -ﬁbred categories. In the case where R is commutativ e, Ho ( R - mo d ) is ev en a symmetric monoidal P -ﬁbred category. Mor eov er , f or any weak equiv alence betw een homotopy cartesian monoids R / / S ov er S , the Quillen mor phism S ⊗ R (−) : R - mo d / / S - mo d induces an equiv alence of P -ﬁbr ed categories ov er S S ⊗ L R (−) : Ho ( R - mo d ) / / Ho ( S - mo d ) . Proof It is suﬃcient to pro ve these asser tions by restricting e v erything o ver S / S , where S runs ov er all the objects of S . In par ticular , w e ma y (and shall) assume that S has a terminal object S . As M is perf ect, it follo ws from condition (c) of Deﬁnition 7.2.3 that we can replace R by an y of its coﬁbrant resolution. In par ticular , w e ma y assume that R S is a coﬁbrant object of Mon ( M )( S ) . W e can thus deﬁne a termwise coﬁbrant car tesian monoid R 0 as the famil y of monoids f ∗ ( R S ) , where f : X / / S runs ov er all the objects of S ' S / S . There is a canonical mor phism of homotopy car tesian monoids R 0 / / R which is a termwise weak equivalence. W e thus get, by condition (c) of Deﬁnition 7.2.3 , an equivalence of ﬁbred categor ies R ⊗ L R 0 (−) : Ho ( R 0 - mo d ) / / Ho ( R - mo d ) . W e can thus replace R b y R 0 , which just means that we can assume that R is car tesian and ter m wise coﬁbrant. The ﬁrst asser tion f ollo ws then easily from Proposition 7.2.11 . In the case where R is commutativ e, we prov e that Ho ( R - mo d ) is a P -ﬁbred symmetric monoidal category as f ollo ws. Let f : X / / Y a mor phism of S . W e w ould like to pro ve that, f or any object M in Ho ( R - mo d )( X ) and any object N in Ho ( R - mo d )( Y ) , the canonical map (7.2.13.1) L f ] ( M ⊗ L R f ∗ ( N )) / / L f ] ( M ) ⊗ L R N is an isomorphism. By adjunction, this is equiv alent to pro v e that, f or any objects N and E in Ho ( R - mo d )( Y ) , the map 7 Basic homotopy commutativ e algebra 239 (7.2.13.2) f ∗ R Hom R ( N , E ) / / R Hom R ( f ∗ ( N ) , f ∗ ( E )) is an isomor phism in Ho ( R - mo d )( X ) (where R Hom R stands f or the internal Hom of Ho ( R - mo d ) ). But the f org etful functors U : Ho ( R - mo d )( X ) / / Ho ( M )( X ) are conservativ e, commute with f ∗ f or any P -mor phism f , and commute with internal Hom: by adjunction, this f ollo ws immediately from the fact that the functors R ⊗ L (−) : Ho ( M )( X ) / / Ho ( R - mo d )( X ) ' Ho ( R 0 - mo d )( X ) are symmetr ic monoidal and deﬁne a mor phism of P -ﬁbred categories (and thus, in particular, commute with f ] f or any P -mor phism f ). Hence, to prov e that ( 7.2.13.2 ) is an isomorphism, it is suﬃcient to prov e that its analog in Ho ( M ) is so, which f ollow s immediately from the fact that the analog of ( 7.2.13.1 ) is an isomor phism in Ho ( M ) by assumption. For the last asser tion, we are also reduced to the case where R and S are car tesian and ter m wise coﬁbrant, in which case this f ollow s easily again from condition (c) of Deﬁnition 7.2.3 .  Proposition 7.2.14 Let M be a combinatorial symmetric monoidal model categor y ov er S which satisﬁes the monoid axiom. Then, for any cartesian monoid R in M ov er S we have a Quillen morphism R ⊗ (−) : M / / R - mo d . If, for any object X of S , the unit object 1 X is coﬁbrant in M ( X ) and the monoid R X is coﬁbrant in Mon ( M )( X ) , then the for g etful functors also deﬁne a Quillen morphism U : R - mo d / / M . Proof The ﬁrst asser tion is obvious. For the second one, note that, for an y object X of S , the monoid R X is also coﬁbrant as an object of M ( X ) ; see Theorem 7.1.3 . This implies that the for getful functor U : R X - mo d / / M ( X ) is a left Quillen functor: by the small object argument and by deﬁnition of the model category str ucture of Theorem 7.2.2 (i), this follo ws from the trivial fact that the endofunctor R X ⊗ (−) : M ( X ) / / M ( X ) is a left Quillen functor itself whenev er R X is coﬁbrant in M ( X ) .  Remar k 7.2.15 The results of the preceding proposition (as well as their proofs) are also tr ue in ter ms of P c art -ﬁbred categor ies ( 3.1.21 ) o v er the category of S / S - diagrams for any object S of S (whence ov er all S -diagrams whenev er S has a terminal object). 240 Construction of ﬁbred categor ies 7.2.16 Consider no w a noether ian scheme S of ﬁnite dimension. W e choose a full subcategory of the category of separated noether ian S -schemes of ﬁnite dimension which is stable by ﬁnite limits, contains separated S -schemes of ﬁnite type, and such that, f or any étale S -mor phism Y / / X , if X is in S / S , so is Y . W e denote b y S / S this chosen category of S -schemes. W e also ﬁx an admissible class P of mor phisms of S / S which contains the class of étale mor phisms. Deﬁnition 7.2.17 A proper ty P of Ho ( M ) , for M a stable combinator ial P -ﬁbred model categor y o v er S / S , is homotopy linear if the follo wing implications are true. (a) If γ : M / / M 0 is a Quillen equiv alence (i.e. a Quillen morphism which is termwise a Quillen equivalence) between stable combinatorial P -ﬁbred model category ov er S / S , then M has proper ty P is and only if M 0 has proper ty P . (b) If M is a stable combinatorial symmetr ic monoidal P -model categor y which satisﬁes the monoid axiom, and such that the unit 1 X of M ( X ) is coﬁbrant, then, f or any car tesian and ter m wise coﬁbrant monoid R in M ov er S / S , R - mo d has proper ty P . Proposition 7.2.18 The f ollowing properties are homo topy linear: A 1 -homotopy invariance, P 1 -stability , the localization property, the pr operty of pr oper tr ansver - sality, separ ability, semi-separability , t -descent (f or a giv en Gro thendiec k topology t on S / S ). Proof Proper ty (a) of the deﬁnition abov e is obvious. Proper ty (b) comes from the fact that the for getful functors U : Ho ( R - mo d ) / / Ho ( M ) are conser v ativ e and commute with all the operations: L f ∗ and R f ∗ f or any mor phism f , as w ell as L f ] f or an y P -mor phism (by Proposition 7.2.14 ). Hence an y prop- erty formulated in ter ms of equations inv olving only these operations is homotop y linear .  P art III Motivic comple x es and relativ e cy cles 8 Relativ e cycles 243 In this entire par t, w e adopt the special conv ention that smooth means smooth separated of ﬁnite type. This concerns also the frame w ork of premotivic categor ies: w e assume the admissible class Sm is made of smooth separated mor phisms of ﬁnite type. This assumption is required by the use of the theory of ﬁnite correspondences (see more precisely Example 9.1.4 ). 8 Relativ e cy cles 8.0.1 In this entire section, S is the categor y of noether ian schemes; any scheme is assumed to be noether ian. W e ﬁx a subr ing Λ ⊂ Q which will be the r ing of coeﬃcients of the algebraic cy cles considered in the follo wing section. When we want to be precise, we say Λ -cycle for "algebraic cy cle with coeﬃcients in Λ ". Otherwise, we simply say cycle and the reader must assume that all algebraic cy cles ha v e their coeﬃcients in the ring Λ . 8.1 Deﬁnitions 8.1.a Category of cy cles 8.1.1 Let X be a scheme. As usual, an element of the underl ying set of X will be called a point and a mor phism Spec ( k ) / / X where k is a ﬁeld will be called a g eometric point . W e often identify a point x ∈ X with the cor responding geometric point Sp ec ( κ x ) / / X . Ho we v er , the e xplicit e xpression "the point Sp ec ( k ) / / X " alwa ys refers to a geometric point. As our schemes are assumed to be noetherian, any immersion f : X / / Y is quasi- compact. Thus, according to [ GD60 , 9.5.10], the schematic closure ¯ X of X in Y e xists which gives a unique factorization of f X j / / ¯ X i / / Y such that i is a closed immersion and j is an open immersion with dense image 82 . Note that when Y is reduced, ¯ X coincide with the topological closure of X in Y with its induced reduced subscheme structure. In this case, w e simply call ¯ Y the closure of Y in X . Deﬁnition 8.1.2 A Λ -cycle is a couple ( X , α ) such that X is a scheme and α is a Λ - linear combination of points of X . A g ener ic point of ( X , α ) is a point which appears 82 Recall the scheme ¯ X is characterized by the proper ty of being the smallest sub-scheme of Y with the e xistence of such a factorization. 244 Motivic comple xes and relative cycles in the Λ -linear combination α with a non zero coeﬃcient. The support Supp ( α ) of α is the closure of the generic points of α , seen as a reduced closed subscheme of X . A mor phism of Λ -cycles ( Y , β ) / / ( X , α ) is a mor phism of scheme f : Y / / X such that f ( Supp ( β )) ⊂ Supp ( α ) . W e say this mor phism is pseudo-dominant if f or an y generic point y of ( Y , β ) , f ( y ) is a generic point of ( X , α ) . When consider ing such a pair ( X , α ) , w e will denote it simpl y by α and ref er to X as the domain of α . W e also use the notation α ⊂ X to mean the domain of the cycle α is the scheme X . The categor y of Λ -cy cle is functor ial in Λ with respect to mor phisms of integral rings. In what f ollow s, cycles are assumed to hav e coeﬃcients in Λ unless e xplicitly stated (follo wing our conv entions for this section, see Paragraph 8.0.1 ). 8.1.3 Giv en a property ( P ) of mor phisms of schemes, we will say that a mor phism f : β / / α of cycles satisﬁes property ( P ) if the induced morphism f | Supp ( α ) Supp ( β ) satisﬁes proper ty ( P ) . Deﬁnition 8.1.4 Let X be a scheme. W e denote by X ( 0 ) the set of generic points of X . W e deﬁne as usual the cycle associated with X as the cy cle with domain X : h X i = Õ x ∈ X ( 0 ) lg ( O X , x ) . x . The integer lg ( O X , x ) , length of an ar tinian local r ing, is called the g eometric multi- plicity of x in X . When no confusion is possible, we usually omit the delimiters in the notation h X i . As an e xample, w e say that α is a cycle ov er X to mean the e xistence of a structural morphism of cy cles α / / h X i . 8.1.5 When Z is a closed subscheme of a scheme X , we denote by h Z i X the cycle h Z i considered as a cy cle with domain X . Consider a cycle α with domain X . Let ( Z i ) i ∈ I be the famil y of the reduced closure of g ener ic points of α . Then we can wr ite α uniquely as α = Í i ∈ I n i . h Z i i X . W e call this writing the standar d form of α f or shor t. Deﬁnition 8.1.6 Let α = Í i ∈ I n i . x i be a cycle with domain X and f : X / / Y be an y mor phism. For any i ∈ I , put y i = f ( x i ) . Then f induces an extension ﬁeld κ ( x i )/ κ ( y i ) betw een the residue ﬁelds. W e let d i be the degree of this e xtension ﬁeld in case it is ﬁnite and 0 otherwise. W e deﬁne the pushforwar d of α by f as the cycle with domain Y f ∗ ( α ) = Õ i ∈ I n i d i . f ( x i ) . Thus, when f is an immersion, f ∗ ( α ) is the same cy cle as α but seen as a cy cle with domain X . Remark also that we obtain the follo wing equality 8 Relativ e cycles 245 (8.1.6.1) f ∗  h X i  =  ¯ X  Y where ¯ X is the schematic closure of X in Y (indeed X is a dense open subscheme in ¯ X ). When f is clear , we sometimes abusivel y put: h X i Y : = f ∗ ( h X i ) . By transitivity of deg rees, we obviousl y ha v e f ∗ g ∗ = ( f g ) ∗ f or a composable pair of mor phisms ( f , g ) . Deﬁnition 8.1.7 Let α = Í i ∈ I n i . x i be a cy cle o v er a scheme S with domain f : X / / S and U ⊂ S be an open subscheme. Let I 0 = { i ∈ I | f ( x i ) ∈ U } . W e deﬁne the res triction of α ov er U as the cy cle α | U = Í i ∈ I 0 n i . x i with domain X × S U considered as a cycle ov er U . If α = Í i ∈ I n i . h Z i i X , then ob viously α | U = Í i ∈ I n i . h Z i × S U i X U . W e state the f ollowing obvious lemma f or conv enience: Lemma 8.1.8 Let S be a scheme, U ⊂ S an open subsc heme and X be an S -sc heme. Let j : X U / / X be the obvious open immersion. (i) F or any cycle ( X U , α 0 ) ,  j ∗ ( α 0 )  | U = α 0 . (ii) Assume ¯ U = S . F or any cycle ( X , α ) pseudo-dominant ov er S , j ∗ ( α | U ) = α . 8.1.b Hilbert cy cles 8.1.9 Recall that a ﬁnite dimensional sc heme X is equidimensional – we will say absolutely equidimensional – if its ir reducible components hav e all the same dimen- sion. W e will sa y that a ﬂat mor phism f : X / / S is equidimensional if it is of ﬁnite type and f or an y connected component X 0 of X , there e xists an integer e ∈ N such that f or any g eneric point η in X 0 , the ﬁber f − 1 [ f ( η )] is absolutely equidimensional of dimension e . Deﬁnition 8.1.10 Let S be a scheme. Let α be a cy cle ov er S with domain X . W e say that α is a Hilbert cycle o v er S if there e xists a ﬁnite famil y ( Z i ) i ∈ I of closed subschemes of X which are ﬂat equidimensional o ver S and a ﬁnite famil y ( n i ) i ∈ I ∈ Λ I such that α = Õ i ∈ I n i . h Z i i X . Example 8.1.11 An y cy cle o ver a ﬁeld k is a Hilber t cy cle o v er Sp ec ( k ) . Let S be the spectr um of a discrete valuation ring. A cy cle α = Í i ∈ I n i . x i o v er S is a Hilber t cy cle if and onl y if each point x i lies ov er the generic points of S . Indeed, an integral S -scheme is ﬂat if and only if it is dominant. The f ollo wing lemma f ollo ws almost directly from a result of [ SV00b ]: 246 Motivic complex es and relativ e cycles Lemma 8.1.12 Let f : S 0 / / S be a morphism of schemes and X be an S -scheme of ﬁnite type. Put X 0 = X × S S 0 . Let ( Z i ) i ∈ I be a ﬁnite family of closed subschemes of X suc h that eac h Z i is ﬂat equidimensional o ver S . W e assume the follo wing relation: (8.1.12.1) Õ i ∈ I n i . h Z i i X = 0 Then w e the follo wing equality holds: Õ i ∈ I n i . h Z i × S S 0 i X 0 = 0 . Proof When we assume that f or any index i ∈ I , Z i / S is equidimensional of dimen- sion e , this lemma is exactl y [ SV00b , Prop. 3.2.2]. W e sho w how to reduce to that case. U p to adding more members to the f amily ( Z i ) , w e can alwa ys assume that Z i is connected. Then, because Z i / S is equidimensional by assumption, there exis ts an integer e i such that f or an y point x ∈ Z ( 0 ) i , the ﬁber f − 1 [ f ( x )] is absolutely equidimensional of dimension e i . In par ticular the transcendence deg ree d x of the residual e xtension κ x / κ f ( x ) satisﬁes the relation: d x = e i . For any integer e ∈ N , we deﬁne the f ollo wing subset of I : I e = { i ∈ I | ∀ x ∈ Z ( 0 ) i , d x = e } . Thus ( I e ) e ∈ N is a par tition of I . One can rewrite the assumption ( 8.1.12.1 ) as follo ws: for any point x ∈ X , Õ i ∈ I | x ∈ Z ( 0 ) i n i . lg ( O Z i , x ) = 0 . In par ticular , given any integer e ∈ N , we deduce that the famil y ( Z i ) i ∈ I e still satisﬁes the relation ( 8.1.12.1 ). As any member of this famil y is equidimensional of dimension e , we can appl y [ SV00b , Prop. 3.2.2] to ( Z i ) i ∈ I e . This concludes.  8.1.13 Consider a Hilber t S -cy cle α ⊂ X and a mor phism of schemes f : S 0 / / S . Put X 0 = X × S S 0 . W e choose a ﬁnite famil y ( Z i ) i ∈ I of ﬂat equidimensional S - schemes and a ﬁnite famil y ( n i ) i ∈ I ∈ Λ I such that α = Í i ∈ I n i . h Z i i X . The previous lemma sa ys exactl y that the cy cle Õ i ∈ I n i . h Z i × S S 0 i X 0 depends only on α and not on the chosen families. Deﬁnition 8.1.14 A dopting the preceding notations and hypothesis, we deﬁne the pullbac k cycle of α along the mor phism f : S 0 / / S as the cycle with domain X 0 8 Relativ e cycles 247 α ⊗ [ S S 0 = Õ i ∈ I n i . h Z i × S S 0 i X 0 . In this setting the follo wing lemma is obvious: Lemma 8.1.15 Let α be a Hilbert cycle ov er S , and S 00 / / S 0 / / S be morphisms of sc hemes. Then ( α ⊗ [ S S 0 ) ⊗ [ S 0 S 00 = α ⊗ [ S S 00 . W e will use another impor tant computation from [ SV00b ] (it is a par ticular case of loc. cit. , 3.6.1). Proposition 8.1.16 Let R be a discret e valuation ring with residue ﬁeld k . Let α ⊂ X be a Hilbert cycle ov er Sp ec ( R ) and f : X / / Y a morphism ov er Sp ec ( R ) . W e denote by f 0 : X 0 / / Y 0 the pullbac k of f ov er Sp ec ( k ) . Suppose that the support of α is proper with r espect to f . Then f ∗ ( α ) is a Hilber t cycle ov er R and the follo wing equality of cycles holds in X 0 : f 0 ∗ ( α ⊗ [ S k ) = f ∗ ( α ) ⊗ [ S k . Deﬁnition 8.1.17 Let p : ˜ S / / S be a birational mor phism. Let C be the minimal closed subset of S such that p induces an isomor phism ( ˜ S − ˜ S × S C ) / / ( S − C ) . Consider α = Í i ∈ I n i . h Z i i X a cy cle o ver S wr itten in standard f or m. W e deﬁne the strict transf or m ˜ Z i of the closed subscheme Z i in X along p as the schematic closure of ( Z i − Z i × S C ) × S ˜ S in X × S ˜ S . W e deﬁne the strict transf orm of α along p as the cycle ov er ˜ S ˜ α = Õ i ∈ I n i . h ˜ Z i i X × S ˜ S . As in [ SV00b ], we remark that a corollar y of the platiﬁcation theorem of Gruson- Ra ynaud is the follo wing: Lemma 8.1.18 Let S be a reduced scheme and α be a pseudo-dominant cycle ov er S . Then there exists a dominant blow-up p : ˜ S / / S such that the strict transf or m ˜ α of α along p is a Hilber t cycle ov er ˜ S . W e conclude this par t by recalling an elementar y lemma about cy cles and Galois descent which will be used e xtensiv ely in the ne xt sections: Lemma 8.1.19 Let L / K be an extension of ﬁelds and X be a K -scheme. W e put X L = X × K Sp ec ( L ) and consider the fait hfully ﬂat mor phism f : X L / / X . Denot e by Cycl ( X ) (resp. Cycl ( X L ) ) the cycles with domain X (resp. X L ). 1. The morphism f ∗ : Cycl ( X ) / / Cycl ( X L ) , β  / / β ⊗ [ K L is a monomorphism. 2. Suppose L / K is ﬁnite. F or any K -cycle β ∈ Cycl ( X ) , f ∗ ( β ⊗ [ K L ) = [ L : K ] . β . 248 Motivic comple xes and relative cycles 3. Suppose L / K is ﬁnite normal with Galois gr oup G . The cycles in the imag e of f ∗ ar e inv ariant under the action of G . F or any cycle β ∈ Cycl ( X L ) G , ther e exists a unique cycle β K ∈ Cycl ( X ) suc h that β K ⊗ [ K L = [ L : K ] i . β wher e [ L : K ] i is the inseparable degree of L / K . 8.1.c Specialization The aim of this section is to giv e conditions on cy cles so that one can deﬁne a relativ e tensor product on them. Deﬁnition 8.1.20 Consider two cy cles α = Í i ∈ I n i . s i and β = Í j ∈ J m i . x j . Let S be the suppor t of α . A mor phism β f / / α of cycles is said to be pre-special if it is of ﬁnite type and f or an y j ∈ J , there e xists i ∈ I such that f ( x j ) = s i and n i | m j in Λ . W e deﬁne the reduction of β / α as the cycle ov er S β 0 = Õ j ∈ J , f ( x j ) = s i m j n i . x j . Example 8.1.21 Let S be a scheme and α a Hilber t S -cycle. Then the canonical morphism of cycles α / / h S i is pre-special. If S is the spectr um of a discrete valuation r ing, an S -cycle α is pre-special if and only if it is a Hilber t S -cy cle. Deﬁnition 8.1.22 Let α be a cy cle. A point (resp. tr ait ) of α will be a mor phism of the form Sp ec ( k ) x / / α (resp. Sp ec ( R ) τ / / α ) such that k is a ﬁeld (resp. R is a discrete v aluation r ing). W e simply say that x (resp. τ ) is dominant if the image of the generic point in the domain of α is a generic point of α . Let x : Sp ec ( k 0 ) / / α be a point. An e xtension of x will be a point y on α of the f or m Sp ec ( k ) / / Sp ec ( k 0 ) x / / α . A fat point of α will be a couple of mor phisms Sp ec ( k ) s / / Sp ec ( R ) τ / / α such that τ is a dominant trait and the image of s is the closed point of Sp ec ( R ) . Giv en a point x : Sp ec ( k ) / / α , a fat point o v er x is a factorization of x through a dominant trait as abov e. In the situation of the last deﬁnition, w e denote simply by ( R , k ) a fat point o v er x , without indicating in the notation the mor phisms s and τ . 8 Relativ e cycles 249 Remar k 8.1.23 With our choice of terminology , a point of α is in general an e xtension of a specialization of a generic point of α . As a fur ther ex ample, a dominant point of α is an extension of a generic point of α . Lemma 8.1.24 F or any cycle α and any non dominant point x : Sp ec ( k 0 ) / / α , ther e exists an extension y : Sp ec ( k ) / / α of x and a fat point ( R , k ) ov er y . Proof Replacing α by its suppor t S , we can assume α = h S i . Let s be the imag e of x in S , κ its residue ﬁeld. W e can assume S is reduced, ir reducible b y taking one irreducible component containing s , and local with closed point s . Let S = Sp ec ( A ) , K = F rac ( A ) . A ccording to [ GD61 , 7.1.7], there e xists a discrete valuation r ing R such that A ⊂ R ⊂ K , and R / A is an extension of local r ings. Then any composite e xtension k / κ of k 0 and the residue ﬁeld of R ov er κ giv es the desired fat point ( R , k ) .  Deﬁnition 8.1.25 Let β / / α be a pre-special mor phism of cy cles. Consider S the support of α and X the domain of β . Let β 0 = Í j ∈ J m j . h Z j i X be the reduction of β / α wr itten in standard f or m. 1. Let Sp ec ( K ) / / α be a dominant point. W e deﬁne the f ollowing cy cle o ver Sp ec ( K ) with domain X K = X × S Sp ec ( K ) : β K = Õ j ∈ J m j . h Z j × S Sp ec ( K ) i X K . 2. Let Sp ec ( R ) τ / / S be a dominant trait, K be the fraction ﬁeld of R and j : X K / / X R be the canonical open immersion. W e deﬁne the f ollo wing cy cle o v er R with domain X R : β R = j ∗ ( β K ) . A ccording to ex ample 8.1.11 , β R is a Hilber t cy cle o v er R . 3. Let x : Spec ( k ) / / α be a point on α and ( R , k ) be a fat point o v er x . W e deﬁne the specialization of β along the f at point ( R , k ) as the cycle β R , k : = β R ⊗ [ R k using the abov e notation and deﬁnition 8.1.14 . It is a cycle o v er Sp ec ( k ) with domain X k = X × S Sp ec ( k ) . Remar k 8.1.26 Let β be an S -cy cle, x : Sp ec ( K ) / / S be a dominant point and U be an open neighborhood of x in S . Then if β is pre-special o v er S , β | U is pre-special o v er U and β K = ( β | U ) K . If τ : Sp ec ( R ) / / S (resp. ( R , k ) ) is a trait (resp. fat p oint) with generic point x , w e also get β R = ( β | U ) R (resp. β R , k = ( β | U ) R , k ). 8.1.27 Let S be a reduced sc heme, and β = Í i ∈ I n i . x i be an S -cycle with domain X . For an y inde x i ∈ I , let κ i be the residue ﬁeld of x i . 250 Motivic complex es and relativ e cycles Consider a dominant point x : Sp ec ( K ) / / S . Let η be its image in S and F be the residue ﬁeld of η . W e put I 0 = { i ∈ I | f ( x i ) = η } where f : X / / S is the structural mor phism. With these notations, we get β K = Õ i ∈ I 0 n i . h Sp ec ( κ i ⊗ F K ) i X K , and f or a dominant trait Sp ec ( R ) / / S with generic point x , (8.1.27.1) β R = Õ i ∈ I 0 n i . h Sp ec ( κ i ⊗ F K ) i X R , where Sp ec ( κ i ⊗ F K ) is seen as a subscheme of X K (resp. X R ). Consider a fat point ( R , k ) with g ener ic point x and wr ite β = Í i ∈ I n i . h Z i i X in standard f orm ( i.e. Z i is the closure of { x i } in X ). Then according to ( 8.1.6.1 ), we obtain 83 β R , k = Õ i ∈ I 0 n i . D Z i , K × R Sp ec ( k ) E X k where Z i , K = Z i × S Sp ec ( K ) is considered as a subscheme of X K and the schematic closure is taken in X R . Considering the description of the schematic closure f or the g eneric ﬁber of an R -scheme ( cf. [ GD67 , 2.8.5]), we obtain the f ollo wing wa y to compute β R , k . By deﬁnition, R is an F -alg ebra. For i ∈ I 0 , let A i be the imag e of the canonical morphism κ i ⊗ F R / / κ i ⊗ F K . It is an R -algebra without R -torsion. Moreo v er , the factorization Sp ec ( κ i ⊗ F K ) / / Sp ec ( A i ) / / Sp ec ( κ i ⊗ F R ) deﬁnes Sp ec ( A i ) as the schematic closure of the left hand side in the r ight hand side ( cf. [ GD67 , 2.8.5]). In particular, we get an immersion Sp ec ( A i ⊗ R k ) / / X k and the nice f ormula: β R , k = Õ i ∈ I 0 n i . h Sp ec ( A i ⊗ R k )i X k . Deﬁnition 8.1.28 Consider a mor phism of cy cles f : β / / α and a point x : Sp ec ( k 0 ) / / α . W e sa y that f is special at x if it is pre-special and for an y extension y : Sp ec ( k ) / / α of x , f or an y fat points ( R , k ) and ( R 0 , k ) ov er y , the equality β R , k = β R 0 , k holds in X k . Equiv alently , we say that β / α is special at x . W e say that f is special (or that β is special ov er α ) if it is special at ev er y point of α . 8.1.29 Here is a dictionar y to compare the abov e deﬁnition with that of Suslin and V oe v odsky in [ SV00b , 3.1.3]. 83 This show s that our deﬁnition coincide with the one given in [ SV00b ] (p. 23, paragraph preceding 3.1.3) in the case where α = h S i , S reduced. 8 Relativ e cycles 251 Consider a pre-special mor phism β / α . Let X be the domain of β , S be the suppor t of α and β 0 be the reduction (see Deﬁnition 8.1.20 ) of β / α , seen as a pre-special S -cy cle. Then the f ollo wing conditions are equivalent: (i) β / α is special; (ii) β 0 / S is special. This follo ws from the v er y deﬁnition of the specialization of β / α along f at points (Deﬁnition 8.1.25 ). Moreo v er , condition (ii) say s e xactl y that β 0 is a relative cycle on X ov er S in the sense of Deﬁnition 3.1.3 of [ SV00b ]. Remar k 8.1.30 1. T r ivially , f is special at ev er y dominant point of α . 2. Giv en an extension y of x , it is equivalent f or f to be special at x or at y (use Lemma 8.1.19 (1)). Thus, in the case where α = h S i , w e can restrict our attention to the points s ∈ S . 3. A ccording to 8.1.26 , the proper ty that β / S is special at s ∈ S depends only on an open neighborhood U of s in S . More precisely , the f ollo wing conditions are equiv alent: (i) β is special at s ov er S . (ii) β | U is special at s ov er U . Example 8.1.31 Let S be a scheme and β be a Hilber t cycle o ver S . W e hav e already seen that β / / h S i is pre-special. The ne xt lemma show s this morphism is in fact special. Lemma 8.1.32 Let S be a scheme and β be a Hilbert cycle ov er S . Consider a point x : Spec ( k ) / / S and a fat point ( R , k ) ov er x . Then β R , k = β ⊗ [ S k . Proof A ccording to the preceding deﬁnition and Lemma 8.1.15 it is suﬃcient to pro v e β R = β ⊗ [ S R . As the two sides of this equation are unchang ed when replacing β b y the reduction β 0 of β / S , we can assume that S is reduced. By additivity , we are reduced to the case where β = h X i is the fundamental cycle associated with a ﬂat S -scheme X . According to 8.1.6.1 , β R =  X K  X R . Applying now [ GD67 , 2.8.5], X K is the unique closed subscheme Z of X R such that Z is ﬂat o ver Sp ec ( R ) and Z × R Sp ec ( K ) = X K . Thus, as X R is ﬂat ov er Sp ec ( R ) , we get X K = X R and this concludes.  Lemma 8.1.33 Let p : ˜ S / / S be a birational mor phism and consider a commuta- tiv e diagr am ˜ S p   Sp ec ( k ) / / Sp ec ( R ) 4 4 * * S suc h that ( R , k ) is a fat point of ˜ S and S . 252 Motivic comple xes and relative cycles Consider a pr e-special cycle β ov er S and ˜ β its strict tr ansform along p . Then, ˜ β is pr e-special and ˜ β R , k = β R , k . Proof Using 8.1.26 , w e reduce to the case where p is an isomor phism which is trivial.  Lemma 8.1.34 Let S be a reduced scheme, x : Spec ( k 0 ) / / S be a point and α be a pr e-special cycle ov er S . Let p : ˜ S / / S be a dominant blow-up suc h that the strict transf orm ˜ α of α along p is a Hilbert cycle ov er ˜ S . Then the f ollowing conditions ar e equiv alent : (i) α is special at x . (ii) f or every couple of points x 1 , x 2 : Sp ec ( k ) / / ˜ S such that p ◦ x 1 = p ◦ x 2 and p ◦ x 1 is an extension of x , ˜ α ⊗ [ ˜ S x 1 = ˜ α ⊗ [ ˜ S x 2 . Proof The case where x is a dominant point follo ws from the deﬁnitions and the f act p is an isomor phism at the generic point. W e thus assume x is non dominant. ( i ) ⇒ ( ii ) : Applying Lemma 8.1.24 to x i , i = 1 , 2 , we can ﬁnd an e xtension x 0 i : Sp ec ( k i ) / / ˜ S of x i and a fat point ( R i , k i ) o v er x 0 i . T aking a composite extension L of k 1 and k 2 o v er k , w e can fur ther assume L = k 1 = k 2 and p ◦ x 0 1 = p ◦ x 0 2 . Then f or i = 1 , 2 , w e get  ˜ α ⊗ [ ˜ S x i  ⊗ [ k L 8 . 1 . 15 ˜ α ⊗ [ ˜ S x 0 i 8 . 1 . 32 ˜ α R i , L 8 . 1 . 33 α R i , L , and this concludes according to 8.1.19 (1). ( ii ) ⇒ ( i ) : Consider an e xtension y : Spec ( k ) / / α o v er x and two fat point ( R 1 , k ) , ( R 2 , k ) ov er y . Fix i ∈ { 1 , 2 } . As p is proper birational, the trait Sp ec ( R i ) on S can be extended (uniquely) to ˜ S . Let x i : Spec ( k ) / / Sp ec ( R i ) / / ˜ S be the induced point. Then the f ollo wing computation allow s concluding: α R i , k 8 . 1 . 33 ˜ α R i , k 8 . 1 . 32 ˜ α ⊗ [ x i  8.1.d Pullback 8.1.35 In this par t, we constr uct a pullback which e xtends the pullback deﬁned by Suslin et V oev odsky in [ SV00b , 3.3.1] to the case of morphism of cycles. Consider the situation of a diag ram of cy cles β f   X   ⊂ α 0 / / α S 0 / / S where the diagram on the r ight is the domain of the one on the left. Let n be e xponential characteristic of Supp ( α 0 ) . The pullbac k of β , considered as an α -cy cle, o ver α 0 will be a Λ [ 1 / n ] -cycle denoted b y β ⊗ α α 0 . It will ﬁts into the follo wing commutative diag ram of cy cles 8 Relativ e cycles 253 β ⊗ α α 0 / /   β   X × S S 0 / /   X   ⊂ α 0 / / α S 0 / / S where the r ight commutativ e square is again the suppor t of the left one. It will be deﬁned under an assumption on β / α and is theref ore non symmetr ic 84 . This assumption will impl y that β / α is pre-special, and the ﬁrst property of β ⊗ α α 0 is that it is pre-special ov er α 0 . W e deﬁne this product in three steps in which the follo wing proper ties 85 will be a guideline: (P1) Let S 0 be the support of α and β 0 be the reduction (see Deﬁnition 8.1.20 ) of β / α , as an S 0 -cy cle. Consider the canonical factorization α 0 / / S 0 / / α . Then, β ⊗ α α 0 = β 0 ⊗ S 0 α 0 . (P2) Consider a commutativ e diagram Sp ec ( E ) / / Sp ec ( R 0 ) / /   (∗) Sp ec ( R )   α 0 / / α such that ( R , E ) (resp. ( R 0 , E ) ) is a fat point on α (resp. α 0 ). Then, ( β ⊗ α α 0 ) R 0 , E = β R , E . Assume α 0 / / α = h S 0 / / S i . (P3) If β is a Hilber t cycle ov er S , β ⊗ S S 0 = β ⊗ [ S S 0 . (P4) Consider a factorization S 0 / / U j / / S such that j is an open immersion. Then β ⊗ S S 0 = β | U ⊗ U S 0 . (P5) Consider a f actor ization S 0 / / ˜ S p / / S such that p is a birational morphism. Then β ⊗ S S 0 = ˜ β ⊗ ˜ S S 0 . Lemma 8.1.36 Consider the hypothesis of 8.1.35 in the case where α 0 = Sp ec ( k ) is a point x of α . W e suppose that f is special at x . Then the pr e-special Λ [ 1 / n ] -cycle β ⊗ α k exists and is uniquely determined by property (P2) abov e. W e also put β k : = β ⊗ α k . The properties (P1) to (P5) ar e fulﬁlled and in addition : (P6) F or any extension ﬁelds L / k , β L = β k ⊗ [ k L . Proof A ccording to Lemma 8.1.24 there alwa ys exis ts a f at point ( R , E ) o v er an e xtension of x . Thus the unicity statement f ollow s from 8.1.19 (1). For the exis tence, we ﬁrst consider the case where α = h S i is a reduced scheme. Applying Lemma 8.1.18 , there exis ts a blo w -up p : ˜ S / / S such that the strict transf or m ˜ β of β along p is a Hilber t cy cle ov er ˜ S . 84 See fur ther 8.2.3 f or this question. 85 All these properties ex cept (P3) will be par ticular cases of the associativity of the pullback. 254 Motivic complex es and relative cycles As p is surjective, the ﬁber ˜ S k is a non-empty algebraic k -scheme. Thus, it admits a closed point giv en b y a ﬁnite extension k 0 0 of k . Let k 0 / k be a normal closure of k 0 0 / k and G be its Galois group. As β / S is special at x by hypothesis, Lemma 8.1.34 implies that ˜ β ⊗ [ ˜ S k 0 is G -inv ar iant. Thus, appl ying Lemma 8.1.19 , there exis ts a unique cy cle β k ⊂ X k with coeﬃcients in Λ [ 1 / n ] such that β k ⊗ [ k k 0 = ˜ β ⊗ [ ˜ S k 0 . W e prov e (P2). Giv en a diag ram (∗) with α 0 = Spec ( k ) , we ﬁrst remark that ( β k ) R 0 , E = β k ⊗ [ k E . As p is proper birational, the dominant trait Sp ec ( R ) / / S lifts to a dominant trait Sp ec ( R ) / / ˜ S . Let E 0 / k be a composite e xtension of k 0 / k and E / k . With these notations, we get the f ollowing computation: β R , E ⊗ [ E E 0 8 . 1 . 33 ˜ β R , E ⊗ [ E E 0 8 . 1 . 32 ˜ β ⊗ [ ˜ S E 0 8 . 1 . 15 ( ˜ β ⊗ [ ˜ S k 0 ) ⊗ [ E E 0 β k ⊗ [ k E 0 , so that we can conclude by appl ying 8.1.19 (1). In the general case, we consider he suppor t S of α abd β 0 / S the reduction of β / α . A ccording to (P1), we are led to put β k : = ( β 0 ) k with the help of the preceding case. Considering the deﬁnition of specialization along f at points, we easily check this cy cle satisﬁes proper ty (P2). Finall y , proper ty (P6) (resp. (P3), (P5)) f ollow s from the unicity statement apply - ing lemmas 8.1.24 , 8.1.19 (1) (resp. and moreov er Lemma 8.1.32 , 8.1.33 ).  Remar k 8.1.37 In the case where x is a dominant point, the cy cle β k deﬁned in the previous proposition ag rees with the one deﬁned in 8.1.25 (1). Lemma 8.1.38 Consider the hypothesis of 8.1.35 in the case where α 0 = Sp ec ( O ) is a trait of α . Let K be the fr action ﬁeld of O and x the corresponding point on α . W e suppose that f is special at x . Then the pre-special Λ [ 1 / n ] -cycle β ⊗ α O exists and is uniquely deﬁned by the property ( β ⊗ α O ) ⊗ [ O K = β K with the notations of the preceding lemma. W e also put β O : = β ⊗ α O . The properties (P1) to (P5) ar e fulﬁlled and in addition : (P6’) F or any extension O 0 / O of discre te valuation rings, β O 0 = β O ⊗ [ O O 0 . Proof Remark that, with the notation of deﬁnition 8.1.7 , β O ⊗ [ O K = β O | Spec ( K ) . For the ﬁrst statement, we simply apply Lemma 8.1.8 and put β O = j ∗ ( β K ) where j : X K / / X O is the canonical open immersion. Then proper ties (P1), (P3), (P4), (P5) and (P6’) of the case considered in this lemma follo ws easily from the uniqueness statement and the cor responding properties in the preceding lemma (applying again 8.1.8 ). It remains to prov e (P2). A ccording to (P1), w e reduce to the case α = h S i f or a reduced scheme S . W e choose a birational mor phism p : ˜ S / / S such that the proper transf or m ˜ β is a Hilbert ˜ S -cy cles. Consider a diag ram of the form (∗) in this case. A ccording to proper ty (P3), we can assume R 0 = O . Remark the trait Sp ec ( R ) / / S admits an e xtension Sp ec ( R ) / / ˜ S as p is proper . The point x admits an e xtension K 0 / K which lifts to a point x 0 : Sp ec ( K 0 ) / / ˜ S – again ˜ S K is a non empty algebraic scheme. The discrete valuation corresponding to 8 Relativ e cycles 255 O ⊂ K extends to a discrete valuation on K 0 as K 0 / K is ﬁnite. Let O 0 ⊂ K 0 be the corresponding v aluation r ing. The cor responding trait Sp ec ( O 0 ) / / S thus admits a lifting to ˜ S cor responding to the point x 0 as p is proper . Consider ing a composite e xtension E 0 / K of K 0 / K and E / K , we hav e obtained a commutativ e diagram Sp ec ( E 0 ) / / Sp ec ( O 0 ) / / Sp ec ( R )   Sp ec ( O 0 ) / / ˜ S which lifts our or iginal diagram (∗) . Let x 1 (resp. x 2 ) be the point Sp ec ( E ) 0 / / ˜ S corresponding to the the composite through the upper wa y (resp. low er wa y) in the preceding diagram. Then, β R , E ⊗ [ E E 0 = ˜ β x 1 . Moreo v er , we get ( β ⊗ S O ) O , E ⊗ [ E E 0 8 . 1 . 32 ( β ⊗ S O ) ⊗ [ O E 0 ( P 5 ) + ( P 6 0 ) ( ˜ β ⊗ ˜ S O 0 ) ⊗ [ O 0 E 0 ( P 3 ) ˜ β x 2 . By h ypothesis, β / α is special at Sp ec ( K 0 ) / / S . Thus Lemma 8.1.34 concludes.  Theorem 8.1.39 Consider the hypot hesis of 8.1.35 . Assume f is special at the g eneric points of α 0 . Then the pre-special Λ [ 1 / n ] -cycle β ⊗ α α 0 exists and is uniquely determined by property (P2). It satisﬁes all the properties (P1) to (P5). Proof A ccording to Lemma 8.1.24 , for an y point s of S 0 with residue ﬁeld κ , there e xists an e xtension E / κ and a fat point ( R , E ) (resp. ( R 0 , E ) ) of α (resp. α 0 ) o v er Sp ec ( E ) / / α (resp. Sp ec ( E ) / / α 0 ). The uniqueness statement f ollo ws by ap- plying Lemma 8.1.19 (1). For the exis tence, we wr ite α 0 = Í i ∈ I n i . h Z i i S 0 in standard f or m. For an y i ∈ I , let K i be the function ﬁeld of Z i and consider the canonical morphism Spec ( K i ) / / α . Let β K i ⊂ X K i be the Λ [ 1 / n ] -cycle deﬁned in lemma 8.1.36 . Let j i : X K i / / X 0 be the canonical immersion and put: (8.1.39.1) β ⊗ α α 0 = Õ i ∈ I n i . j i ∗ ( β K i ) . Then proper ties (P1), (P3), (P4) and (P5) are direct consequences of this deﬁnition and of the cor responding proper ties of Lemma 8.1.36 . W e check proper ty (P2). Given a diagram of the form (∗) , there exis ts a unique i ∈ I such that Spec ( R 0 ) dominates Z i . Thus w e g et f or this choice of i ∈ I that ( β ⊗ α α 0 ) R 0 , E =  j i ∗ ( β K i )  R 0 , E . Let K 0 be the fraction ﬁeld of R 0 and consider the open immersion j 0 : X K 0 / / X R 0 . The f ollo wing computation then concludes:  j i ∗ ( β K i )  R 0 , E j 0 ∗  j i ∗ ( β K i ) K 0 ) ⊗ [ R 0 E 8 . 1 . 26 j 0 ∗ ( β K 0 ) ⊗ [ R 0 E 8 . 1 . 38 β R 0 ⊗ [ R 0 E 8 . 1 . 38 ( P 2 ) β R , E . 256 Motivic complex es and relative cycles Deﬁnition 8.1.40 In the situation of the previous theorem, we call the Λ [ 1 / n ] -cy cle β ⊗ α α 0 the pullback of β / α by α 0 . 8.1.41 By construction, the cycle β ⊗ α α 0 is bilinear with respect to addition of cycles in the f ollo wing sense: (P7) Consider the hypothesis of 8.1.35 . Let α 0 1 , α 0 2 be cy cles with domain S 0 such that α = α 0 1 + α 0 2 . If β / α is special at the g ener ic points of α 1 and α 2 , then the f ollowing cycles are equal in X × S S 0 : β ⊗ α ( α 0 1 + α 0 2 ) = β ⊗ α α 0 1 + β ⊗ α α 0 2 . (P7’) Consider the h ypothesis of 8.1.35 . Let β 1 , β 2 be cycles with domain X such that β = β 1 + β 2 . If β 1 and β 2 are special ov er α at the generic points of α 0 , then β / α is special at the g ener ic points of α 0 and the f ollowing cy cles are equal in X × S S 0 : ( β 1 + β 2 ) ⊗ α α 0 = β 1 ⊗ α α 0 + β 2 ⊗ α α 0 . In the theorem abo v e, we can assume that X (resp. S , S 0 ) is the suppor t of β (resp. α , α 0 ). Thus the suppor t of β ⊗ α α 0 is included in X × S S 0 . More precisely : Lemma 8.1.42 Consider the hypothesis of 8.1.35 and assume that X (resp. S , S 0 ) is the support of β (resp. α , α 0 ). Then, if β / α is special at the g eneric points of α 0 , we obtain: (i) Let ( X × S S 0 ) ( 0 ) be the g eneric points of X × S S 0 . Then, we can write β ⊗ α α 0 = Õ x ∈( X × S S 0 ) ( 0 ) m x . x (ii) F or any g eneric point x of X × S S 0 , if m x , 0 , the image of x in S 0 is a g eneric point s 0 and the multiplicity of s 0 in α 0 divides m x in Λ [ 1 / n ] . Proof Point (ii) is just a traduction that β ⊗ α α 0 is pre-special ov er α 0 . For point (i), w e reduce easily to the case where α is the scheme S and S is reduced. W e can also assume that α 0 is the spectrum of a ﬁeld k . It is suﬃcient to check point (i) after an e xtension of k . Thus w e can apply Lemma 8.1.18 to reduce to that case where β is a Hilbert cy cle o ver S . This case is obvious.  Deﬁnition 8.1.43 In the situation of the previous lemma, we put m SV ( x ; β ⊗ α α 0 ) : = m x ∈ Λ [ 1 / n ] and we call them the Suslin- V oev odsky multiplicities (in the operation of pullback). Remar k 8.1.44 Consider the notations of the previous lemma: 1. Assume that α is the spectrum of a ﬁeld k . Then the product β ⊗ k α 0 is alwa ys deﬁned and agrees with the classical exterior pr oduct (according to (P3)). 8 Relativ e cycles 257 2. A ccording to the previous lemma, the ir reducible components of X × S S 0 which does not dominate an irreducible component of S 0 ha v e multiplicity 0 : they correspond to the "non proper components" with respect to the operation β ⊗ α α 0 . 3. Assume α 0 / / α = h S 0 p / / S i , β = Í i ∈ I n i . x i . Let y be a generic point of X × S S 0 lying ov er a generic point s 0 of S 0 . Let S 0 0 be the ir reductible component of S 0 corresponding to s 0 . Consider any irreducible component S 0 of S which contains p ( s 0 ) and let β 0 = Í i n i . x i where the sum runs o v er the inde x es i such that x i lies o v er S 0 . Then, according to ( 8.1.39.1 ), m SV ( y ; β ⊗ S h S 0 i ) = m SV ( y ; β 0 ⊗ S 0 h S 0 0 i ) . This is a ke y proper ty of the Suslin- V oev odsky multiplicities which e xplains wh y we hav e to consider the property that β / α is special at s 0 (see 8.3.25 f or a reﬁned statement). Lemma 8.1.45 Consider a mor phism of cycles α 0 / / α and a pre-special morphism f : β / / α which is special at the g eneric points of α . Consider a commutative squar e Sp ec ( k 0 ) x 0 / /   α 0   Sp ec ( k ) x / / α suc h that k and k 0 ar e ﬁelds. Then the follo wing conditions are equivalent : (i) f is special at x . (ii) β ⊗ α α 0 / / α 0 is special at x 0 . Proof This f ollo ws easily from Lemma 8.1.24 and proper ty (P2).  Corollary 8.1.46 Let f : β / / α be a special mor phism. Then f or any mor phism α 0 / / α , β ⊗ α α 0 / / α 0 is special. Deﬁnition 8.1.47 Let f : β / / α be a mor phism of cycles and x : Spec ( k ) / / α be a point. W e sa y that f is Λ -univ ersal at x if it is special at x and the cycle β ⊗ α k has coeﬃcients in Λ . In the situation of this deﬁnition, let s be the image of x in the suppor t of α , and κ s be its residue ﬁeld. Then according to (P6), β k = β κ s ⊗ [ κ s k . Thus f is Λ -univ ersal at x if and only if it is Λ -universal at s . Fur thermore, the f ollowing lemma f ollow s easily : Lemma 8.1.48 Let f : β / / α be a morphism of cycles. The follo wing conditions ar e equivalent : (i) F or any point s ∈ α , f is Λ -universal at s . (ii) F or any point x : Sp ec ( k ) / / α , f is Λ -universal at x . (iii) F or any mor phism of cycles α 0 / / α , β ⊗ α α 0 has coeﬃcients in Λ . 258 Motivic comple xes and relative cycles Deﬁnition 8.1.49 W e sa y that a morphism of cycles f is Λ -univ ersal if it satisﬁes the equiv alent proper ties of the preceding lemma. Of course, Λ -univ ersal mor phisms are stable by base c hange. These deﬁnitions will be applied similarl y to mor phisms of schemes b y consider ing the associated morphism of cy cles. Example 8.1.50 A ccording to property (P3) of the pullback, a ﬂat equidimensional morphism of schemes is Λ -univ ersal. 8.1.51 Let β / α be a mor phism of Λ -cycles. Let S be the suppor t of α and consider the obvious mor phism of cy cles S / / α . Recall from proper ty (P1) of Paragraph 8.1.35 that the cycle β 0 : = β ⊗ α S is the reduction of β / α (Deﬁnition 8.1.20 ). This is a special Λ -cy cle o v er S (see Paragraph 8.1.29 ) Moreo v er , it f ollo ws from the deﬁnition of the product that the follo wing condi- tions are equiv alent: (i) β / α is Λ -universal; (ii) β 0 / S is Λ -univ ersal. In par ticular , condition (ii) appear in Lemma 3.3.9 of [ SV00b ] (with a restr iction on the relativ e dimension that is not needed in fact). Remar k 8.1.52 Though Lemma 3.3.9 of [ SV00b ] does not giv e rise to an y deﬁnition in loc. cit. , it is central in the theor y of Suslin and V oev odsky . In par ticular , it appears in the deﬁnition of the g roups z ( X / S , r ) , c ( X / S , r ) ,... that takes place r ight after Lemma 3.3.9. Our deﬁnition has the advantag e to: • w ork properl y o v er non reduced schemes; • ha v e a local formulation (this is essential for the theorems of constructibility in subsection 8.3.a ); • being free of unnecessary assumptions such has the relativ e dimension of ﬁbers (the integer r that appear in z ( X / S , r ) ). Besides, the categor ical language introduced, obviousl y inspired by E.G.A., is v er y natural and will prov e to be useful in the treatment of ﬁnite cor respondences (see for e xample the deﬁnition of the composition product, 9.1.5 , and the shor t proof of the properties of this composition product, 9.1.7 ). The f ollo wing proposition sho ws that one can bound the denominators that can happen after an arbitrar y number of base chang es. Proposition 8.1.53 Let β / α be a special morphism of Λ -cycles. Then ther e exists an integ er N > 0 suc h that N . β is Λ -universal. 8 Relativ e cycles 259 Proof A ccording to Paragraph 8.1.51 , one can reduce to the case where α is a reduced scheme S . W e then prov e by noetherian induction on S the f ollo wing asser tion: for an y closed subscheme Z ⊂ S , and an y special Λ -cycle α on S , there e xists an integer N > 0 suc h that N . α is Λ -special T ake a special Λ -cy cle α on S . According to Lemma 8.1.18 , there exis ts a birational morphism p : ˜ S / / S such that the strict transform ˜ α of α along p is a Hilber t cy cle, thus Λ -universal. Let U be a dense open subscheme of S abov e which p is an isomorphism. Thus f or any point s ∈ U , with in v erse image t in p − 1 ( U ) , we obtain that the cy cle α s = ˜ α t has Λ -coeﬃcients. Let Z be the complement of U in S , with its reduced schematic structure. Then, by construction, the pullback α ⊗ S Z is an Λ [ 1 / N ] -cy cle. In par ticular α 0 = N . α ⊗ S Z is a special Λ -cy cle ov er Z . As Z is a proper closed subscheme of S , we can apply to the N oether ian induction hypothesis to Z and α 0 . W e ﬁnd an integer N 0 > 0 such that N 0 .α 0 is Λ -universal. By transitivity of pullbacks (which f ollo ws easily from the uniqueness statement of Theorem 8.1.39 ; see Proposition 8.2.4 ), w e thus obtain that ( N N 0 ) . α is Λ -universal ov er S .  Recall that Λ is a sub-r ing of the ring of rationals. One easil y deduce from the preceding proposition the f ollo wing result. Corollary 8.1.54 F or any Λ -cycle α special ov er a (noetherian) sc heme S , there exists an integ er N > 0 suc h that N . α is Z -univ ersal ov er S . 8.2 Intersection theoretic properties 8.2.a Commutativity Lemma 8.2.1 Consider morphisms of cycles with suppor t in the left diagram β   X f   ⊂ γ / / α T g / / S suc h that β / α is pre-special and γ / α is pseudo-dominant. Assume α = Õ i ∈ I n i . s i , β = Õ j ∈ J m j . x j , γ = Õ l ∈ H p l . t l and denote by κ s i (r esp. κ x j , κ t l ) the residue ﬁeld of s i (r esp. x j , t l ) in S (r esp. X , T ). Considering ( i , j , l ) ∈ I × J × H suc h that f ( x j ) = g ( t l ) = s i , we denote by ν j , l : Sp ec  κ x j ⊗ κ s i κ t l  / / X × S T the canonical immersion. Then the f ollowing assertions hold: (i) β is special at the g eneric points of γ . (ii) The cycle β ⊗ α γ has coeﬃcients in Λ . 260 Motivic comple xes and relative cycles (iii) The f ollowing equality of cycles holds β ⊗ α γ = Õ i , j , l m j n i p l . ν j , l ∗  h Sp ec  κ y j ⊗ κ x i κ z l  i  wher e the sum runs ov er ( i , j , l ) ∈ I × J × H such that f ( x j ) = g ( t j ) = s i . Proof Asser tion (i) is in fact the ﬁrst point of 8.1.30 . Assertion (ii) follo ws from assertion (iii), which is a consequence of the deﬁning f or mula ( 8.1.39.1 ) and remark 8.1.37 .  Corollary 8.2.2 Let g : T / / S be a ﬂat mor phism and β = Í j ∈ J m j . h Z j i X be a pr e-special S -cycle written in standard form. Then β / S is pre-special at the g eneric points of T and β ⊗ S h T i = Õ j ∈ J m j . h Z j × S T i . The pullback β ⊗ α γ , at it is deﬁned onl y when β / α is special, is in general non symmetr ic in β and γ . Ho we ver the pre vious lemma implies it is symmetr ic whenev er it makes sense: Corollary 8.2.3 Consider pr e-special mor phisms of cycles β / / α and γ / / α . Then β (resp. γ ) is special at the g eneric points of γ (resp. β ) and the follo wing equality holds: β ⊗ α γ = γ ⊗ α β . 8.2.b Associativity Proposition 8.2.4 Consider mor phism of cycles β f / / α , α 00 / / α 0 / / α suc h that f is special at the g eneric points of α 0 and of α 00 . Let n be the exponential c haract eristic of α 00 . Then the f ollowing assertions hold: (i) The r elative cycle ( β ⊗ α α 0 )/ α 0 is special at the g eneric points of α 00 . (ii) The cycle ( β ⊗ α α 0 ) ⊗ α 0 α 00 has coeﬃcients in Λ [ 1 / n ] . (iii) ( β ⊗ α α 0 ) ⊗ α 0 α 00 = β ⊗ α α 00 . Proof Asser tion (i) is a corollar y of Lemma 8.1.45 . Asser tion (ii) is in fact a corollary of assertion (iii), which in tur n f ollow s easily from the uniqueness statement in theorem 8.1.39 .  Lemma 8.2.5 Let γ g / / β f / / α be tw o pr e-special morphisms of cy cles with domains Y / / X / / S . Consider a fat point ( R , k ) ov er α suc h that γ / β is special at the g eneric points of β R , k . Then γ / α is pre-special and the follo wing equality of cycles holds in Y k : γ R , k = γ ⊗ β ( β R , k ) . 8 Relativ e cycles 261 Proof The ﬁrst statement is obvious. W e ﬁrst prov e: γ R = γ ⊗ β β R . Remark that β R / / β is pseudo-dominant. Thus γ / β is special at the generic points of β R and the r ight hand side of the preceding equality is well deﬁned. Moreo ver , according to Lemma 8.2.1 , we can restrict to the case where α = s , β = x and γ = y , with multiplicity 1 . Let κ s , κ x , κ y be the corresponding respective residue ﬁelds, and K be the fraction ﬁeld of R . Then, according to ( 8.1.27.1 ), γ R = h κ y ⊗ κ s K i Y R and β R = h κ x ⊗ κ s K i X R . But Lemma 8.2.1 implies that γ ⊗ β β R = h κ y ⊗ κ x ( κ x ⊗ κ s K )i X R . Thus the associativity of the tensor product of ﬁelds allow s concluding. From this equality and Proposition 8.2.4 , we deduce that: γ R ⊗ β R β R , k = ( γ ⊗ β β R ) ⊗ β R β R , k = γ ⊗ β β R , k . Thus, the equality we hav e to pro v e can be written γ R ⊗ [ R k = γ R ⊗ β R ( β R ⊗ [ R k ) and w e are reduced to the case α = Sp ec ( R ) . In this case, w e can assume β = h X i with X integral. Let us consider a blow -up ˜ X p / / X such that the proper transform ˜ γ of γ along p is a Hilber t cy cle o v er ˜ X ( 8.1.18 ). W e easily get (from (P3) and 8.1.15 ) that ˜ γ k = ˜ γ ⊗ ˜ X h ˜ X k i . Let Y (resp. ˜ Y ) be the support of γ (resp. ˜ γ ), q : ˜ Y / / Y the canonical projection. W e consider the car tesian square obtained b y pullback along Sp ec ( k ) / / Sp ec ( R ) : ˜ Y k q k / /   Y k   ˜ X k p k / / X k . As X k ⊂ X (resp. Y k ⊂ Y ) is purely of codimension 1 , the proper mor phism p k (resp. q k ) is still birational. As a consequence, q k ∗ ( ˜ γ ) = γ . Let y be a point in ˜ Y ( 0 ) k ' Y ( 0 ) k which lies abo v e a point x in ˜ X ( 0 ) k ' X ( 0 ) k Then, according to (P5) and using the notations of 8.1.43 , we get m SV ( y ; ˜ γ ⊗ ˜ X h ˜ X k i ) = m SV ( y ; γ ⊗ X h X k i ) . This readily implies q k ∗ ( ˜ γ ⊗ ˜ X h ˜ X k i ) = γ ⊗ X h X k i and allow s us to conclude.  As a corollar y of this lemma using the uniqueness statement in Theorem 8.1.39 , w e obtained: Corollary 8.2.6 Let γ g / / β f / / α be pre-special mor phisms of cycles. Let x : Sp ec ( k ) / / α be a point. If β / α is special (r esp. Λ -univer sal) at x and γ / β is special (resp. Λ -univer sal) at the g eneric points of β k , then γ / α is special at x . 262 Motivic complex es and relative cycles Let α 0 / / α be any morphism of cycles with domain S 0 / / S and n be the exponential char acteristic of α 0 . Then, whenever it is w ell deﬁned, the follo wing equality of Λ [ 1 / n ] -cycles holds: γ ⊗ β ( β ⊗ α α 0 ) = γ ⊗ α α 0 . A consequence of the transitivity f or mulas is the associativity of the pullback: Corollary 8.2.7 Suppose giv en the follo wing morphisms of cycles α   β f     γ g   δ σ suc h that f and g are pr e-specials. Then, whenev er it is well deﬁned, the f ollowing equality of cycles hold: γ ⊗ σ ( β ⊗ δ α ) = ( γ ⊗ σ β ) ⊗ δ α Proof Indeed, b y the transitivity f ormulas 8.2.4 and 8.2.6 , both members of the equation are equal to ( γ ⊗ σ β ) ⊗ β ( β ⊗ δ α ) .  8.2.c Projection formulas Proposition 8.2.8 Consider morphisms of cycles with support in the left diagr am β   X   ⊂ α 0 / / α S 0 q / / S suc h that β / α is special at the g eneric points of α 0 . Consider a fact orization S 0 g / / T / / S . Then β / α is special at the g eneric points of g ∗ ( α ) and the f ollowing equality of cycles holds in X × S T : β ⊗ α g ∗ ( α 0 ) = ( 1 X × S g ) ∗ ( β ⊗ α α 0 ) . Proof The ﬁrst assuption is obvious. By linear ity , w e can assume S 0 is integral and α 0 is the generic point s of S 0 with multiplicity 1 . Let L (resp. E ) be the residue ﬁeld of s (resp. g ( s ) ). Consider the pullback square X L g 0 / / j   X E i   X × S S 0 g X / / X × S T where i and j are the natural immersions. Let d be the degree of L / E if it is ﬁnite and 0 other wise. W e are reduced to pro v e the eq uality g X ∗ ( j ∗ ( β L )) = d . i ∗ ( β E ) . Using the functoriality of pushf or ward 8 Relativ e cycles 263 and proper ty (P6), it is suﬃcient to pro v e the equality g 0 ∗ ( β E ⊗ [ E L ) = d . β E . If d = 0 , the morphism g 0 induces an inﬁnite e xtension of ﬁelds on an y point of X L which concludes. If L / E is ﬁnite, g 0 is ﬁnite ﬂat and β E ⊗ [ E L is the usual pullback b y g 0 . Then the needed equality f ollo ws easily (see [ Ful98 , 1.7.4]).  Lemma 8.2.9 Let β / / α be a pre-special morphism of cycles with domain X p / / S . Let ( R , k ) a fat point ov er α and X f / / Y / / S be a factorization of p . Let f k be the pullbac k of f ov er Sp ec ( k ) . Suppose that the support of β is proper with respect to f . Then f ∗ ( β ) is pre-special ov er α and the equality of cycles  f ∗ ( β )  R , k = f k ∗ ( β R , k ) holds in Y k . Proof As usual, consider ing the suppor t S of α , w e reduce to the case where α = h S i . Let K be the fraction ﬁeld of R . As Sp ec ( K ) maps to a generic point of S , we can assume S is integ ral. Let F be its function ﬁeld. W e can assume b y linear ity that β is a point x in X with multiplicity 1 . Let L (resp. E ) be the residue ﬁeld of x (resp. y = f ( x ) ). Let d be the degree of L / E if it is ﬁnite and 0 otherwise. Consider the follo wing pullback square Sp ec ( L ⊗ F K ) j / / f 0   X × S Sp ec ( R ) = X R f R   Sp ec ( E ⊗ F K ) i / / Y × S Sp ec ( R ) = Y R . A ccording to the formula ( 8.1.27.1 ), we obtain: f R ∗ ( β R ) = f R ∗ j ∗ ( h L ⊗ F K i ) = i ∗ f 0 ∗ ( h L ⊗ F K i ) = i ∗ f 0 ∗ ( f ∗ 0 ( h E ⊗ F K i ) = i ∗ ( d . h E ⊗ F K i ) = h f ∗ ( β )i R . W e are ﬁnally reduced to the case S = Sp ec ( R ) and β is a Hilber t cycle ov er Sp ec ( R ) . Note that f ∗ ( β ) is still a Hilber t cycle ov er Sp ec ( R ) . As β R , k = β ⊗ [ R k , the result f ollo ws now from Proposition 8.1.16 .  Corollary 8.2.10 Consider morphisms of cycles with support in the left diagr am β   X p   ⊂ α 0 / / α S 0 / / S suc h that β / α is special at the g eneric points of α 0 (r esp. Λ -univer sal). Consider a fact orization X f / / Y / / S of p . Suppose that the support of β is proper with respect to f . Then f ∗ ( β )/ α is special at the g eneric points of α 0 (r esp. Λ -univer sal) and the follo wing equality of cycles holds in X × S S 0 : ( f × S 1 S 0 ) ∗ ( β ⊗ α α 0 ) =  f ∗ ( β )  ⊗ α α 0 . 264 Motivic complex es and relativ e cycles 8.3 Geometric properties 8.3.1 W e introduce a notation which will often come in the ne xt section. Let S be a scheme and α = Í i ∈ I n i . h Z i i X an S -cycle wr itten in standard f orm. Let s be a point of S and Sp ec ( k ) ¯ s / / S be a geometric point of S with k separably closed. Let S 0 be one of the f ollo wing local schemes: the localization of S at s , the Hensel localization of S at s , the strict localization of S at ¯ s . W e then deﬁne the cy cle with coeﬃcients in Λ and domain X × S S 0 as: α | S 0 = Õ i ∈ I n i h Z i × S S 0 i X × S S 0 . Remar k 8.3.2 The canonical mor phism S 0 / / S is ﬂat. In par ticular , α / S is special at the generic points of S 0 and w e easily g et: α | S 0 = α ⊗ S S 0 . 8.3.a Constructibility Deﬁnition 8.3.3 Let S be a scheme and s ∈ S a point. W e sa y that a pre-special S -cy cle α is emphtr ivial at s if it is special at s and α ⊗ S s = 0 . Naturall y , w e sa y that α is trivial if it is zero. Thus α is trivial if and only if it is trivial at the generic points of S . Recall from [ GD67 , 1.9.6] that an ind-constructible subset of a noether ian scheme X is a union of locally closed subset of X . Lemma 8.3.4 Let S be a noetherian scheme, and α / S be a pre-special cycle. Then the set T =  s ∈ S | α / S is special (resp. trivial, Λ -univ ersal) at s  is ind-constructible in S . Proof Let s be a point of T , and Z be its closure in S with its reduced subscheme structure. Put α Z = α ⊗ S Z , deﬁned because α is special at the generic point of Z . Giv en an y point t of Z , w e kno w that α / S is special at t if and onl y if α Z / Z is special at t ( cf. 8.1.45 ). But there e xists a dense open subset U s of Z such that α Z | U Z is a Hilbert cy cle o ver U Z . Thus, α / S is special at each point of U s and U s ⊂ T . This concludes and the same argument pro v es the respective statements.  8.3.5 Let I be a left ﬁlter ing category and ( S i ) i ∈ I be a projective sys tem of noether ian schemes with aﬃne transition morphisms. W e let S be the projectiv e limit of ( S i ) and w e assume the follo wings: 1. S is noether ian. 2. There e xists an inde x i ∈ I such that f or all j ≥ i , the canonical projection S p i / / S j is dominant. 8 Relativ e cycles 265 In this case, there exis ts an index j / i such that f or an y k / j , the map p k induces an isomorphism S ( 0 ) / / S ( 0 ) k on the generic points ( cf. [ GD67 , 8.4.2, 8.4.2.1]). Thus, replacing I by I / j , w e can assume that this proper ty is satisﬁed f or all index i ∈ I . As a consequence, the f ollo wing proper ties are consequences of the previous ones: (3) For any i ∈ I , p i : S / / S i is pseudo-dominant and p i induces an isomor phism S ( 0 ) / / S ( 0 ) i . (4) For any ar row j / / i of I , p j i : S j / / S i is pseudo-dominant and p j i induces an isomor phism S ( 0 ) j / / S ( 0 ) i . Proposition 8.3.6 Consider the notations and hypothesis abov e. Assume w e ar e giv en a projectiv e syst em of cycles ( α i ) i ∈ I suc h that α i is a pr e-special cycle ov er S i and f or any j / / i , α j = α i ⊗ S i S j . Put α = α i ⊗ S i S for an index i ∈ I . 86 The f ollowing conditions ar e equivalent : (i) α / S is special (resp. Λ -univer sal). (ii) Ther e exists i ∈ I suc h that α i / S i is special (resp. Λ -univer sal). (iii) Ther e exists i ∈ I suc h that for all j / i , α j / S j is special (resp. Λ -univer sal). Let s be point of S and s i its imag e in S i . Then the follo wing conditions are equiv alent : (i) α / S is special (resp. Λ -univer sal) at s . (ii) Ther e exists i ∈ I suc h that α i / S i is special (resp. Λ -univer sal) at s i . (iii) Ther e exists i ∈ I such that for all j / i , α j / S j is special (resp. Λ -univ ersal) at s j . Proof Let P be one of the respectiv e proper ties: “special”, “tr ivial”, “ Λ -univ ersal”. Using the fact that being P at s is an ind-constr uctible proper ty (from Lemma 8.3.4 ), it is suﬃcient to apply [ GD67 , th. 8.3.2] to the follo wing famil y of sets: F i = { s i ∈ S i | α i satisﬁes P at s i } , F = { s ∈ S | α satisﬁes P at s } . T o get the tw o sets of equivalent conditions of the statement from op. cit. we hav e to pro v e the f ollo wing relations: ( 1 ) : ∀ ( j / / i ) ∈ Fl ( I ) , p − 1 j i ( F i ) ⊂ F j , ( 2 ) : F = ∪ i ∈ I p − 1 i ( F i ) . W e consider the case where P is the proper ty “special”. F or relation (1), we apply 8.1.45 which implies the strong er relation p − 1 j i ( F i ) = F j . For relation (2), another application of 8.1.45 gives in fact the strong er relation F = p − 1 i ( F i ) f or an y i ∈ I . Consider a point s j ∈ S and put s i = p j i ( s j ) . Assume α i is special at s i . Then, applying 8.2.4 and (P3), we g et: (8.3.6.1) α j ⊗ S j s j = ( α i ⊗ S i s i ) ⊗ [ κ ( s i ) κ ( s j ) . 86 The pullback is well deﬁned because of point (3) and (4) of the hypothesis abov e. 266 Motivic complex es and relative cycles Similarl y , given s ∈ S j , s i = p i ( s ) , and assuming α i is special at s i , w e get: (8.3.6.2) α ⊗ S s = ( α i ⊗ S i s i ) ⊗ [ κ ( s i ) κ ( s ) . W e consider no w the case where P is the proper ty “tr ivial”. Then relation (1) f ollow s from ( 8.3.6.1 ). Relation (2) f ollo ws from ( 8.3.6.1 ) and 8.1.19 (1). W e ﬁnally consider the case P is the proper ty “ Λ -universal”. Relation (1) in this case is again a consequence of ( 8.3.6.1 ). A ccording to ( 8.3.6.2 ), w e get the inclusion ∪ i ∈ I f − 1 i ( F i ) ⊂ F . W e ha v e to prov e the reciprocal inclusion. Consider a point s ∈ S with residue ﬁeld k such that α / S is Λ -universal at s . For an y i ∈ I , w e put s i = p i ( s ) and denote by k i its residue ﬁeld. It is suﬃcient to ﬁnd an inde x i ∈ I such that α i ⊗ S i s i has coeﬃcients in Λ . Thus we are reduced to the f ollowing lemma: Lemma 8.3.7 Let ( k i ) i ∈ I o p be an ind-ﬁeld and put: k = lim / / i ∈ I o p k i . Consider a family ( β i ) i ∈ I suc h that β i is a k i -cycle of ﬁnite type with coeﬃcients in Q and for any j / i , β j = β i ⊗ [ k i k j . W e put β = β i ⊗ [ k i k . If f or an index i ∈ I , β i ⊗ [ k i k has coeﬃcients in Λ , then ther e exists j / i such that β j has coeﬃcients in Λ .  W e can assume that for an y j / i , β j has positiv e coeﬃcients. Let X j (resp. X ) be the support of β j (resp. β ). W e obtain a pro-scheme ( X j ) j / i such that X = lim o o i ∈ I X i . The transition maps of ( X j ) j / i are dominant. Thus, b y enlarging i , we can assume that for an y j / i , the induced map π 0 ( X i ) / / π 0 ( X j ) is a bi jection. Thus we can consider each element of π 0 ( X ) separately and assume that all the X i are integrals: f or any j / i , β j = n j . h X j i f or a positiv e element n j ∈ Q . Arguing g ener ically , w e can fur ther assume X j = Spec  L j  f or a ﬁeld extension of ﬁnite type L j of k j . By assumption now , for an y j / i , L i ⊗ k i k j is an Ar tinian r ing whose reduction is the ﬁeld L j . Moreo v er , n j = n i . lg ( L i ⊗ k i k j ) and we know that n : = n i . lg ( L i ⊗ k i k ) belongs to Λ . Let p be a prime not inv er tible in Λ such that v p ( n i ) < 0 where v p denotes the p -adic valuation on Q . It is suﬃcient to ﬁnd an inde x j / i such that v p ( n j ) ≥ 0 . Let L = ( L i ⊗ k i k ) r e d . Remark that L = lim / / i ∈ I o p L i . It is a ﬁeld e xtension of ﬁnite type of k . Consider elements a 1 , . . . , a n algebraicall y independent ov er k suc h that L is a ﬁnite extension of k ( a 1 , . . . , a n ) . By enlarging i , we can assume that a 1 , . . . , a n belongs to L i . Thus L i is a ﬁnite extension of k i ( a 1 , . . . , a n ) : replacing k i b y k i ( a 1 , . . . , a n ) , w e can assume that L i / k i is ﬁnite. Let L 0 be the sube xtension of L ov er k generated b y the p -th roots of elements of k . As L / k is ﬁnite, L 0 / k is ﬁnite, generated by elements b 1 , . . . , b r ∈ L . consider an index j / i such that b 1 , . . . , b r belongs to L j . It follo ws that v p ( lg ( L i ⊗ k i k j )) = v p ( lg ( L i ⊗ k i k )) . Thus v p ( n j ) = v p ( n ) ≥ 0 and w e are done.  Corollary 8.3.8 Let S be a sc heme and α be a pr e-special S -cycle. Let ¯ s be a g eometric point of S , with imag e s in S , and S 0 be the strict localization of S at ¯ s . Then the f ollowing conditions ar e equiv alent: 8 Relativ e cycles 267 (i) α / S is special at s . (i’) α / S is special at ¯ s . (ii)  α | S 0  / S 0 is special at ¯ s (notation of 8.3.1 ). (iii) Ther e exists an étale neighborhood V of ¯ s in S such that ( α ⊗ S V )/ V is special at ¯ s . Proof The equivalence of (i) and (i’) f ollow s tr iviall y from deﬁnition ( cf. 8.1.30 ). Recall from 8.3.1 that α | S 0 = α ⊗ S S 0 . Thus ( i 0 ) ⇒ ( ii ) is easy (see 8.1.45 ). Moreov er , ( ii ) ⇒ ( ii i ) is a consequence of the previous proposition applied to the pro-scheme of étale neighborhood of ¯ s . Finall y , ( iii ) ⇒ ( i ) f ollo ws from Lemma 8.1.45 .  Proposition 8.3.9 Consider the notations and hypothesis of 8.3.5 . Assume that S and S i ar e reduced for any i ∈ I . Suppose given a projectiv e sy stem ( X i ) i ∈ I of S i -sc hemes of ﬁnite type suc h that f or any j / i , X j = X i × S i S j . W e let X be the projectiv e limit of ( X i ) . Then for any pr e-special (resp. special, Λ -univer sal) S -cycle α ⊂ X , ther e exists i ∈ I and a pre-special (resp. special, Λ -univer sal) S i -cycle α i ⊂ X i suc h that α = α i ⊗ S i S . 87 Proof Using Proposition 8.3.6 , we are reduced to consider the ﬁrst of the respective cases of the proposition. W r ite α = Í r ∈ Θ n r . h Z r i X in standard f or m. Consider r ∈ Θ . As X is noetherian, there e xists an inde x i ∈ I and a closed subscheme Z r , i ⊂ X i such that Z r = Z r , i × S i S . Moreov er , replacing Z r , i b y the reduced closure of the image of the canonical map Z r (∗) / / Z r , i , we can assume that the map (∗) is dominant. For an y j ∈ I / i , we put Z r , j = Z r , i × S i S j . The limit of the pro-scheme ( Z r , j ) j ∈ I / i o p is the integral scheme Z r . Thus, applying [ GD67 , 8.2.2], w e see that by enlarging i , w e can assume that f or any j ∈ I / i , Z r , j is ir reducible (but not necessar il y reduced). W e repeat this cons tr uction for ev er y r ∈ Θ , enlarging i at eac h step. Fix no w an element j ∈ I / i . The scheme Z r , j ma y not be reduced. How ev er, its reduction Z 0 r , j is an integral scheme such that Z 0 r , j × S j S = Z r . W e put α j = Õ r ∈ Θ n r h Z 0 r , j i X j . Let z r , j be the generic point of Z 0 r , j , and s r , j be its image in S j . It is a generic point and corresponds uniquely to a generic point s r of S according to the point (3) of the h ypothesis 8.3.5 . Thus α j / S j is pre-special. Moreo v er , w e g et from the abo v e that κ ( z r , j ) ⊗ κ ( s r , j ) κ ( s r ) = κ ( z r ) where z r is the generic point of Z r . Thus the relation α j ⊗ S j S = α follo ws from lemma 8.2.1 .  87 This pullback is deﬁned in any case because of point (3) of the hypothesis abov e. 268 Motivic complex es and relativ e cycles 8.3.b Samuel multiplicities 8.3.10 W e giv e some recall on Samuel multiplicities, f ollowing as a general reference [ Bou93 , VIII.§7]. Let A be a noether ian local ring with maximal ideal m . Let M , 0 be a A -module of ﬁnite type and q ⊂ m an ideal of A such that M / q M has ﬁnite length. Let d be the dimension of the suppor t of M . Recall from loc. cit. that Samuel multiplicity of M at q is deﬁned as the integer: e A q ( M ) : = lim n / / ∞  d ! n d lg A ( M / q n M )  In the case M = A , we simpl y put e q ( A ) : = e A q ( A ) and e ( A ) : = e A m ( A ) . W e will use the follo wing proper ties of these multiplicities that we recall for the con v enience of the reader; let A be a local noether ian ring with maximal ideal m : Let Φ be the generic points p of Sp ec ( A ) such that dim ( A / p A ) = dim A . Then according to proposition 3 of loc. cit. : ( S 1 ) e q ( A ) = Õ p ∈ Φ lg ( A p ) . e q ( A / p ) . Let B be a local ﬂat A -algebra such that B / m B has ﬁnite length o ver B . Then according to proposition 4 of loc. cit. : ( S 2 ) e m B ( B ) e ( A ) = lg B ( B / m B ) . Let B be a local ﬂat A -algebra such that m B is the maximal ideal of B . Let q ⊂ A be an ideal suc h that A / q A has ﬁnite length. Then according to the corollar y of proposition 4 in loc. cit. : ( S 3 ) e q B ( B ) = e q ( A ) . Assume A is integral with fraction ﬁeld K . Let B be a ﬁnite local A -alg ebra such that B ⊃ A . Let k B / k A be the extension of the residue ﬁelds of B / A . Then, according to proposition 5 and point b) of the corollar y of proposition 4 in loc. cit. , ( S 4 ) e m B ( B ) e ( A ) = dim K ( B ⊗ A K ) [ k B : k A ] . Deﬁnition 8.3.11 (i) Let S = Sp ec ( A ) be a local scheme, s = m the closed point of S . Let Z be an S -scheme of ﬁnite type with special ﬁber Z s . For any generic point z of Z s , denoting by B the local ring of Z at z , w e deﬁne the Samuel multiplicity of Z at z ov er S as the rational number: 8 Relativ e cycles 269 m S ( z , Z / S ) = e m B ( B ) e ( A ) . In the case where Z is integral, we deﬁne the Samuel specialization of the S -cycle h Z i at s as the cy cle with rational coeﬃcients and domain Z s : h Z i ⊗ S S s = Õ z ∈ Z ( 0 ) s m S ( z , Z / S ) . z . Consider an S -cycle of ﬁnite type α = Í i ∈ I n i . h Z i i X written in standard f or m. W e deﬁne the Samuel specialization of the S -cycle α at s as the cycle with domain X s : α ⊗ S S s = Õ i ∈ I n i . h Z i i ⊗ S S s . (ii) Let S be a scheme. For any point s of S , w e let S ( s ) be the localized scheme of S at s . Let f : Z / / S be an S -scheme of ﬁnite type, and z a point of Z whic h is generic in its ﬁber . Put s = f ( z ) . W e deﬁne the Samuel multiplicity of Z / S at z as the integer m S ( z , Z / S ) : = m S ( z , Z × S S ( s ) / S ( s ) ) . Consider an S -cy cle of ﬁnite type α with domain X and a point s of S . W e deﬁne the Samuel specialization of the S -cycle α at s as the cycle with rational coeﬃcients: α ⊗ S S s =  α | S ( s )  ⊗ S S ( s ) s . Lemma 8.3.12 Let S be a sc heme, and p : Z 0 / / Z an S -morphism whic h is a birational univer sal homeomorphism. Then for any point s ∈ S , h Z 0 i ⊗ S S s = h Z i ⊗ S S s in ( Z 0 s ) r e d = ( Z s ) r e d . Proof By hypothesis, p induces an isomorphism Z 0( 0 ) ' Z ( 0 ) betw een the generic points. Given any ir reducible component T 0 of Z 0 corresponding to the ir reducible component T of Z , we g et by hypothesis: T 0 r e d ' T r e d (as schemes), lg  O Z 0 , T 0  = lg  O Z , T  . Thus, w e easily conclude from the deﬁnition.  8.3.13 Let Z f / / S be a mor phism of ﬁnite type and a z a point of Z , s = f ( z ) . Assume z is a generic point of Z s . W e introduce the f ollo wing condition: D ( z , Z / S ) :  For any ir reducible component T of Z ( z ) , T s = ∅ or dim ( T ) = dim ( Z ( z ) ) . 270 Motivic comple xes and relative cycles Remar k 8.3.14 This condition is in par ticular satisﬁed if Z ( z ) is absolutely equidi- mensional (and a fortiori if Z is absolutely equidimensional). An immediate translation of ( S 1 ) giv es: Lemma 8.3.15 Let S be a local scheme with closed point s and Z be an S -scheme of ﬁnite type suc h that Z s is irr educible with g eneric point z . If the condition D ( z , Z / S ) is satisﬁed, then h Z i ⊗ S S s = m S ( z , Z / S ) . z . W e get directly from ( S 2 ) the f ollo wing lemma: Lemma 8.3.16 Let S be a scheme, s be a point of S , and α = Í i ∈ I n i . h Z i i X be an S -cycle in standard form such that Z i is a ﬂat S -scheme of ﬁnite type. Then α is a Hilber t S -cycle and α ⊗ S S s = α ⊗ [ S s . With the notations of 8.3.1 , we get from ( S 3 ): Lemma 8.3.17 Let S be a sc heme, s a point of S with r esidue ﬁeld k and α an S -cycle of ﬁnite type. (i) Let S 0 be the Hensel localization of S at s . Then, α ⊗ S S s =  α | S 0  ⊗ S S 0 s . (ii) Let ¯ k a separable closure corr esponding and ¯ s the corr esponding g eometric point of S . Let S ( ¯ s ) be the strict localization of S at ¯ s . Then,  α ⊗ S S s  ⊗ [ k ¯ k =  α | S ( ¯ s )  ⊗ S S ( ¯ s ) ¯ s . Let us recall from [ GD67 , 13.3.2] the f ollo wing deﬁnition: Deﬁnition 8.3.18 Let f : X / / S be a mor phism of ﬁnite type between noether ian schemes, and x a point of X . W e sa y f is equidimensional at x if there exis ts an open neighborhood U of x in X and a quasi-ﬁnite pseudo-dominant S -mor phism U / / A d S f or d ∈ N . The integer d is independent of the choice of U : it is called the relativ e dimension of f at x . W e say f is equidimensional if it is equidimensional at ev ery point of X . Remar k 8.3.19 A quasi-ﬁnite mor phism is equidimensional if and only if it is pseudo- dominant. According to [ GD67 , 12.1.1.5], this deﬁnition agrees with the conv ention stated in paragraph 8.1.9 in the case of ﬂat mor phisms. Note that a direct translation of ( S 4 ) gives: Lemma 8.3.20 Let S = Sp ec ( A ) be an integr al local sc heme with closed point s and fr action ﬁeld K . Let Z be a ﬁnite equidimensional S -sc heme and z a g eneric point of Z s . Let B be the local ring of Z at z . Then, m S ( z , Z / S ) = dim K ( B ⊗ A K ) [ κ ( z ) : κ ( s )] . 8.3.21 Recall that a scheme S is said to be unibr anch ( resp. g eometrically unibr anch) at a point s ∈ S if the henselisation (resp. str ict henselisation) of the local ring O S , s is irreducible (see [ GD67 , 6.15.1, 18.8.16]). The sc heme S is said to be unibranc h (resp. g eometrically unibranc h ) if it is so at any point s ∈ S . The f ollo wing result is the ke y point of this subsection. 8 Relativ e cycles 271 Proposition 8.3.22 Consider a cartesian squar e Z 0 g 0 / / f 0   Z f   S 0 g / / S and a point s 0 of S 0 , s = g ( s 0 ) . Let k (r esp. k 0 ) be the residue ﬁeld of s (resp. s 0 ). W e assume the f ollowing conditions: 1. S (r esp. S 0 ) is g eometrically unibranc h at s (resp. s 0 ). 2. f and f 0 ar e equidimensional of dimension n . 3. F or any g eneric point z of Z s (r esp. z 0 of Z s 0 ) the condition D ( z , Z / S ) (r esp. D ( z 0 , Z 0 / S 0 ) ) is satisﬁed. Then, the f ollowing equality holds in Z s 0 : h Z 0 i ⊗ S S 0 s 0 = ( h Z i ⊗ S S s ) ⊗ [ k k 0 . Proof A ccording to Lemma 8.3.15 , w e hav e to prov e the equality: (8.3.22.1) Õ z 0 ∈ Z ( 0 ) s 0 m S ( z 0 , Z 0 / S 0 ) . z 0 = Õ z ∈ Z ( 0 ) s m S ( z , Z / S ) . h Sp ec ( κ ( z ) ⊗ k k 0 ) i Z s 0 . As f is equidimensional of dimension n , we can assume according to 8.3.18 that there exis ts a quasi-ﬁnite pseudo-dominant S -mor phism p : Z / / A n S . For any generic point z of Z s , t = p ( z ) is the generic point of A n s . Thus applying ( S 3 ), we get: m S ( z , Z / S ) = m S ( z , Z / A n S ) . Consider the S 0 morphism p 0 : Z 0 / / A n Z 0 obtained by base chang e. It is quasi- ﬁnite. As Z 0 / S 0 is eq uidimensional of dimension n , p 0 must be pseudo-dominant. For any generic point z 0 of Z s 0 , t 0 = p 0 ( z 0 ) is the generic point of A n s 0 and as in the preceding paragraph, we g et m S ( z 0 , Z 0 / S 0 ) = m S ( z 0 , Z 0 / A n S 0 ) . Moreo v er , the residue ﬁeld κ t of t (resp. κ t 0 of t 0 ) is k ( t 1 , . . . , t n ) (resp. k 0 ( t 1 , . . . , t n ) ) and this implies Sp ec  κ ( z ) ⊗ κ t κ t 0  is homeomor phic to Sp ec ( κ ( z ) ⊗ k k 0 ) and has the same geometric multiplicities. Putting this and the tw o preceding relations in ( 8.3.22.1 ), w e get reduced to the case n = 0 – indeed, according to [ GD67 , 14.4.1.1], A n S (resp. A n S 0 ) is geometrically unibranch at t (resp. t 0 ). Assume no w n = 0 , so that f and f 0 are quasi-ﬁnite pseudo-dominant. Let ¯ k be a separable closure of k and ¯ k 0 a separable closure of a composite of ¯ k and k 0 . It is suﬃcient to pro ve relation ( 8.3.22.1 ) after extension to ¯ k 0 (Lemma 8.1.19 ). Thus according to 8.3.17 and hypothesis (3), w e can assume S and S 0 are integral strictly local schemes. 272 Motivic complex es and relative cycles For any z ∈ Z ( 0 ) s , the extension κ ( z )/ k is totally inseparable. Moreov er , z corre- sponds to a unique point z 0 ∈ Z ( 0 ) s 0 and w e hav e to prov e for any z ∈ Z ( 0 ) s : m S ( z 0 , Z 0 / S 0 ) = m S ( z , Z / S ) . lg ( κ ( z ) ⊗ k k 0 ) . Let S = Sp ec ( A ) , K = F rac ( A ) and B = O Z , z (resp. S 0 = Sp ec ( A 0 ) , K 0 = F rac ( A 0 ) and B 0 = O Z 0 , z 0 ). As B is quasi-ﬁnite dominant ov er A and A is henselian, B / A is necessarily ﬁnite dominant. The same is true f or B 0 / A 0 and ( S 4 ) giv es the f or mulas: m S ( z , Z / S ) = dim K ( B ⊗ A K ) [ κ ( z ) : k ] , m S ( z 0 , Z 0 / S 0 ) = dim K 0 ( B 0 ⊗ A 0 K 0 ) [ κ ( z 0 ) : k 0 ] . As B 0 ⊗ A 0 K 0 = ( B ⊗ A K ) ⊗ K K 0 , the numerator of these tw o rationals are the same. T o conclude, we are reduced to the easy relation [ κ ( z 0 ) : k 0 ] . lg ( κ ( z ) ⊗ k k 0 ) = [ κ ( z ) : k ] . Deﬁnition 8.3.23 Let S be a scheme and α = Í i ∈ I n i . h Z i i X be an S -cy cle in standard f or m. W e say α / S is pseudo-equidimensional ov er s if it is pre-special and for an y i ∈ I , the structural map Z i / / S is equidimensional at the g ener ic points of the ﬁber Z i , s . Proposition 8.3.24 Let S be a strictly local integ ral sc heme with closed point s and r esidue ﬁeld k and α be an S -cycle pseudo-equidimensional at s . Then for any extension Sp ec ( k 0 ) s 0 / / S of s and any fat point ( R , k 0 ) of S ov er s 0 , the f ollowing r elation holds: α R , k 0 =  α ⊗ S S s  ⊗ [ k k 0 . Proof W e put S 0 = Sp ec ( R ) and denote by s 0 its closed point. Reductions .– By additivity , we reduce to the case α = h Z i , Z is integral and the structural mor phism f : Z / / S is equidimensional at the g ener ic points of Z s . Any generic point of S 0 s 0 dominate a g ener ic point of Z s so that w e can argue locally at each generic point x of Z s . Thus we can assume Z s is ir reducible with generic point x . Moreo v er , as Z is equidimensional at x , we can assume according to 8.3.18 there e xists a quasi-ﬁnite pseudo-dominant S -mor phism (8.3.24.1) Z p / / A n S . Note that S is geometricall y unibranch at s . Thus, applying [ GD67 , 14.4.1] ("cr itère de Chev alley"), f is univ ersally open at x . As S 0 is a trait whose close point goes to s in S , it f ollow s from [ GD67 , 14.3.7] that the base chang e f 0 : Z 0 / / S 0 of f along S 0 / S is pseudo-dominant. Let T be an ir reducible component of Z 0 , with special ﬁber T s 0 and generic ﬁber T K 0 o v er S 0 . Then T / / S 0 is a dominant morphism of ﬁnite type. Thus, according 8 Relativ e cycles 273 to [ GD67 , 14.3.10], either T s 0 = ∅ or dim ( T s 0 ) = dim ( T K 0 ) . Moreov er, the dimension of T η is equal to the transcendental degree of the function ﬁeld of T ov er K 0 , which is equal to the transcendental degree of Z o v er K . This is n according to ( 8.3.24.1 ). Thus, in any case, T is equidimensional of dimension n ov er S 0 and this implies Z 0 is equidimensional of dimension n o v er S 0 . Moreov er , either T s 0 = ∅ or dim ( T ) = n + 1 = dim ( Z 0 ) . Note this implies that for any generic point z 0 of Z s 0 , the condition D ( z 0 , Z 0 / S 0 ) is satisﬁed. Middle step .– W e prov e: α R , k = h Z 0 i ⊗ S S 0 s 0 . A ccording to Lemma 8.3.16 , α R , k = h Z 0 K i ⊗ [ R k 0 = h Z 0 K i ⊗ S S 0 s 0 . But the canonical map Z 0 K / / Z 0 is a birational univ ersal homeomor phism so that w e conclude this step by Lemma 8.3.12 . F inal step .– W e hav e only to point out that the conditions of Proposition 8.3.22 are fulﬁlled f or the obvious square; this is precisely what we need.  Corollary 8.3.25 Let S be a reduced sc heme, s a point of S and α an S -cycle whic h is pseudo-equidimensional ov er s . Let ¯ s be a g eometric point of S with imag e s in S and S 0 be the strict localization of S at ¯ s . W e let S 0 = ∪ i ∈ I S 0 i be the irr educible components of S 0 and α i be the cycle made by the part of the cycle α ⊗ [ S S 0 whose points dominate S 0 i . Then the f ollowing conditions ar e equiv alent: (i) α / S is special at s . (ii) the cycle α λ ⊗ S S 0 i ¯ s does not depend on i ∈ I . Mor eov er , when these conditions are fulﬁlled, α ⊗ S ¯ s = α λ ⊗ S S 0 i ¯ s . Proof A ccording to Corollar y 8.3.8 , w e reduce to the case S = S 0 . Then this f ollo ws directly from the preceding proposition.  Corollary 8.3.26 Let S be a reduced scheme, g eometrically unibranc h at a point s ∈ S , and α an S -cycle. The f ollowing conditions ar e equivalent : (i) α / S is pseudo-equidimensional ov er s . (ii) α / S is special at s . U nder these conditions, α ⊗ S s = α ⊗ S S s . Remar k 8.3.27 In par ticular , o v er a reduced geometrically unibranch scheme S , ev er y cy cle whose suppor t is equidimensional o v er S is special. Corollary 8.3.28 Let S be a r educed scheme and s ∈ S a point such that S is g eometrically unibr anch at s and e ( O S , s ) = 1 . Then f or any S -cycle α , the follo wing conditions ar e equivalent : (i) α / S is pseudo-equidimensional ov er s . (ii) α / S is Λ -univ ersal at s . 274 Motivic comple xes and relative cycles Remar k 8.3.29 In par ticular , o v er a regular scheme S , ev ery cy cle whose suppor t is equidimensional o ver S is Λ -univ ersal. Remark also the f ollowing theorem: Theorem 8.3.30 Let S be an excellent scheme, s ∈ S a point. The follo wing condi- tions ar e equivalent : (i) S is regular at s . (ii) S is g eometrically unibranc h at s and e ( O S , s ) = 1 . (iii) S is unibranc h at s and e ( O S , s ) = 1 . Bibliogr aphical r efer ences f or the proof . W e can assume S is the spectr um of an e x cellent local r ing A with closed point s . The implication ( i ) ⇒ ( ii ) f ollow s from the f act that a nor mal local r ing is geometrically unibranch (at its closed point) and from [ Bou93 , A C.VIII.§7, prop. 2]. ( ii ) ⇒ ( iii ) is tr ivial. Concerning the implication ( ii i ) ⇒ ( i ) , let ˆ A be the completion of the local ring A . W e know from [ Bou93 , A C.VIII.108, e x. 24] that when e ( A ) = 1 and ˆ A is integ ral, A is regular . Note e ( A ) = 1 implies A is reduced. T o conclude, w e refer to [ GD67 , 7.8.3, (vii)] which established that if A is local ex cellent reduced, ˆ A is integ ral if and only if A is unibranch. Finall y , w e get the f ollo wing theorem already prov ed b y Suslin and V oev odsky ([ SV00b , 3.5.9]): Theorem 8.3.31 Let S be a scheme and s a point with residue ﬁeld κ s suc h that the local ring A of S at s is regular . Then for any equidimensional S -sc heme Z and any g eneric point z of Z s , m SV ( z , h Z i ⊗ S s ) = Õ i (− 1 ) i lg A T or A i ( O Z , z , κ s ) . Proof W e reduce to the case S = Sp ec ( A ) . Then Z is absolutely equidimensional, and w e can apply Lemma 8.3.15 tog ether with Corollar y 8.3.26 to get that m SV ( z , h Z i ⊗ S s ) = m S ( z , Z / S ) . Then the result f ollo ws from a theorem of Ser re [ Ser75 , IV .12, th. 1].  Remar k 8.3.32 Let S be a regular scheme, X a smooth S -scheme and α ⊂ X an S - cy cle whose suppor t is equidimensional o ver S . Let s be a point of S and i : X s / / X the closed immersion of the ﬁber of X at s . Then the cy cle i ∗ ( α ) of [ Ser75 , V -28, par . 7] is well-deﬁned and w e get: α ⊗ S s = i ∗ ( α ) . 9 Finite correspondences 9.0.1 In this section, S is the categor y of all noetherian schemes. W e ﬁx an admis- sible class P of mor phisms in S and assume in addition that P is contained in the class of separated mor phisms of ﬁnite type. 9 Finite cor respondences 275 Consider tw o S -schemes X and Y . T o clarify cer tain f or mulas, we will denote X × S Y simply b y X Y and let p X XY : XY / / X be the canonical projection mor phism. W e ﬁx a r ing of coeﬃcients Λ ⊂ Q . 9.1 Deﬁnition and composition 9.1.1 Let S be a base scheme. For an y P -scheme X / S , w e let c 0 ( X / S , Λ ) be the Λ -module made of the ﬁnite and Λ -univ ersal S -cycles with domain X . 88 Consider a mor phism f : Y / / X of P -schemes o v er S . Then the pushf or ward of cy cles induces a well-deﬁned mor phism: f ∗ : c 0 ( Y / S , Λ ) / / c 0 ( X / S , Λ ) . Indeed, consider a cy cle α ∈ c 0 ( Y / S ) . Let us denote by Z its support in Y and b y f ( Z ) ⊂ X imag e of the latter b y f . W e consider these subsets as reduced subschemes. Note that f ( Z ) is separated and of ﬁnite type o ver S because X / S is noetherian, separated, and of ﬁnite type, by assumption 9.0.1 . Because Z / S is proper, [ GD61 , 5.4.3(ii)] show s that f ( Z ) is indeed proper ov er S . Thus, the cy cle f ∗ ( α ) is Λ -univ ersal according to Corollar y 8.2.10 . Finally , Z / S is ﬁnite, w e deduce that f ( Z ) is quasi-ﬁnite, thus ﬁnite, o v er S . This implies the result. Deﬁnition 9.1.2 Let X and Y be tw o P -schemes o ver S . A ﬁnite S -cor respondence from X to Y with coeﬃcients in Λ is an element of c S ( X , Y ) Λ : = c 0 ( X × S Y / X ) . W e denote such a cor respondence b y the symbol X • α / / Y . In the case Λ = Z , we simply put c S ( X , Y ) : = c S ( X , Y ) Z . Through the rest of this section, unless explicitl y stated, any cycle and any ﬁnite S -cor respondence are assumed to hav e coeﬃcients in Λ . Remar k 9.1.3 1. According to proper ties (P7) and (P7’) ( cf. 8.1.41 ) of the pullback, c S ( X , Y ) Λ commutes with ﬁnite sums in X and Y . 2. Consider α ∈ c S ( X , Y ) Λ . Let Z be the support of α . Then, Z is ﬁnite pseudo- dominant ov er X (by deﬁnition 8.1.20 ). This means that Z is ﬁnite equidimen- sional o v er X . When X is regular (resp. X is reduced geometrically unibranch and char ( X ) ⊂ Λ × ), a cy cle α ⊂ X × S Y wr itten in standard f or m: α = Õ i n i h Z i i X × S Y 88 With the notations of [ SV00b ], c 0 ( X / S , Z ) = c e q u i ( X / S , 0 ) when S is reduced. 276 Motivic complex es and relativ e cy cles deﬁnes a ﬁnite S -correspondence from X to Y if and only if f or any inde x i ∈ I , the scheme Z i is ﬁnite equidimensional ov er X ( i.e. ﬁnite and dominant o ver an irreducible component of X ) – cf. 8.3.29 (resp. 8.3.27 ). Moreo v er , in each respectiv e case, c S ( X , Y ) Λ is the free Λ -module generated b y the closed integral subschemes Z of X × S Y which are ﬁnite equidimensional o v er X . 3. By deﬁnition, we get an inclusion: c S ( X , Y ) ⊂ c S ( X , Y ) Λ which induces an injectiv e map: c S ( X , Y ) ⊗ Z Λ / / c S ( X , Y ) Λ . A ccording to Corollar y 8.1.54 , this map is a bijection. Indeed, giv en any ﬁnite Λ - linear S -cor respondence α : X • / / Y , applying the mentioned corollar y , there e xists an integer N > 0 suc h that N . α is Z -universal, so in par ticular an element of c S ( X , Y ) . If we assume that N is minimal, as α is Λ -univ ersal b y assumption, N mus t be in v er tible in Λ . Theref ore, ( N . α ) ⊗ 1 N belongs to c S ( X , Y ) ⊗ Z Λ and is sent to α by the preceding map, which concludes. Giv en more generall y inclusions of r ings Λ ⊂ Λ 0 ⊂ Q , w e get an inclusion of groups c S ( X , Y ) Λ ⊂ c S ( X , Y ) Λ 0 which induces an injection: (9.1.3.1) c S ( X , Y ) Λ ⊗ Λ Λ 0 / / c S ( X , Y ) Λ 0 . Applying Proposition 8.1.53 and the same argument as abov e, w e g et that this map is in fact sur jectiv e, and theref ore a bijection. Example 9.1.4 1. Let f : X / / Y be a mor phism in P / S . Because X / S is separated (assumption 9.0.1 ), the graph Γ f of f is a closed subscheme of X × S Y . The canonical projection Γ f / / X is an isomorphism. Thus h Γ f i XY is a Hilber t cycle o v er X . In par ticular , it is Λ -univ ersal and also ﬁnite o v er X , thus it deﬁnes a ﬁnite S -cor respondence from X to Y . 2. Let f : Y / / X be a ﬁnite S -mor phism which is Λ -univ ersal (as a mor phism of the associated cy cles). Then the graph Γ f of f is closed in X × S Y and the projection Γ f / / X is isomor phic to f . Thus the cycle h Γ f i XY is a ﬁnite Λ - univ ersal cycle o v er X which theref ore deﬁne a ﬁnite S -cor respondence t f : X • / / Y called the transpose of the ﬁnite Λ -univ ersal mor phism f . Suppose w e are giv en ﬁnite S -correspondences X • α / / Y • β / / Z . Consider the f ollowing diag ram of cy cles : 9 Finite cor respondences 277 β ⊗ Y α / /   β / /   Z . α / /   Y X (9.1.4.1) The pullback cycle is w ell-deﬁned and has coeﬃcients in Λ as β is Λ -universal o v er Y . Moreo v er , according to the deﬁnition of pullback ( cf. 8.1.39 ) and Corollar y 8.2.6 , β ⊗ Y α is a ﬁnite Λ -universal cycle o ver X with domain X Y Z . N ote ﬁnally that according to 9.1.1 , the pushforw ard of this latter cy cle b y p X Z XY Z is an element of c S ( X , Z ) Λ . Deﬁnition 9.1.5 Using the preceding notations, we deﬁne the composition product of β and α as the ﬁnite S -cor respondence β ◦ α = p X Z XY Z ∗ ( β ⊗ Y α ) : X • / / Z . Remar k 9.1.6 In the case where S is regular and X , Y , Z are smooth ov er S , the composition product deﬁned abov e agree with the one deﬁned in [ Dég07 , 4.1.16] in terms of the T or -f or mula of Ser re. In fact, this is a direct consequence of 8.3.31 after reduction to the case where α and β are represented by closed integ ral subschemes (see also point (2) of remark 9.1.3 ). W e sum up the main proper ties of the composition f or ﬁnite cor respondences in the f ollo wing proposition : Proposition 9.1.7 Let X , Y , Z be P -schemes ov er S . 1. F or any ﬁnite S -corr espondences X • α / / Y • β / / Z • γ / / T , w e have ( γ ◦ β ) ◦ α = γ ◦ ( β ◦ α ) . 2. F or any X • α / / Y g / / Z , h Γ g i Y Z ◦ α = ( 1 X × S g ) ∗ ( α ) . 3. F or any X f / / Y • β / / Z , β ◦ h Γ f i XY = β ⊗ Y h X i . Mor eov er , if f is ﬂat, β ◦ h Γ f i XY = ( f × S 1 Z ) ∗ ( β ) considering the ﬂat pullbac k of cycles in the classical sense. 4. F or any X f o o Y • β / / Z suc h that f is ﬁnite Λ -univer sal, β ◦ t f = ( f × S 1 Z ) ∗ ( β ) . 5. F or any X • α / / Y g o o Z suc h that g is ﬁnite Λ -univer sal, t g ◦ α = h Z i ⊗ Y α . If w e suppose that g is ﬁnite ﬂat, then t g ◦ α = ( 1 X × S g ) ∗ ( α ) . Proof (1) Using respectivel y the projection f or mulas 8.2.10 and 8.2.8 , we obtain ( γ ◦ β ) ◦ α = p XT XY Z T ∗  ( γ ⊗ Z β ) ⊗ Y α  γ ◦ ( β ◦ α ) = p XT XY Z T ∗  γ ⊗ Z ( β ⊗ Y α )  . 278 Motivic complex es and relative cycles Thus this f ormula is a direct consequence of the associativity 8.2.7 . (2) Let  : Γ g / / Y and p X Z X Γ g : X Γ g / / X Z be the canonical projections. As  is an isomor phism, we hav e tautologically h Y i =  ∗ ( h Γ g i ) . W e conclude by the f ollowing computation : ( 1 X × S g ) ∗ ( α ) = ( 1 X × S g ) ∗ ( h Y i ⊗ Y α ) = ( 1 X × S g ) ∗ (  ∗ h Γ g i ⊗ Y α ) (∗) = ( 1 X × S g ) ∗ ( 1 X × S  ) ∗ ( h Γ g i ⊗ Y α ) = p X Z X Γ g ∗ ( h Γ g i ⊗ Y α ) (∗) = p X Z XY Z ∗ ( h Γ g i Y Z ⊗ Y α ) The equalities labeled (∗) f ollo w from the projection formula of 8.2.10 . (3) The ﬁrst asser tion f ollow s from projection f ormula of 8.2.8 and the fact that Γ f is isomor phic to X : β ◦ h Γ f i XY = p X Z XY Z ∗ ( β ⊗ Y h Γ f i XY ) = β ⊗ Y p X XY ∗ ( h Γ f i XY ) = β ⊗ Y h X i The second asser tion f ollow s from Corollar y 8.2.2 . (4) and (5): The proof of these asser tions is str ictl y similar to that of (2) and (3) instead that w e use the projection f or mula of 8.2.8 (and do not need the commutativity 8.2.3 ).  As a corollary , we obtain that the composition of S -morphisms coincide with the composition of the associated g raph considered as ﬁnite S -cor respondences. For any S -mor phism f : X / / Y , w e will still denote by f : X • / / Y the ﬁnite S -correspondence equal to h Γ f i XY . Note moreov er that f or an y P -scheme X / S , the identity mor phism of X is the neutral element for the composition of ﬁnite S -correspondences. Deﬁnition 9.1.8 W e let P c or Λ , S be the category of P -schemes ov er S with mor phisms the ﬁnite S -cor respondences and the composition product of deﬁnition 9.1.5 . An object of P c or Λ , S will be denoted by [ X ] . The categor y P c or Λ , S is additive, and the direct sum is given b y the disjoint union of P -schemes ov er S . W e ha v e a canonical faithful functor (9.1.8.1) γ : P / S / / P c or Λ , S which is the identity on objects and the graph on morphisms. W e call it the gr aph functor . 9.1.9 Giv en extension of rings Λ ⊂ Λ 0 ⊂ Q , w e get according to Remark 9.1.3 (3) and the deﬁnition of composition of ﬁnite correspondences a functor of Λ 0 -linear categories: (9.1.9.1) P c or Λ , S ⊗ Λ Λ 0 / / P c or Λ 0 , S 9 Finite cor respondences 279 which is the identity on objects and the maps of the f or m ( 9.1.3.1 ) on mor phisms. A ccording to Remark 9.1.3 (3), the later maps are bi jections and we get the f ollo wing result about changing coeﬃcients. Proposition 9.1.10 With the notations abov e, the functor ( 9.1.9.1 ) is an equiv alence of categories. 9.1.11 Giv en two S -mor phisms f : Y / / X and g : X 0 / / X such that g is ﬁnite Λ -univ ersal, we get from the previous proposition the equality of cy cles in Y X 0 : t g ◦ f = h X 0 i ⊗ X h Y i Y X where Y is seen as a closed subscheme of Y X through the graph of f . In particular, when either f or g is ﬂat, we g et (use proper ty (P3) of 8.1.35 or Corollary 8.2.2 ): t g ◦ f = h X 0 × X Y i Y X 0 . T o state the next formulas (the g eneralized degree f or mulas), w e introduce the f ollowing notion: Deﬁnition 9.1.12 Let f : X 0 / / X be a ﬁnite pseudo-dominant mor phism (recall Deﬁnition 8.1.2 ). For any generic point x of X , w e deﬁne the deg ree of f at x as the integer: deg x ( f ) = Õ x 0 / x [ κ x 0 : κ x ] where the sum r uns o v er the generic points of X 0 lying abov e x . Proposition 9.1.13 Let X be a connected S -scheme and f : X 0 / / X be a ﬁnite S -morphism. If f is special then ther e exists an integ er d ∈ N ∗ suc h that for any g eneric point x of X , deg x ( f ) = d . Mor eov er , f ◦ t f = d . 1 X . W e simply call d the degr ee of the ﬁnite special mor phism f . Proof Let ∆ 0 be the diagonal of X 0 / S . F or any generic point x of X , we let ∆ x be the diagonal of the cor responding ir reducible component of X , seen as a closed subscheme of X . According to Proposition 9.1.7 , and the deﬁnition of pushf or w ards, w e get α : = f ◦ t f = ( f × S f ) ∗ ( h ∆ 0 i X 0 X 0 ) = Õ x ∈ X ( 0 ) deg x ( f ) . h ∆ x i X X . Considering generic points x , y of X , we prov e deg x ( f ) = deg y ( f ) . By induction, w e can reduce to the case where x and y ha v e a common specialization s in X because X is connected and noether ian. Then, as α / X is special, we get by deﬁnition of the pullback (see more precisely 8.1.44 ) α ⊗ S s = deg x ( f ) . s = deg y ( f ) . s 280 Motivic complex es and relative cycles as required. The remaining asser tion then f ollo ws.  Proposition 9.1.14 Let f : X 0 / / X be an S -mor phism which is ﬁnite, radicial and Λ -univ ersal. Assume X is connected, and let d be the degr ee of f . Then t f ◦ f = d . 1 X 0 . In particular , if d is invertible in Λ , f is an isomor phism in P c or Λ , S . Proof A ccording to 9.1.11 , t f ◦ f = h X 0 i ⊗ X h X 0 i as cycles in X 0 X 0 . Let x be the generic point of X and k be its residue ﬁeld. Let { x 0 i , i ∈ I } be the set of generic points of X , and f or an y i ∈ I , k 0 i be the residue ﬁeld of x 0 i . According to 8.2.1 , we thus obtain: t f ◦ f = Õ ( i , j ) ∈ I 2 h Sp ec  k 0 i ⊗ k k 0 j  i X 0 X 0 . The result now follo ws by the deﬁnition of the deg ree and the fact that for any i ∈ I , k 0 i / k is radicial.  9.2 Monoidal structure Fix a base scheme S . Let X , X 0 , Y , Y ’ be P -schemes o ver S . Consider ﬁnite S -cor respondences α : X • / / Y and α 0 : X 0 • / / Y 0 . Then α X 0 : = α ⊗ X h X X 0 i and α 0 X : = α 0 ⊗ X 0 h X X 0 i are both ﬁnite Λ -universal cy cles ov er X X 0 . Using stability b y composition of ﬁnite Λ -universal mor phisms ( cf. Corollar y 8.2.6 ), the cy cle ( α X 0 ) ⊗ X X 0 ( α 0 X ) is ﬁnite Λ -universal o ver X X 0 . Deﬁnition 9.2.1 Using the abov e notation, we deﬁne the tensor product of α and α 0 o v er S as the ﬁnite S -cor respondence α ⊗ t r S α 0 = ( α X 0 ) ⊗ X X 0 ( α 0 X ) : X X 0 • / / Y Y 0 . Let us ﬁrst remark that this tensor product is commutativ e (use commutativity of the pullback 8.2.3 ) and associativ e (use associativity of the pullback 8.2.7 ). Moreo v er , it is compatible with composition : Lemma 9.2.2 Suppose given ﬁnite S -correspondences : α : X / / Y , β : Y / / Z , α 0 : X 0 / / Y 0 , β 0 : Y 0 / / Z 0 . Then ( β ◦ α ) ⊗ t r S ( β 0 ◦ α 0 ) = ( β ⊗ t r S β 0 ) ◦ ( α ⊗ t r S α 0 ) . Proof W e put α X 0 = α ⊗ X h X X 0 i , α 0 X = α 0 ⊗ X h X X 0 i and β Y 0 = β ⊗ Y h Y Y 0 i , β 0 Y = β 0 ⊗ Y h Y Y 0 i . W e can compute the r ight hand side of the abo ve equation as f ollow s : 9 Finite cor respondences 281 p X X 0 Z Z 0 X X 0 Y Y 0 Z Z 0 ∗  ( β Y 0 ⊗ Y Y 0 β 0 Y ) ⊗ Y Y 0 ( α X 0 ⊗ X X 0 α 0 X )  ( 1 ) = p X X 0 Z Z 0 X X 0 Y Y 0 Z Z 0 ∗  ( β Y 0 ⊗ Y Y 0 β 0 Y ) ⊗ Y Y 0 ( α 0 X ⊗ X X 0 α X 0 )  ( 2 ) = p X X 0 Z Z 0 X X 0 Y Y 0 Z Z 0 ∗  β Y 0 ⊗ Y Y 0 (( β 0 Y ⊗ Y Y 0 α 0 X ) ⊗ X X 0 α X 0 )  ( 3 ) = p X X 0 Z Z 0 X X 0 Y Y 0 Z Z 0 ∗  ( β Y 0 ⊗ Y Y 0 α X 0 ) ⊗ X X 0 ( β 0 Y ⊗ Y Y 0 α 0 X ))  . Equality (1) f ollow s from commutativity 8.2.3 , equality (2) from associativity 8.2.7 and equality (3) b y both commutativity and associativity . For the left hand side, we note that using the projection formula 8.2.10 , the left hand side is equal to p X X 0 Z Z 0 X X 0 Y Y 0 Z Z 0 ∗   ( β ⊗ Y α ) ⊗ X h X X 0 i  ⊗ X X 0  ( β 0 ⊗ Y 0 α 0 ) ⊗ X 0 h X X 0 i   . W e are left to remark that ( β ⊗ Y α ) ⊗ X h X X 0 i =  ( β Y 0 ) ⊗ Y Y 0 α  ⊗ X h X X 0 i = β Y 0 ⊗ Y Y 0 α X 0 , using transitivity 8.2.4 and associativity 8.2.7 . W e thus conclude by symmetr y of the other par t in the left hand side.  Deﬁnition 9.2.3 W e deﬁne a symmetr ic monoidal structure on the category P c or Λ , S b y putting [ X ] ⊗ t r S [ Y ] = [ X × S Y ] on objects and using the tensor product of the previous deﬁnition for mor phisms. 9.2.4 Note that the functor γ : P / S / / P c or Λ , S is monoidal for the cartesian structure on the source categor y . Indeed, this is a consequence of proper ty (P3) of the relativ e product (see 8.1.35 ) and the remark that for any mor phisms f : X / / Y and f 0 : X 0 / / Y 0 , ( Γ f × S X 0 ) × X X 0 ( Γ 0 f × S X ) = Γ f × S f 0 . 9.3 Functoriality Fix a mor phism of schemes f : T / / S . For any P -scheme X / S , we put X T = X × S T . For a pair of P -schemes o v er S (resp. T -schemes) ( X , Y ) , w e put X Y = X × S Y (resp. X Y T = X × T Y ). 9.3.a Base chang e Consider a ﬁnite S -cor respondence α : X • / / Y . The cy cle α ⊗ X h X T i deﬁnes a ﬁnite T -correspondence from X T to Y T denoted b y α T . 282 Motivic complex es and relative cycles Lemma 9.3.1 Consider ﬁnite S -correspondences X • α / / Y • β / / Y . Then ( β ◦ α ) T = β T ◦ α T . Proof This follo ws easily using the projection f or mula 8.2.10 , the associativity f or - mula 8.2.7 and the transitivity formula 8.2.4 : p X Z XY Z ∗ ( β ⊗ Y α ) ⊗ X h X T i = p X Z T XY Z T ∗  ( β ⊗ Y α ) ⊗ X h X T i  = p X Z T XY Z T ∗  β ⊗ Y ( α ⊗ X h X T i )  = p X Z T XY Z T ∗  ( β ⊗ Y h Y T i ) ⊗ Y T ( α ⊗ X h X T i )  . Deﬁnition 9.3.2 Let f : T / / S be a mor phism of sc hemes. Using the preceding lemma, w e deﬁne the base chang e functor f ∗ : P c or Λ , S / / P c or Λ , T [ X / S ]  / / [ X T / T ] c S ( X , Y ) Λ 3 α  / / α T . W e sum up the basic properties of the base change for cor respondences in the f ollowing lemma. Lemma 9.3.3 T ake the notation and hypothesis of the previous deﬁnition. 1. The functor f ∗ is symmetric monoidal. 2. Let f ∗ 0 : P / S / / P / T be the classical base chang e functor on P -schemes ov er S . Then the follo wing diagr am is commutative: P / S γ S / / f ∗ 0   P c or Λ , S f ∗   P / T γ T / / P c or Λ , T . 3. Let σ : T 0 / / T be a mor phism of schemes. Through the canonical isomor - phisms ( X T ) T 0 ' X T 0 , equality ( f ◦ σ ) ∗ = σ ∗ ◦ f ∗ holds. Proof (1) This point f ollow s easily using the associativity f or mula 8.2.7 and the transitivity f ormulas 8.2.4 , 8.2.6 . (2) This point f ollo ws from the fact that for an y S -mor phism f : X / / Y , there is a canonical isomor phism Γ f T / / Γ f × S T . (3) This point is a direct application of the transitivity 8.2.4 .  Lemma 9.3.4 Let f : T / / S be a universal homeomor phism. Then f ∗ : P c or Λ , S / / P c or Λ , T is fully fait hful. Proof Let X and Y be P -schemes ov er S . Then X T / / X is a univ ersal homeomor - phism. An y g ener ic point x of X cor responds uniquel y to a g eneric point of X T . Let m x (resp. m 0 x ) be the geometric multiplicity of x in X (resp. X T ). Consider a ﬁnite S -correspondence α = Í i ∈ I n i . z i . For each i ∈ I , let x i be the g ener ic point of X dominated b y z i . Then we get b y deﬁnition: 9 Finite cor respondences 283 f ∗ ( α ) = Õ i ∈ I m 0 x i n i m x i . z i and the lemma is clear .  9.3.b Restriction Consider a P -mor phism p : T / / S . For any pair of T -schemes ( X , Y ) , w e denote b y δ XY : X × T Y / / X × S Y the canonical closed immersion deduced by base chang e from the diagonal immersion of T / S . Consider a ﬁnite T -cor respondence α : X • / / Y . The cycle δ XY ∗ ( α ) is the cycle α considered as a cycle in X × S Y . It deﬁnes a ﬁnite S -cor respondence from X to Y . Lemma 9.3.5 Let X , Y and Z be T -schemes. The f ollowing r elations are true : 1. F or any T -mor phism f : X / / Y , δ XY ∗  h Γ f i XY T  = h Γ f i XY . 2. F or all α ∈ c T ( X , Y ) Λ and β ∈ c T ( Y , Z ) Λ , δ X Z ∗ ( β ◦ α ) = ( δ Y Z ∗ ( β )) ◦ ( δ XY ∗ ( α )) . Proof The ﬁrst assertion is obvious. The second asser tion is a consequence of the projection f or mulas 8.2.8 and 8.2.10 , and the functor iality of pushf or wards : ( δ Y Z ∗ ( β )) ◦ ( δ XY ∗ ( α )) = p X Z XY Z ∗  δ Y Z ∗ ( β ) ⊗ Y δ XY ∗ ( α )  = p X Z XY Z ∗ δ XY Z ∗ ( β ⊗ Y α ) = δ X Z ∗ p X Z T XY Z T ∗ ( β ⊗ Y α ) . Deﬁnition 9.3.6 Let p : T / / S be a P -mor phism. Using the preceding lemma, we deﬁne a functor p ] : P c or Λ , T / / P c or Λ , S [ X / / T ]  / / [ X / / T p / / S ] c T ( X , Y ) Λ 3 α  / / δ XY ∗ ( α ) . This functor enjo ys the follo wing proper ties: Lemma 9.3.7 Let p : T / / S be a P -morphism. 1. The functor p ] is left adjoint to the functor p ∗ . 2. F or any composable P -morphisms Z q / / T p / / S , ( pq ) ] = p ] q ] . 3. Let p 0 ] : P / T / / P / S be the functor induced by composition with p . Then the f ollowing diagr am is commutative: P / T γ T / / p 0 ]   P c or Λ , T p ]   P / S γ S / / P c or Λ , S . 284 Motivic comple xes and relative cycles Proof For point (1), we ha v e to construct f or schemes X / T and Y / S a natural iso- morphism c S  p ] X , Y  Λ ' c T ( X , p ∗ Y ) Λ . It is induced b y the canonical isomor phism of schemes ( p ] X ) × S Y ' X × T ( p ∗ Y ) . Point (2) follo ws from the associativity of the pushf or ward functor on cy cles. Note also that this identiﬁcation is compatible with the transposition of the identiﬁcation of 9.3.3 (3) according to the adjunction proper ty just obtained. Point (3) is a reformulation of 9.3.5 (2).  9.3.c A ﬁniteness property 9.3.8 W e assume here that P is the class of all separated mor phisms of ﬁnite type in S . Let I be a left ﬁlter ing categor y and ( X i ) i ∈ I be a projectiv e system of separated S -schemes of ﬁnite type with aﬃne dominant transition morphisms. W e let X be the projectiv e limit of ( X i ) i and assume that X is Noetherian o v er S . Proposition 9.3.9 Let Y be a P -scheme of ﬁnite type ov er S . Then the canonical morphism ϕ : lim / / i ∈ I o p c S ( X i , Y ) Λ / / c 0 ( X × S Y / X , Λ ) . is an isomorphism. Proof Note that according to [ A GV73 , IV , 8.3.8(i)], w e can assume the conditions (2) of 8.3.5 is veriﬁed f or ( X i ) i ∈ I . Thus conditions (1) to (4) of loc. cit. are veriﬁed. Then the surjectivity of ϕ follo ws from 8.3.9 and the injectivity from 8.3.6 .  9.4 The ﬁbred category of correspondences W e can summarize the preceding constructions: Proposition 9.4.1 The 2 -functor P c or Λ : S  / / P c or Λ , S equipped with the pullback deﬁned in 9.3.2 and with the tensor product of 9.2.3 is a monoidal P -ﬁbred categor y such that the functor γ : P / / P c or Λ (see ( 9.1.8.1 ) ) is a mor phism of monoidal P -ﬁbred category. Proof A ccording to Lemma 9.3.7 , f or any P mor phisms p , p ∗ admits a left adjoint p ] . W e hav e check ed that γ is symmetr ic monoidal and commutes with f ∗ and p ] (see respectiv ely 9.2.4 , 9.3.3 and 9.3.7 ). But γ is essentially surjective. Thus, to prov e 9 Finite cor respondences 285 the proper ties ( P -BC) and ( P -PF) f or the ﬁbred categor y P c or Λ , we are reduced to the case of P which is easy (see e xample 1.1.28 ). This concludes.  Remar k 9.4.2 Consider the deﬁnition abov e. 1. The categor y P c or Λ is Λ -linear . For any scheme S , P c or Λ , S is additive. For an y ﬁnite famil y of schemes ( S i ) i ∈ I which admits a sum S in S , the canonical map P c or Λ , S / / Ê i ∈ I P c or Λ , S i is an isomor phism. 2. The functor γ : P / / P c or Λ is nothing else than the canonical geometric sections of P c or Λ (see deﬁnition 1.1.35 ). W e will apply these deﬁnitions in the par ticular cases P = Sm (resp. P = S f t ) the class of smooth separated (resp. separated) mor phisms of ﬁnite type. Note that w e get a commutativ e square Sm γ / /   S m c or Λ   S f t γ / / S f t , c or Λ where the vertical maps are the obvious embeddings of monoidal Sm -ﬁbred cate- gories. 9.4.3 Consider extensions of r ings Λ ⊂ Λ 0 ⊂ Q . The functors ( 9.1.9.1 ) f or various schemes S in S are compatible with the operations of a P -ﬁbred category be- cause it is just concerned with adding denominators in the coeﬃcients of the ﬁnite correspondences considered. Thus the y induce a morphism of monoidal P -ﬁbred categories ov er S : (9.4.3.1) P c or Λ ⊗ Λ Λ 0 / / P c or Λ 0 . A ccording to Proposition 9.1.10 , we immediatel y get the f ollowing result: Proposition 9.4.4 Then the mor phism ( 9.4.3.1 ) is an eq uivalence of monoidal P - ﬁbr ed categories. Remar k 9.4.5 The restriction of the category P c or Z to the categor y of regular schemes was already deﬁned in [ Dég07 ]. Indeed, one can check using the compar ison of Suslin- V oev odsky’ s multiplicities with Ser re’ s intersection multiplicities (using T or - f or mulas ; cf. 8.3.31 ), that the operations τ ∗ , τ ] , and ⊗ t r deﬁned here coincide with that of [ Dég07 ]. This remark extends 9.1.6 . 286 Motivic complex es and relativ e cycles 10 Shea v es with transf ers 10.0.1 The categor y S is the categor y of noether ian schemes of ﬁnite dimension. W e ﬁx an admissible class P of morphisms in S satisfying the f ollo wing assumptions: (a) An y mor phism in P is separated of ﬁnite type. (b) An y étale separated mor phism of ﬁnite type is in P . W e ﬁx a topology t on S which is P -admissible and such that: (c) For any scheme S , there is a class of cov ers E of the form ( p : W / / S ) with p a P -mor phism such that t is the topology generated b y E and the cov ers of the f or m ( U / / U t V , V / / U t V ) for any schemes U and V in S . W e ﬁx a r ing of coeﬃcients Λ . Whenev er we speak of Λ -cycles (or the premotivic category P c or Λ , etc...), we mean cycles with coeﬃcients in the localization of Z with respect to inv er tible primes in Λ . Note that in sections 10.4 and 10.5 , w e will apply the conv entions of section 1.4 b y replacing the class of smooth morphisms of ﬁnite type (resp. morphisms of ﬁnite type) there by the class of smooth separated mor phisms of ﬁnite type (resp. separated morphisms of ﬁnite type). 10.1 Preshea v es with transfers W e consider the additiv e category P c or Λ , S of deﬁnition 9.1.8 and the graph functor γ : P / S / / P c or Λ , S of ( 9.1.8.1 ). Deﬁnition 10.1.1 A pr esheaf with transf ers F ov er S is an additive presheaf of Λ -modules o v er P c or Λ , S . W e denote by PSh  P c or Λ , S  the cor responding category . If X is a P -scheme ov er S , we denote b y Λ t r S ( X ) the presheaf with transfers represented b y X . W e denote b y ˆ γ ∗ the functor (10.1.1.1) PSh  P c or Λ , S  / / PSh ( P / S , Λ ) , F  / / F ◦ γ . Note that PSh  P c or Λ , S  is obviousl y a Grothendieck abelian categor y generated b y the objects Λ t r S ( X ) f or a P -scheme X / S . Moreov er , the follo wing proposition is straightf or w ard: Proposition 10.1.2 There is an essentially unique Gro thendiec k abelian P -premo tivic category PSh  P c or Λ  whic h is g eometrically g enerated (cf. 1.1.41 ), whose ﬁber ov er a sc heme S is PSh  P c or Λ , S  and suc h that the functor Λ t r S induces a mor phism of additiv e monoidal P -ﬁbr ed categories. 10 Sheav es with transf ers 287 (10.1.2.1) P c or Λ / / PSh  P c or Λ  . Mor eov er , the functor ( 10.1.1.1 ) induces a morphism of abelian P -premo tivic cate- gories ˆ γ ∗ : PSh ( P , Λ ) / / o o PSh  P c or Λ  : ˆ γ ∗ . Proof T o help the reader , we recall the f ollo wing consequence of Y oneda’ s lemma: Lemma 10.1.3 Let F : ( P c or Λ , S ) o p / / Λ - mo d be a presheaf with transf ers. Let I be the category of repr esentable presheav es with transf ers ov er F . Then the canonical map lim / / Λ t r S ( X ) / / F Λ t r S ( X ) / / F is an isomorphism. The limit is taken in PSh  P c or Λ , S  and runs ov er I .  This lemma allo ws us to deﬁne the str uctural left adjoint of PSh  P c or Λ  (recall f ∗ , p ] f or p a P -mor phism and the tensor product) because they are indeed determined b y ( 10.1.2.1 ). The exis tence of the structural right adjoints is f ormal. The same lemma allow s to get the adjunction ( ˆ γ ∗ , ˆ γ ∗ ) .  Remar k 10.1.4 Note that for any presheaf with transf ers F ov er S , and an y mor phism f : T / / S (resp. P -mor phism p : S / / S 0 ), we get as usual f ∗ F = F ◦ f ∗ (resp. p ∗ F = F ◦ p ] ) where the functor f ∗ (resp. p ] ) on the r ight hand side is taken with respect to the P -ﬁbred category P c or Λ . Let us state the f ollowing lemma f or future use. Lemma 10.1.5 Let Let ( S α ) α ∈ A be a pr ojective syst em of sc hemes in S , with domi- nant aﬃne transition maps, and suc h that S = lim o o α ∈ A S α is r epresentable in S . Consider an index α 0 ∈ A and a presheaf with tr ansfers F ov er S α 0 . F or any index α / α 0 , we denote by F α (r esp. F ) the pullbac k of F α 0 ov er S α (r esp. S ) in the sense of the pr emotivic structur e on PSh  P c or Λ  . Then the canonical map: lim / / α ∈ A / α 0 F α ( S α ) / / F ( S ) is an isomorphism. Proof The presheaf F α 0 can be written as an inductive limit of representable sheav es of the form Λ t r S α 0 ( X α 0 ) of a P -scheme X α 0 / S α 0 . As the global section functor on preshea v es with transfers commute with inductiv e limit, w e are reduced to the case where F = Λ t r S α 0 ( X α 0 ) . In this case, the lemma f ollo ws directly from Proposition 9.3.9 .  288 Motivic complex es and relative cycles 10.2 Shea ves with transfers Deﬁnition 10.2.1 A t -sheaf with transf ers ov er S is a presheaf with transf ers F such that the functor F ◦ γ S is a t -sheaf. W e denote b y Sh t  P c or Λ , S  the full subcategor y of PSh ( P c or Λ , S , Λ ) of sheav es with transfers. A ccording to this deﬁnition, we g et a canonical faithful functor γ ∗ : Sh t  P c or Λ , S  / / Sh t ( P / S , Λ ) , F  / / F ◦ γ . Example 10.2.2 A par ticularl y impor tant case f or us is the case when t = Nis is the Nisnevic h topology . According to the original deﬁnition of V oev odsky , a Nisnevich sheaf with transf ers will be called simply a sheaf with transf ers . Remar k 10.2.3 Later on, in the case P = S f t , we will use the notation Λ tr S ( X ) to denote the presheaf on the big site S f t , c or Λ , S represented by a separated S -scheme of ﬁnite type. Proposition 10.2.4 Let X be an P -scheme o ver S . 1. The pr esheaf Λ t r S ( X ) is an étale sheaf with tr ansfer s. 2. If char ( X ) ⊂ Λ × , Λ t r S ( X ) is a qfh -sheaf with transf ers. Proof For point (1), w e f ollo w the proof of [ Dég07 , 4.2.4]: the computation of the pullback by an étale map is giv en in our context b y point (3) of Proposition 9.1.7 . Moreo v er , the proper ty for a cycle α / Y to be Λ -universal is étale-local on Y according to 8.3.8 . For point (2), w e refer to [ SV00b , 4.2.7].  W e can actuall y descr ibe e xplicitly representable presheav es with transfers in the f ollowing case: Proposition 10.2.5 Let S be a scheme and X be a ﬁnite étale S -sc heme. Then for any P -scheme Y ov er S , Γ ( Y , Λ t r S ( X )) = π 0 ( Y × S X ) . Λ . This readily follo ws from the follo wing lemma: Lemma 10.2.6 Let f : X / / S be an étale separat ed morphism of ﬁnite type. Let π f i ni t e 0 ( X / S ) be the set of connected components V of X suc h that f ( V ) is equal to a connected component of S ( i.e. f is ﬁnite ov er V ). Then c 0 ( X / S , Λ ) = π f i ni t e 0 ( X / S ) . Λ . Proof W e can assume that S is reduced and connected. W e ﬁrst treat the case where X = S . Consider a ﬁnite Λ -univ ersal S -cycle α with domain S . W r ite α = Í i ∈ I n i . h Z i i S in standard f or m. By deﬁnition, Z i dominates an irreducible component of S thus Z i is equal to that ir reducible component. Consider S 0 an ir reducible component of S and an index i ∈ I such that S 0 ∩ Z i is not 10 Sheav es with transf ers 289 empty . Consider a point s ∈ S 0 ∩ Z i . W e hav e obviousl y α s = n i . h Sp ec ( κ ( s ) ) i , 0 . Thus there exis ts a component of α which dominates S 0 i.e. ∃ j ∈ I such that Z j = S 0 . Moreo v er , computing α s using alternativel y Z i and Z j giv es n i = n j . As S is noether ian, we see inductiv ely { Z i | i ∈ I } is the set of ir reducible components of S and f or any i , j ∈ I , n i = n j . Thus c 0 ( S / S , Λ ) = Z . Consider no w the case of an étale S -scheme X . By additivity of c 0 , we can assume that X is connected. Consider the f ollo wing canonical map: c 0 ( X / S , Λ ) / / c 0 ( X × S X / X , Λ ) , α  / / α ⊗ [ S X . Note that considering the projection p : X × S X / / X , by deﬁnition, α ⊗ [ S X = p ∗ ( α ) . Consider the diagonal δ : X / / X × S X of X / S . Because X / S is étale and separated, δ is a direct factor of X × S X and w e can write X × S X = X t U . Because c 0 is additiv e, c 0 ( X × S X / X , Λ ) = c 0 ( X / X , Λ ) ⊕ c 0 ( U / X , Λ ) . Moreo v er , the projection on the ﬁrst f actor is induced by the map δ ∗ on cycles. Because δ ∗ p ∗ = 1 , w e deduce that a cy ck e in c 0 ( X / S , Λ ) corresponds uniquel y to a cycle in c 0 ( X / X , Λ ) . According to the preceding case, this latter group is the free group generated by the cy cle h X i . This latter cy cle is Λ -universal o v er S , because X / S is ﬂat. Thus, if X / S is ﬁnite, it is an element of c 0 ( X / S , Λ ) so that c 0 ( X / S , Λ ) = Λ . Otherwise, not any of the Λ -linear combination of h X i belongs to c 0 ( X / S , Λ ) so that c 0 ( X / S , Λ ) = 0 .  10.3 Associated sheaf with transfers 10.3.1 Recall from 3.2.1 that we denote b y ( P / S ) q the category of I -diag rams of objects in P / S index ed by a discrete category I . Giv en any simplicial object X of ( P / S ) q , we will consider the comple x Λ t r S ( X ) of PSh  P c or Λ , S  applying the deﬁnition of 5.1.8 to the Grothendieck P -ﬁbred categor y PSh ( P ) . Consider a t -co v er p : W / / X in P / X . W e denote b y W n X the n -f old product of W o ver X (in the categor y P / X ). W e denote b y ˇ S ( W / X ) the Čech simplicial object of P c or Λ , S such that ˇ S n ( W / X ) = W n + 1 X . The canonical mor phism ˇ S ( W / X ) / / X is a t -hyperco v er according to deﬁnition 3.2.1 . W e will call these par ticular type of t -hyperco vers the Čec h t -hyper cov ers of X . Deﬁnition 10.3.2 W e will sa y that the admissible topology t on P is compatible with tr ansfer s (resp. weakly compatible with transf ers ) if for any scheme S and any t -hyperco ver (resp. any Čech t -hyperco v er) X / / X in the site P / S , the canonical morphism of comple xes (10.3.2.1) Λ t r S ( X ) / / Λ t r S ( X ) induces a quasi-isomorphism of the associated t -shea v es on P / S . 290 Motivic comple xes and relative cycles Obviousl y , if t is compatible with transfers then it is w eakly compatible with transf ers. Recall from 10.2.4 that, in the cases t = Nis , ´ e t , ( 10.3.2.1 ) is actuall y a mor phism of complex es of t -sheav es with transf ers. The f ollowing proposition is a generaliza- tion of [ V oe96 , 3.1.3] but its proof is in fact the same. Proposition 10.3.3 The Nisnevich (r esp étale) topology t on P is w eakly compatible with tr ansfer s. Proof W e consider a t -cov er p : W / / X , the associated Čech hyperco ver X = ˇ S ( W / X ) of X and we prov e that the map ( 10.3.2.1 ) is a quasi-isomorphism of t - shea v es. Recall that a point of P / S for the topology t is giv en by an essentially aﬃne pro-object ( V i ) i ∈ I of P / S . Moreov er , its projective limit V in the categor y of schemes is in par ticular a local henselian noether ian scheme. It will be suﬃcient to chec k that the ﬁber of ( 10.3.2.1 ) at the point ( V i ) i ∈ I is a quasi- isomorphism. Thus, according to Proposition 9.3.9 , we can assume that S = V is a local henselian scheme and we are to reduce to pro v e that the comple x of abelian groups . . . / / c 0 ( W × X W / S , Λ ) / / c 0 ( W / S , Λ ) p ∗ / / c 0 ( X / S , Λ ) / / 0 is acy clic. W e denote this complex by C . Recall that the abelian g roup c 0 ( X / S ) is cov ariantly functor ial in X with respect to separated mor phisms of ﬁnite type f : X 0 / / X ( cf. paragraph 9.1.1 ). Moreov er , if f is an immersion, f ∗ is obviousl y injective. Let F 0 be the set of closed subschemes Z of X such that Z / S is ﬁnite. Given a closed subscheme Z in F 0 , w e let C Z be the comple x of abelian g roups (10.3.3.1) . . . / / c 0 ( W Z × Z W Z / S , Λ ) / / c 0 ( W Z / S , Λ ) p Z ∗ / / c 0 ( Z / S , Λ ) / / 0 where p Z is the t -co v er obtained by pullback along Z / / X . From what we hav e just recalled, we can identify C Z with a subcomplex of C . The set F 0 can be ordered b y inclusion, and C is the union of its subcomple xes C Z . If F 0 is empty , then C = 0 and the proposition is clear . Other wise, F 0 is ﬁltered and we can wr ite: C = lim / / Z ∈ F 0 C Z . Thus, it will be suﬃcient to prov e that C Z is acyclic f or any Z ∈ F 0 . Because S is henselian and Z is ﬁnite o v er S , Z is indeed a ﬁnite sum of local henselian schemes. This implies that the t -co v er p Z , which is in par ticular étale sur jectiv e, admits a splitting s : Z / / W Z . Then the comple x ( 10.3.3.1 ) is contractible with contracting homotop y deﬁned by the famil y of mor phisms ( s × Z 1 W n Z ) ∗ : c 0 ( W n Z / S , Λ ) / / c 0 ( W n + 1 Z / S , Λ ) . 10.3.4 Considering an additiv e abelian presheaf G on P / S , the natural transforma- tion 10 Sheav es with transf ers 291 X  / / Hom PSh ( P / S ) ( ˆ γ ∗ Λ t r S ( X ) , G ) deﬁnes a presheaf with transf ers o v er S . 89 W e will denote b y G τ its restriction to the site P / S . Note that this deﬁnition can be applied in the case where G is a t -sheaf on P / S , because under the assumption 10.0.1 on t , it is in par ticular an additive presheaf. Deﬁnition 10.3.5 W e will sa y that t is mildly compatible with transfers if f or an y scheme S and an y t -sheaf F on P / S , F τ is a t -sheaf on P / S . If t is weakl y compatible with transf ers then is it mildly compatible with transfers. Remar k 10.3.6 Assume t is mildly compatible with transfers. Then for any scheme S , an y t -co v er p : W / / X in P / S induces a mor phism p ∗ : Λ t r S ( W ) / / Λ t r S ( X ) which is an epimor phism of the associated t -shea v es on P / S . This means that f or an y correspondence α ∈ c S ( Y , X ) , there exis ts a t -co v er q : W 0 / / Y and a correspondence α 0 ∈ c S ( W 0 , W ) making the f ollo wing diagram commutative: W 0 ˆ α / / • / / q   W p   Y α / / • / / X (10.3.6.1) Lemma 10.3.7 Assume t is mildly compatible with tr ansfer s. Let S be a scheme and P t r be a presheaf with transf ers ov er S . W e put P = P t r ◦ γ as a presheaf on P / S . W e denote by F the t -sheaf associated with P and by η : P / / F the canonical natur al transf or mation. Then ther e exists a unique pair ( F t r , η t r ) suc h that : 1. F t r is a sheaf with transf ers ov er S suc h that F t r ◦ γ = F . 2. η t r : P t r / / F t r is a natural transf ormation of pr esheav es with transf ers such that the induced transf ormation P = ( P t r ◦ γ ) / / ( F t r ◦ γ ) = F coincides with η . Proof As a preliminary obser vation, we note that giv en a presheaf G on P / S , the data of a presheaf with transfers G t r such that G t r ◦ γ = G is equivalent to the data f or an y P -schemes X and Y of a bilinear product (10.3.7.1) G ( X ) ⊗ Z c S ( Y , X ) / / G ( Y ) , ρ ⊗ α  / / h ρ , α i such that: 89 Actually , this deﬁnes a r ight adjoint to the functor ˆ γ ∗ . 292 Motivic complex es and relative cycles (a) For any mor phism f : Y 0 / / Y in P / S , f ∗ h ρ , α i = h ρ , α ◦ f i . (b) For any mor phism f : X / / X 0 in P / S , if ρ = f ∗ ( ρ 0 ) , h ρ , α i = h ρ 0 , f ◦ α i . (c) When X = Y , f or any ρ ∈ F ( X ) , h ρ , 1 X i = ρ . (d) For any ﬁnite S -cor respondence β ∈ c S ( Z , Y ) , h h ρ , α i , β i = h ρ , α ◦ β i . On the other hand, the data of products of the form ( 10.3.7.1 ) f or an y P -schemes X and Y o v er S which satisfy the conditions (a) and (b) abov e is equivalent to the data of a natural transf or mation φ : G / / G τ b y putting h ρ , α i φ = [ φ X ( ρ )] Y .α . Applying this to the presheaf with transf ers P t r , w e obtain a canonical natural transf or mation ψ : P / / P τ . By assumption on t , F τ is a t -sheaf. Thus, there e xists a unique natural transf ormation ψ such that the f ollo wing diagram commutes: P ψ / / η   P τ η τ   F φ / / F τ Thus we get products of the form 10.3.7.1 associated with φ which satisﬁes (a) and (b). The commutativity of the abo ve diag ram asser ts they are compatible with the ones cor responding to P t r and the unicity of the natural transf ormation φ implies the uniqueness statement of the lemma. T o ﬁnish the proof of the e xistence, we must show (c) and (d) for the product h ., i φ . Consider a couple ( ρ , α ) ∈ F ( X ) × c S ( Y , X ) . Because F is the t -sheaf associated with P , there exis ts a t -co ver p : W / / X and a section ˆ ρ ∈ P ( W ) such that p ∗ ( ρ ) = η W ( ˆ ρ ) . Moreo v er , according to Remark 10.3.6 , we g et a t -co v er q : W 0 / / Y and a cor respondence ˆ α ∈ c S ( W 0 , W ) making the diag ram ( 10.3.6.1 ) commutative. Then w e get using (a) and (b): q ∗ h ρ , α i φ = h ρ , α ◦ q i φ = h ρ , p ◦ ˆ α i φ = h p ∗ ρ , ˆ α i φ = h η W ( ˆ ρ ) , ˆ α i φ = h ˆ ρ , ˆ α i ψ . Because q ∗ : F ( X ) / / F ( W ) is injective, we deduce easily from this pr inciple the properties (c) and (d) and this concludes.  10.3.8 Let us consider the canonical adjunction a ∗ t : PSh ( P / S , Λ ) / / o o Sh t ( P / S , Λ ) : O t where O t is the canonical f org etful functor . W e also denote by O t r t : Sh t  P c or Λ , S  / / PSh  P c or Λ , S  the obvious for getful functor . T rivially , the f ollo wing relation holds: (10.3.8.1) ˆ γ ∗ a t , ∗ = a t , ∗ γ ∗ . 10 Sheav es with transf ers 293 Proposition 10.3.9 Using the notations abo ve, the f ollowing condition on the ad- missible topology t ar e equivalent : (i) t is mildly compatible with tr ansfer s. (ii) F or any scheme S , the functor O t r t admits a lef t adjoint a ∗ t : PSh  P c or Λ , S  / / Sh t  P c or Λ , S  whic h is exact and such that the exchang e transf or mation (10.3.9.1) a ∗ t ˆ γ ∗ / / γ ∗ a ∗ t induced by the identiﬁcation ( 10.3.8.1 ) is an isomor phism. U nder these conditions, the f ollowing pr operties hold f or any scheme S : (iii) The category Sh t  P c or Λ , S  is a Gro thendiec k abelian category. (iv) The functor γ ∗ commutes with all limits and colimits. Proof The f act (i) implies (ii) f ollow s from the preceding lemma as w e can put a t r t ( F ) = F t r with the notation of the lemma. The fact this deﬁnes a functor , as well as the proper ties stated in (ii), f ollo ws from the uniqueness statement of loc. cit. Let us assume (ii). Then (iii) follo ws f or mally from (ii), from the e xistence, adjunction proper ty and ex actness of a ∗ t , as PSh  P c or Λ , S  is a Grothendieck abelian category . Moreov er , w e deduce from the isomor phism ( 10.3.9.1 ) that γ ∗ is e xact: indeed, a ∗ t and ˆ γ ∗ are exact. As γ ∗ commutes with arbitrar y direct sums, w e g et (iv). From this point, w e deduce the e xistence of a r ight adjoint γ ! to the functor γ ∗ . Using again the isomorphism ( 10.3.9.1 ), we obtain f or an y t -sheav es F on P / S and an y P -scheme X / S a canonical isomor phism F τ ( X ) = γ ! F ( X ) . This pro v es (i).  10.3.10 U nder the assumption of the previous proposition, giv en an y P -scheme X / S , w e will put Λ t r S ( X ) t = a ∗ t Λ t r S ( X ) . The abo v e proposition show s that the famil y Λ t r S ( X ) t f or P -schemes X / S is a generating f amily in Sh t  P c or Λ , S  . Moreov er , we get easily the follo wing corollar y of the preceding proposition and Proposition 10.1.2 : Corollary 10.3.11 Assume that t is mildly compatible with tr ansfer s. Then there exists an essentially unique Grot hendiec k abelian P -pr emotivic cat- egor y Sh t  P c or Λ  whic h is g eometrically g enerat ed, whose ﬁber o v er a sc heme S is Sh t  P c or Λ , S  and suc h that the t -sheaﬁﬁcation functor induces an adjunction of abelian P -premo tivic categories: a ∗ t : PSh  P c or Λ  / / o o Sh t  P c or Λ  : O t r t . Mor eov er , the functor γ ∗ induces an adjunction of abelian P -pr emotivic categories: (10.3.11.1) γ ∗ : Sh t ( P , Λ ) / / o o Sh t  P c or Λ  : γ ∗ . 294 Motivic comple xes and relative cycles Remar k 10.3.12 Notice moreo ver that γ ∗ a ∗ t = a ∗ t ˆ γ ∗ . Proof In fact, using the e xactness of a ∗ t , giv en any sheaf F with transfers F ov er S , w e get a canonical isomor phism F = lim / / Λ t r S ( X ) t / / F Λ t r S ( X ) t where the limit is taken in Sh t  P c or Λ , S  and r uns ov er the representable t -sheav es with transf ers ov er F . As in the proof of 10.1.2 , this allo ws deﬁning uniquely the structural left adjoints of Sh t  P c or Λ  . The e xistence (and uniqueness) of the structural right adjoints then f ollow s f or mall y . The same remark allow s to construct the functor γ ∗ .  Remar k 10.3.13 Let us explicit the meaning of the preceding Corollar y for a topology t which is compatible with transfers. Giv en a complex C with coeﬃcients in the category Sh t  P c or Λ , S  , the f ollo wing conditions are equivalent: (i) C is local (Deﬁnition 5.1.9 ), (i’) γ ∗ ( C ) is local, (i”) giv en any P -scheme X / S and an y integer n ∈ Z , the canonical map H n ( C ( X )) / / H n t ( X , γ ∗ ( C )) is an isomor phism, (ii) C is t -ﬂasque (Deﬁnition 5.1.9 ), (ii’) γ ∗ ( C ) is t -ﬂasque, (ii”) giv en any t -hyperco ver p : X / / X in P / S and any integer n ∈ Z , the canonical map p ∗ : H n ( C ( X )) / / H n ( C ( X )) is an isomor phism. More precisel y , the equiv alence of (i) and (ii) is the preceding corollary , while the equiv alence of (i) and (i’) (resp. (ii) and (ii’)) f ollow s from the e xistence of the adjunction ( 10.3.11.1 ) and the fact γ ∗ is ex act. The equivalence between (i’) and (i”) (resp. (ii’) and (ii”)) is a simple translation of Deﬁnition 5.1.9 . 10.3.14 Recall from Deﬁnition 5.1.9 we say that the abelian P -premotivic cate- gory Sh t  P c or Λ  satisﬁes cohomological t -descent if f or any scheme S , and any t -hyperco ver X / / X in P / S , the induced mor phism of complex es in Sh t  P c or Λ , S  Λ t r S ( X ) t / / Λ t r S ( X ) t is a quasi-isomorphism. The preceding corollar y thus giv es the f ollo wing one: Corollary 10.3.15 Assume t is mildly compatible with tr ansfer s. Then the f ollowing conditions ar e equiv alent: 10 Sheav es with transf ers 295 (i) The topology t is compatible with transf ers. (ii) The abelian P -premo tivic categor y Sh t  P c or Λ  satisﬁes cohomological t - descent. (iii) The abelian P -premotivic category Sh t  P c or Λ  is compatible with t (see 5.1.9 ). Proof The equivalence of (i) and (ii) f ollo ws easily from the isomorphism ( 10.3.9.1 ). The equivalence of (ii) and (iii) is Proposition 5.1.26 applied to the adjunction ( 10.3.11.1 ), in view of 10.3.9 (iv).  10.3.16 Recall from Paragraph 2.1.10 that a cd-structure P on S is the data of a famil y of commutative squares, called P -distinguished, of the f or m B k / / g   Q Y f   A i / / X (10.3.16.1) which is stable by isomorphisms. Further, we will consider the f ollowing assumptions on P : (a) P is complete, regular and bounded in the sense of [ V oe10c ]. (b) An y P -distinguished square as abo v e is made of P -mor phisms and k is an immersion. (c) An y square as abov e which is car tesian and such that X = A t Y , i and f being the obvious immersions, is P -distinguished. Then the topology t P associated with P (see 2.1.10 ) is P -admissible and satisfy assumption 10.0.1 (c). Moreov er, according to [ V oe10c , 2.9], w e obtain the f ollowing properties: (d) A presheaf F on P / S is a t P -sheaf if and onl y if F ( ∅ ) = 0 and f or an y P -distinguished square ( 10.3.16.1 ) in P / S , the sequence 0 / / F ( X ) f ∗ + e ∗ / / F ( Y ) ⊕ F ( A ) k ∗ − g ∗ / / F ( B ) is e xact. (e) For an y P -distinguished square ( 10.3.16.1 ) the sequence of representable pre- shea v es on P / S 0 / / Λ S ( B ) k ∗ − g ∗ / / Λ S ( Y ) ⊕ Λ S ( A ) f ∗ + e ∗ / / Λ S ( X ) / / 0 becomes e xact on the associated t P -shea v es. Proposition 10.3.17 Consider a cd-structur e P satisfying properties (a) and (b) abov e and assume that t = t P is the topology associated with P . Then the follo wing conditions ar e equivalent : (i) The topology t is compatible with transf ers. 296 Motivic complex es and relativ e cy cles (ii) The topology t is mildly compatible with transf ers. (iii) F or any scheme S and any P -distinguished squar e ( 10.3.16.1 ) in P / S , the shor t sequence of pr esheav es with transf ers ov er S 0 / / Λ t r S ( B ) k ∗ − g ∗ / / Λ t r S ( Y ) ⊕ Λ t r S ( A ) f ∗ + e ∗ / / Λ t r S ( X ) / / 0 becomes exact on the associated t -sheaves on P / S . Proof The implication (i) ⇒ (ii) is obvious. The implication (ii) ⇒ (iii) f ollo ws from point (e) abo v e and the f ollow - ing facts: γ ∗ is r ight e xact (Corollary 10.3.11 ), γ ∗ a t = a t r t ˆ γ ∗ (remark 10.3.12 ), k ∗ : Λ t r S ( B ) / / Λ t r S ( Y ) is a monomorphism of presheav es with transfers (use 9.1.7 (2) and the fact k is an immersion from assumption (b)). Assume (iii). Then we obtain (ii) as a direct consequence of the point (d) abov e. Thus, to prov e (i), w e ha ve only to prov e that the abelian P -premotivic category Sh t  P c or Λ  satisﬁes cohomological t -descent according to 10.3.15 . Let S be a scheme. Recall that the category D  Sh t ( P / S , Λ )  has a canonical DG-structure (see for ex ample 5.0.2 ). The cohomological t -descent for Sh t  P c or Λ , S  can be ref or mulated b y saying that f or any comple x K of t -sheav es on P / S , and an y t -hyperco ver X / / X , the canonical map of D ( Λ - mo d ) R Hom • D ( Sh t ( P / S , Λ ) ) ( γ ∗ Λ t r S ( X ) t , K ) / / R Hom • D ( Sh t ( P / S , Λ ) ) ( γ ∗ Λ t r S ( X ) t , K ) is an isomor phism. Recall also that there is the injective model structure on the cate- gory C ( Sh t ( P / S , Λ ) ) f or which e v er y object is coﬁbrant, with quasi-isomor phisms as weak equiv alences (see [ CD09 , 2.1] f or more details). Replacing K b y a ﬁbrant resolution f or the injective model structure, we get f or an y simplicial objects X of P / S q that: R Hom • D ( Sh t ( P / S , Λ ) ) ( γ ∗ Λ t r S ( X ) t , K ) = Hom • D ( Sh t ( P / S , Λ ) ) ( γ ∗ Λ t r S ( X ) t , K ) . Thus it is suﬃcient to prov e that the presheaf E : P / S o p / / C ( Λ - mo d ) , X  / / Hom • D ( Sh t ( P / S , Λ ) ) ( γ ∗ Λ t r S ( X ) t , K ) satisﬁes t -descent in the sense of 3.2.5 . W e der iv e from (iii) that E sends a P -distinguished square to a homotop y car tesian square in D ( Λ - mo d ) . Thus the asser tion f ollow s from the arguments on t -descent from [ V oe10b , V oe10c ].  Remar k 10.3.18 It f ollo ws from Remark 10.3.13 that under the equivalent conditions (i), (ii), (iii) of the abo ve corollar y , the admissible topology t = t P is bounded in Sh t  P c or Λ  in the sense of Deﬁnition 5.1.28 . Over a scheme S , a bounded g enerating famil y is given by the f ollowing comple xes of Sh t  P c or Λ , S  : · · · / / 0 / / Λ t r S ( B ) k ∗ − g ∗ / / Λ t r S ( Y ) ⊕ Λ t r S ( A ) f ∗ + e ∗ / / Λ t r S ( X ) / / 0 / / · · · 10 Sheav es with transf ers 297 induced b y a P -distinguished square of the form ( 10.3.16.1 ) – see also Example 5.1.29 . W e end-up this section with a compatibility of cer tain sheav es with transfers with projectiv e limits of sc hemes. This will be the k ey point to establish continuity f or motivic comple xes. Proposition 10.3.19 Let t be one of the topologies Nis , ´ e t , c dh . Let ( S α ) α ∈ A be a projectiv e sys tem of sc hemes in S , with dominant aﬃne transi- tion maps, and such that S = lim o o α ∈ A S α is r epresentable in S . Consider an index α 0 ∈ A and a t -sheaf with transf ers F ov er S f t , c or Λ , S 0 . F or any index α / α 0 , we denote by F α (r esp. F ) the pullbac k of F α 0 ov er S α (r esp. S ) in the sense of the pr emotivic structur e on Sh t  P c or Λ  (obtained in Corollary 10.3.11 ). Then the canonical map: lim / / α ∈ A / α 0 F α ( S α ) / / F ( S ) is an isomorphism. Proof W e consider the f orgetful functor: O t r t : Sh t  S f t , c or Λ  / / PSh  S f t , c or Λ  . It is full y faithful and it commutes with the global section functor . W e want to pro v e the proposition by using Lemma 10.1.5 . Thus it is suﬃcient to pro ve that, f or an y morphism f : X / / S in S , the functor O t r t commutes with f ∗ . In other w ords, the pullback functor ˆ f ∗ f or preshea ves with transf ers on S f t , c or Λ preserves t -sheav es with transf ers: f or any t -sheaf with transfers F ov er S , ˆ f ∗ ( F ) is a t -sheaf with transf ers. Let us ﬁrst treat the case where f is separated of ﬁnite type. Then ˆ f ∗ admits a left adjoint ˆ f ] which preser v es t -co v ers. Thus the proper ty is clear . In the general case, we write f as a projectiv e limit of mor phisms of sc hemes ( f α : X α / / S ) α ∈ A such that the transition morphisms of the projective scheme ( X α ) α ∈ A are aﬃne and dominant and each f α is separated of ﬁnite type. 90 T o chec k that ˆ f ∗ ( F ) is a t -sheaf, we consider a t -co v er p : W / / X of an S -scheme separated of ﬁnite type. Because of our choice of topology t , there e xists an index α 1 / α 0 such that p : W / / X can be lifted as a t -cov er p 1 : W α 1 / / X α 1 o v er S α 1 . Using Lemma 10.1.5 again, w e no w are reduced to prov e that for an y α / α 1 , ˆ f ∗ α 1 ( F ) satisﬁes the t -sheaf proper ty with respect to the pullback of p 1 o v er S α / S α 1 . This f ollow s from the ﬁrst case treated.  Remar k 10.3.20 The previous proposition g eneralizes [ Dég07 , Prop. 2.19]. 90 W r ite the O S -algebra f ∗ ( O X ) as the ﬁltered union of its ﬁnite type sub- O S -algebras, ordered by inclusion. 298 Motivic complex es and relative cycles 10.4 Examples 10.4.1 Assume that t is the Nisnevic h topology . A ccording to Lemma 10.3.3 and Proposition 10.3.17 , t is then compatible with transf ers. With the notation of Corollar y 10.3.11 , w e get the f ollowing deﬁnition: Deﬁnition 10.4.2 W e denote by Sh tr (− , Λ ) , Sh tr (− , Λ ) the respectiv e abelian premotivic and generalized abelian premotivic categories deﬁned in Corollar y 10.3.11 in the respectiv e cases P = S m , P = S f t . From now on, the objects of Sh tr ( S , Λ ) (resp. Sh tr ( S , Λ ) ) are called sheaves with transf ers o ver S (resp. g eneralized sheav es with transf ers o v er S ). Let X be a separated S -scheme of ﬁnite type. W e let Λ tr S ( X ) be the generalized sheaf with transf ers represented by X ( cf. 10.2.4 ). If X is S -smooth, w e denote b y Λ t r S ( X ) its restriction to S m c or Λ , S – i.e. the sheaf with transf ers ov er S represented by X . An impor tant proper ty of sheav es with transf ers is that the abelian premotivic category Sh tr (− , Λ ) (resp. Sh tr (− , Λ ) ) is compatible with the Nisnevic h topology on S m (resp. S f t ) according to Proposition 10.3.17 . Note moreo v er that it is compactly geometricall y generated. 10.4.3 W e also obtained an adjunction (resp. generalized adjunction) of premotivic abelian categories γ ∗ : Sh ( S m , Λ ) / / o o Sh tr (− , Λ ) : γ ∗ γ ∗ : Sh  S f t , Λ  / / o o Sh tr (− , Λ ) : γ ∗ . Note that in each case γ ∗ is conservativ e and e xact according to 10.3.9 (iv). Remar k 10.4.4 An impor tant application of the exis tence of the pair of adjoint func- tors ( γ ∗ , γ ∗ ) is the f ollo wing computation: giv en any comple x K of sheav es with transf ers o v er S and any smooth S -scheme X , Hom D ( Sh tr ( S , Λ )) ( Λ t r S ( X ) , K [ n ]) = Hom D ( Sh tr ( S , Λ )) ( L γ ∗ Λ S ( X ) , K [ n ]) = Hom D ( Sh ( S m , Λ ) ) ( Λ S ( X ) , γ ∗ ( K )[ n ]) = H n Nis ( X , γ ∗ ( K )) . This is a g eneralization of [ V SF00 , c hap. 5, 3.1.9] to unbounded comple xes and arbitrary bases. 10.4.5 Let S be a scheme. Consider the inclusion functor ϕ : S m c or Λ , S / / S f t , c or Λ , S . It induces a functor ϕ ∗ : Sh tr ( S , Λ ) / / Sh tr ( S , Λ ) , F  / / F ◦ ϕ 10 Sheav es with transf ers 299 which is obviousl y ex act and commute with arbitrary direct sums. In par ticular , it commutes with arbitrar y colimits. Lemma 10.4.6 Wit h the notations abov e, the functor ϕ ∗ admits a left adjoint ϕ ! suc h that for any smooth S -scheme X , ϕ ! ( Λ t r S ( X )) = Λ tr S ( X ) . The functor ϕ ! is fully fait hful. In other w ords, w e hav e deﬁned an enlarg ement of premotivic abelian categories ( cf. deﬁnition 1.4.13 ) (10.4.6.1) ϕ ! : Sh tr (− , Λ ) / / Sh tr (− , Λ ) : ϕ ∗ . Proof Let F be a sheaf with transf ers. Let { X / F } be the category of representable sheaf Λ t r S ( X ) o v er F f or a smooth S -scheme X . W e put ϕ ! ( F ) = lim / / { X / F } Λ tr S ( X ) . The adjunction property of ϕ ! is immediate from the Y oneda lemma. W e pro v e that f or any sheaf with transfers F , the unit adjunction mor phism F / / ϕ ∗ ϕ ! ( F ) is an isomorphism. As already remarked, ϕ ∗ commutes with colimits so that we are restricted to the case where F = Λ t r S ( X ) which f ollow s by deﬁnition.  10.4.7 Assume now that t = c dh is the cdh-topology , and P = S f t is the class of separated mor phisms of ﬁnite type. Recall the topology t is associated with the lower cd-structur e – see Example 2.1.11 . Then the assumptions of Proposition 10.3.17 with respect to the lo w er cd-structure are fulﬁlled according to [ SV00b , 4.3.3] combined with [ SV00b , 4.2.9]. Thus we g et the follo wing result: Proposition 10.4.8 The admissible topology c dh on S f t is compatible with trans- f ers. As a corollary , we get a g eneralized premotivic abelian categor y whose ﬁber o ver a scheme S is the categor y Sh tr c dh ( S , Λ ) of c dh -shea v es with transfers on S f t . It is compatible with the cdh-topology . Moreo v er , the restriction of a c dh to Sh tr ( S , Λ ) induces a morphism of generalized premotivic categor ies; we get the f ollo wing commutativ e diagram of such mor phisms: Sh ( − , Λ ) γ ∗   a ∗ c dh / / Sh c dh ( − , Λ ) γ ∗ c dh   Sh tr (− , Λ ) a ∗ c dh / / Sh tr c dh (− , Λ ) 300 Motivic comple xes and relative cycles 10.5 Comparison results 10.5.a Change of coeﬃcients 10.5.1 Assume the topology t is mildly compatible with transfers and consider a localization Λ 0 of Λ . Then the mor phism ( 9.4.3.1 ) of P -premotivic categor ies extends to an adjunction of abelian P -premotivic categories: (10.5.1.1) Sh t  P c or Λ  ⊗ Λ Λ 0 / / o o Sh t  P c or Λ 0  Proposition 9.4.4 immediately yields the f ollo wing result: Proposition 10.5.2 Consider the abov e notations. Then the adjuction ( 10.5.1.1 ) is an equiv alence of P -premotivic categories. Remar k 10.5.3 Remark 9.4.5 can be e xtended to shea v es with transf ers: f or an y regular scheme S , the category Sh tr ( S , Z ) = Sh Nis  S m c or Z , S  deﬁned here coincides with that deﬁned in [ Dég07 ], as well as its operations of a P -premotivic categor y when restricted to regular schemes. Remar k 10.5.4 In a previous v ersion of this text, the preceding proposition was ob- tained under restrictive hypothesis. W e hav e been able to remov e these unnecessar y assumptions thanks to point (3) of Remark 9.1.3 , which is a consequence of Propo- sition 8.1.53 (bound on the denominators of intersection multiplicities). 10.5.b Repr esentable qfh -sheav es 10.5.5 Let us denote b y Sh qfh ( S , Λ ) the category of qfh -sheav es of Λ -modules ov er S f t / S . Remark that for an S -scheme X , the Λ -presheaf represented by X is not a sheaf f or the qfh -topology . W e denote the associated sheaf b y Λ qfh S ( X ) . W e let a qfh be the associated qfh -sheaf functor . Recall that f or any S -scheme X , the graph functor ( 10.4.3 ) induces a mor phism of sheav es Λ S ( X ) γ X / S / / Λ tr S ( X ) . W e recall the f ollo wing theorem of Suslin and V oev odsky (see [ SV00b , 4.2.7+4.2.12]): Theorem 10.5.6 Let S be a scheme such that c har ( S ) ⊂ Λ × . Then, for any S -sc heme X , the application of a qfh to the map γ X / S giv es an isomor phism in Sh qfh ( S , Λ ) : Λ qfh S ( X ) γ qfh X / S / / Λ tr S ( X ) . 10 Sheav es with transf ers 301 10.5.7 Assume char ( S ) ⊂ Λ × . Using the previous theorem, we associate to any qfh -sheaf F ∈ Sh qfh ( S , Λ ) a presheaf with transfers ρ ( F ) : X  / / Hom Sh qfh ( S , Λ ) ( Λ tr S ( X ) , F ) . W e obviousl y get γ ∗ ρ ( F ) = F as a presheaf o v er S f t / S so that ρ ( F ) is a sheaf with transf ers and we hav e deﬁned a functor ρ : Sh qfh ( S , Λ ) / / Sh tr ( S , Λ ) . For any S -scheme X , ρ ( Λ qfh S ( X )) = Λ tr S ( X ) according to the previous proposition. Corollary 10.5.8 Assume char ( S ) ⊂ Λ × . Let f : X 0 / / X be a mor phism of S - sc hemes. If f is a univ ersal homeomorphism, then the map f ∗ : Λ tr S ( X 0 ) / / Λ tr S ( X ) is an isomorphism in Sh tr ( S , Λ ) . Proof Indeed, according to [ V oe96 , 3.2.5], Λ qfh S ( X 0 ) / / Λ qfh S ( X ) is an isomor phism in Sh qfh ( S , Λ ) and w e conclude by appl ying the functor ρ .  10.5.c qfh -shea ves and transf ers Proposition 10.5.9 Assume char ( S ) ⊂ Λ × . Any qfh -sheaf of Λ -modules ov er the category of S -schemes of ﬁnit e type is natur ally endow ed with a unique structure of a sheaf with transf ers, and any morphism of such qfh -sheaves is a mor phism of sheav es with transf ers. In par ticular , the qfh -sheaﬁﬁcation functor deﬁnes a lef t exact functor lef t adjoint to the for g etful functor ρ : Sh qfh ( S , Λ ) / / Sh tr ( S , Λ ) introduced in 10.5.7 . Proof It follo ws from Theorem 10.5.6 that the categor y of Λ -linear ﬁnite cor respon- dences is canonically equiv alent to the full subcategory of the categor y of qfh -sheav es of Λ -modules spanned b y the objects of shape Λ qfh S ( X ) f or X separated of ﬁnite type o v er S . The ﬁrst asser tion is thus an immediate consequence of Theorem 10.5.6 and of the (additive) Y oneda lemma. The fact that the qfh -sheaﬁﬁcation functor deﬁnes a left adjoint to the restr iction functor ρ is then obvious, while its left e xactness is a consequence of the facts that it is left ex act (at the lev el of sheav es without trans- f ers) and that for getting transfers deﬁnes a conser v ative and e xact functor from the category of Nisnevich sheav es with transfers to the categor y of Nisnevic h shea v es.  Recall the follo wing theorem: Theorem 10.5.10 Assume Λ is a Q -alg ebra. Let F be an étale Λ -sheaf on S f t / S . Then f or any S -scheme X , and any integ er i , the canonical map H i Nis ( X , F ) / / H i ´ e t ( X , F ) is an isomorphism. 302 Motivic complex es and relative cycles Proof Using the compatibility of étale cohomology with projectiv e limits of schemes, w e are reduced to pro ve that H i ´ e t ( X , F ) = 0 whene v er X is henselian local and i > 0 . Let k be the residue ﬁeld of X , G its absolute Galois group and F 0 the restriction of F to Spec ( k ) . Then F 0 is a G -module and according to [ A GV73 , 8.6], H i ´ e t ( X , F ) = H i ( G , F 0 ) . As G is proﬁnite, this group must be torsion so that it vanishes by assumption.  Remar k 10.5.11 The preceding theorem also f ollow s f ormally from Theorem 3.3.23 . Proposition 10.5.12 Assume Λ is a Q -alg ebra. Le t S be an excellent sc heme and F be a qfh -sheaf of Λ -modules on S f t / S . Then f or any g eometrically unibranc h S -sc heme X of ﬁnite type, and any integ er i , the canonical map H i Nis ( X , F ) / / H i qfh ( X , F ) is an isomorphism. Proof A ccording to 10.5.10 , H i Nis ( X , F ) = H i ´ e t ( X , F ) . Let p : X 0 / / X be the normalization of X . As X is an e x cellent geometrically unibranc h scheme, p is a ﬁnite universal homeomor phism. It f ollow s from [ A GV73 , VII, 1.1] that H i ´ e t ( X , F ) = H i ´ e t ( X 0 , F ) and from [ V oe96 , 3.2.5] that H i qfh ( X , F ) = H i qfh ( X 0 , F ) . Thus w e can assume that X is normal, and the result is now ex actly [ V oe96 , 3.4.1].  Corollary 10.5.13 Assume Λ is a Q -alg ebra. Let S be an excellent sc heme. 1. Let X be a g eometrically unibranc h S -scheme of ﬁnite type. F or any point x of X , the local henselian sc heme X h x is a point for the category of sheav es Sh qfh ( S , Λ ) ( i.e. evaluating at X h x deﬁnes an exact functor). 2. The f amily of points X h x of t he pr evious type is a conser vativ e f amily f or Sh qfh ( S , Λ ) . Proof The ﬁrst point f ollo ws from the previous proposition. F or an y e x cellent scheme X , the normalization mor phism X 0 / / X is a qfh -co v er . Thus the cate- gory Sh qfh ( S , Λ ) is equiv alent to the category of qfh -sheav es on the site made of geometricall y unibranch S -schemes of ﬁnite type. This implies the second asser- tion.  10.5.14 Giv en any scheme S , we introduce the f ollo wing composite functor using the notations of 10.5.7 and 10.4.5 : ψ ∗ : Sh qfh ( S , Λ ) ρ / / Sh tr ( S , Λ ) ϕ ∗ / / Sh tr ( S , Λ ) . Theorem 10.5.15 Assume Λ is a Q -algebr a and let S be a g eometrically unibr anch excellent scheme. Considering the abov e notation, the follo wing conditions are true: (i) F or any S -scheme X of ﬁnite type, ψ ∗  Λ qfh S ( X )  = Λ t r S ( X ) . (ii) The functor ψ ∗ admits a lef t adjoint ψ ! . (iii) F or any smooth S -scheme X , ψ !  Λ t r S ( X )  = Λ qfh S ( X ) . 11 Motivic complex es 303 (iv) The functor ψ ∗ is exact and preserves colimits. (v) The functor ψ ! is fully fait hful. A ccording to proper ty (iii), the functor ψ ! commutes with pullbacks. In other words, w e ha ve deﬁned an enlarg ement of abelian premotivic categor ies ( cf. deﬁnition 1.4.13 ) o v er the categor y of (noether ian) g eometr icall y unibranch schemes: (10.5.15.1) ψ ! : Sh tr (− , Λ ) / / o o Sh qfh ( − , Λ ) : ψ ∗ Proof Point (i) follo ws from Theorem 10.5.6 . Recall the enlargement of ( 10.4.6.1 ): ϕ ! : Sh tr (− , Λ ) / / Sh tr (− , Λ ) : ϕ ∗ . W e deﬁne the functor ψ ! as the composite: Sh tr ( S , Λ ) ϕ ! / / Sh tr ( S , Λ ) γ ∗ / / Sh ( S , Λ ) a qfh / / Sh qfh ( S , Λ ) . A ccording to the proper ties of the functors in this composite, ψ ! is ex act and preser v es colimits. Moreov er , for any smooth S -scheme X , as Λ t r S ( X ) is a qfh -sheaf ov er S f t / S according to 10.2.4 , ψ ! ( Λ t r S ( X )) = Λ qfh S ( X ) which prov es (iii). Proper ty (ii) f ollo ws from (iii) and the fact ψ ! commutes with colimits, while the sheav es Λ t r S ( X ) for X / S smooth generate Sh tr ( S , Λ ) . For any smooth S -scheme X , Γ ( X ; ψ ∗ ( F )) = F ( X ) . Thus the e xactness of ψ ∗ f ollow s from Corollar y 10.5.13 — here we use the assumption that S is geometrically unibranch and ex cellent, as it implies that X satisﬁes the same proper ties, so that w e can apply the mentioned corollary . As ψ ∗ obviousl y preser v es direct sums, we get (iv). T o chec k that f or any sheaf with transf ers F the unit map F / / ψ ∗ ψ ! ( F ) is an isomorphism, we thus are reduced to the case where F = Λ t r S ( X ) f or a smooth S -scheme X which follo ws from (i) and (iii).  11 Motivic complex es 11.0.1 In this section, S is the categor y of noether ian ﬁnite dimensional schemes. It is adequate in the sense of 2.0.1 . Given a scheme S , we denote b y S m S the categor y smooth separated S -schemes of ﬁnite type. It is admissible in the sense of 1.0.1 . W e ﬁx a r ing of coeﬃcients Λ . 304 Motivic complex es and relativ e cycles 11.1 Deﬁnition and basic properties 11.1.a Premotivic categories A ccording to Proposition 10.3.17 and Corollar y 10.3.15 , the abelian premotivic cate- gory Sh tr (− , Λ ) constructed in 10.4.2 is compatible with Nisnevich topology . Thus w e can appl y to it the general deﬁnitions of section 5 . This giv es the f ollowing deﬁnition: Deﬁnition 11.1.1 W e deﬁne the ( Λ -linear) category of motivic complexes (resp. stable mo tivic complexes or simpl y motiv es ) follo wing Deﬁnition 5.2.16 (resp. Deﬁnition 5.3.22 ) as DM eﬀ Λ = D eﬀ A 1  Sh tr (− , Λ )  resp. DM Λ = D A 1  Sh tr (− , Λ )  . Giv en a scheme S , we will put: DM eﬀ ( S , Λ ) = DM eﬀ Λ ( S ) , DM ( S , Λ ) = DM Λ ( S ) . 11.1.2 Let us unf old the preceding deﬁnition. Given a scheme S in S , the tr iangu- lated category DM eﬀ ( S , Λ ) is equal to the A 1 -localization of the derived category D ( Sh tr ( S , Λ )) of the category of sheav es with transfers ov er S . Giv en a smooth scheme S -scheme X of ﬁnite type, w e hav e denoted b y Λ t r S ( X ) the sheaf with transf ers represented by X o v er S . W e will see this sheaf as an object of DM eﬀ ( S , Λ ) , as a complex concentrated in degree 0 , and call it the eﬀectiv e motivic comple x associated with X / S . Recall the follo wing operations as par t of the premotivic str ucture: • Giv en any mor phism f : T / / S in S , there e xists an adjunction of the f or m: L f ∗ : DM eﬀ ( S , Λ ) / / o o DM eﬀ ( T , Λ ) : R f ∗ . • Giv en a separated smooth mor phism of ﬁnite type f : T / / S in S , there exis ts an adjunction of the form: L f ] : DM eﬀ ( S , Λ ) / / o o DM eﬀ ( T , Λ ) : f ∗ = L f ∗ . • Giv en any noether ian ﬁnite dimensional scheme S , the categor y DM eﬀ ( S , Λ ) is symmetric closed monoidal. These operations are subject to the proper ties of a premotivic categor y: functoriality , smooth base chang e f or mula, smooth projection f or mula – see section 1 for more details. By constr uction, the tr iangulated premotivic categor y DM eﬀ Λ satisﬁes the homotop y proper ty and the Nisnevic h descent proper ties. By constr uction (cf. ( 5.3.23.2 )), we g et an adjunction of tr iangulated premotivic categories (11.1.2.1) Σ ∞ : DM eﬀ Λ / / o o DM Λ : Ω ∞ . 11 Motivic complex es 305 Considering the T ate motivic complex (11.1.2.2) Λ t r S ( 1 ) : = Λ t r S ( P 1 S /{ 1 } ) , the object Σ ∞ ( Λ t r S ( 1 )) is ⊗ -in vertible in DM ( S , Λ ) and this property characterizes uniquel y the homotopy category DM ( S , Λ ) – see R emark 5.3.29 . Giv en a smooth separated S -scheme X of ﬁnite type, we put: M S ( X ) : = Σ ∞ Λ t r S ( X ) and simply call it the motive associated with X / S . Usuall y we denote b y 1 S the unit of the monoidal categor y DM ( S , Λ ) . By constr uction, the premotivic category DM Λ satisﬁes the homotopy , stability and Nisnevic h descent proper ties (see Paragraph 5.3.23 ). Example 11.1.3 • Let k be a perf ect ﬁeld. Then DM eﬀ ( k , Z ) contains as a full subcategory the categor y DM eﬀ − ( k ) deﬁned b y V oev odsky (cf [ V SF00 , Chap. 5]). This is the content of the proof of [ V SF00 , Chap. 5, Prop. 3.2.3]. Indeed, recall from Paragraph 5.2.18 that DM eﬀ ( k , Z ) is equiv alent to the full subcategor y of D ( Sh tr ( k , Z )) made by the complex es which are A 1 -local. Ov er a per f ect ﬁeld, Theorem 3.1.12 of [ V SF00 , Chap. 5] implies that a comple x of sheav es with transf ers is A 1 -local if and only if its homotopy sheav es are A 1 -in variant. • Let S be a regular scheme. The tr iangulated categor ies DM eﬀ ( S , Z ) and DM ( S , Z ) introduced here coincide with that constructed in [ CD09 ]. The same is tr ue concerning the operations of premotivic tr iangulated categories (see Remark 10.5.3 ). 11.1.4 Let Λ 0 be a localization of Λ . The premotivic adjunction (11.1.4.1) Sh tr (− , Λ ) ⊗ Λ Λ 0 / / o o Sh tr (− , Λ 0 ) obtained as a par ticular case of ( 10.5.1.1 ) gives the f ollowing adjunctions of triangu- lated premotivic categor ies: DM Λ ⊗ Λ Λ 0 / / o o DM Λ 0 , DM eﬀ Λ ⊗ Λ Λ 0 / / o o DM eﬀ Λ 0 . (11.1.4.2) Proposition 10.5.2 immediately yields the f ollo wing result: Proposition 11.1.5 The premotivic adjunctions ( 11.1.4.2 ) are equivalences of trian- gulated premotivic categories. In other w ords, for an y sc heme S , the triangulated monoidal categor y DM ( S , Λ 0 ) (resp. DM eﬀ ( S , Λ 0 ) ) is the naive localization of the category DM ( S , Λ ) (resp. DM eﬀ ( S , Λ ) ) with respect to integers inv er tible in Λ 0 . 306 Motivic complex es and relative cycles 11.1.b Constructible and geometric motiv es 11.1.6 The premotivic tr iangulated category DM eﬀ Λ is geometrically generated: given an y scheme S , the essentially small set G e f f S of motivic comple xes of the f orm Λ t r S ( X ) f or a smooth separated S -scheme X of ﬁnite type form a set of g enerators in the triangulated category DM eﬀ ( S , Λ ) . Similarl y , the premotivic triangulated categor y DM Λ is Z -generated where Z is the set of twists cor responding to the T ate twist: given an y scheme S , the essentially small set G S of motiv es of the f or m M S ( X )( n ) f or a smooth separated S -scheme X of ﬁnite type and an integer n ∈ Z f or m a set of g enerators in the triangulated categor y DM ( S , Λ ) . Follo wing the g eneral con v entions about premotivic tr iangulated category (Deﬁ- nition 1.4.9 ), we deﬁne the notion of constructibility for motives as follo ws: Deﬁnition 11.1.7 Giv en any scheme S , w e deﬁne the categor y of constructible mo- tiv es (resp. constructible motivic complexes ) ov er S as the thick tr iangulated sub- category of DM ( S , Λ ) (resp. DM eﬀ ( S , Λ ) ) generated by G S (resp. G e f f S ). W e denote it b y DM c ( S , Λ ) (resp. DM eﬀ c ( S , Λ ) ). Remar k 11.1.8 Recall that DM c , Λ (resp. DM eﬀ c , Λ ) is Sm -ﬁbred monoidal subcategor y of DM Λ (resp. DM eﬀ Λ ) o v er S . In other w ords, constructible motives (resp. motivic comple x es) are stable by the operations f ∗ , p ] f or p smooth and tensor product. This is obvious from deﬁnitions. 11.1.9 Let S be a scheme. Consider the tr iangulated subcategory V S of K b ( S m c or Λ , S ) generated by complex es of one the f ollo wing f orms : 1. f or an y distinguished square W k / / g   V f   U j / / X of smooth S -schemes, [ W ] g ∗ − k ∗ / / [ U ] ⊕ [ V ] j ∗ + f ∗ / / [ X ] 2. f or an y smooth S -scheme X , p : A 1 X / / X the canonical projection. [ A 1 X ] p ∗ / / [ X ] . Deﬁnition 11.1.10 W e deﬁne the categor y DM eﬀ g m ( S , Λ ) of geome tric eﬀective mo- tiv es ov er S as the pseudo-abelian en v elope of the tr iangulated categor y K b ( S m c or Λ , S )/ V S . W e deﬁne the categor y DM g m , Λ ( S ) of g eometric motiv es o v er S as the tr iangulated category obtained from DM eﬀ g m ( S , Λ ) by f or mally inv er ting the T ate complex [ P 1 S ] / / [ S ] . 11 Motivic complex es 307 Remar k 11.1.11 The categor ies of geometric motiv es (resp. eﬀectiv e g eometr ic mo- tiv es) ov er an arbitrar y base, as deﬁned here, already appears in the w ork of Iv or ra [ Iv o07 , sec. 1.3]. 11.1.12 A ccording to this deﬁnition, w e can construct f or any scheme S a commuta- tiv e diagram of functors: DM eﬀ g m ( S , Λ ) / /   DM eﬀ ( S , Λ ) Σ ∞   DM g m ( S , Λ ) / / DM ( S , Λ ) (11.1.12.1) where the r ight v er tical map is the left adjoint of ( 11.1.2.1 ). Recall from Remark 10.3.18 that the Nisnevic h topology is bounded in Sh tr (− , Λ ) . Thus, as a corollary of Proposition 5.2.38 , Corollary 5.2.39 and Corollar y 5.3.42 w e get the follo wing result: Theorem 11.1.13 The horizontal functors of the squar e ( 11.1.12.1 ) are fully faithful and their essential images consist of constructible objects in the sense of Deﬁnition 11.1.7 . Giv en any motiv e (resp. motivic complex) M ov er S , the following conditions are equiv alent : (i) M is g eometric ( i.e. in the imag e of the horizontal map of diagram ( 11.1.12.1 ) ), (ii) M is constructible, (iii) M is compact. The triangulated categor y DM ( S , Λ ) (resp. DM eﬀ ( S , Λ ) ) is compactly g enerated. Mor e pr ecisely, the objects of the se t of g enerat ors G S (r esp. G e f f S ) deﬁned in P arag raph 11.1.6 are compact. Remar k 11.1.14 If S = Sp ec ( k ) is the spectrum of a per f ect ﬁeld, then the categor ies DM g m ( S , Λ ) and DM eﬀ g m ( S , Λ ) coincide with the categor ies introduced by V oe v odsky in [ VSF00 , chap. 5, Sec. 2.1]. The abov e theorem is a generalization of [ V SF00 , chap. 5, Th. 3.2.6] to an arbitrar y base (and the non-eﬀective case). 11.1.c Enlargement, descent and continuity 11.1.15 W e can apply the deﬁnitions of section 5 to the generalized abelian premo- tivic category Sh tr (− , Λ ) constructed in 10.4.2 Deﬁnition 11.1.16 W e deﬁne the ( Λ -linear) categor y of g eneralized mo tivic com- plexes (resp. g eneralized motives ) f ollo wing deﬁnition 5.3.22 (resp. deﬁnition 5.2.16 ) as DM eﬀ Λ = D eﬀ A 1  Sh tr (− , Λ )  resp. DM Λ = D A 1  Sh tr (− , Λ )  . 308 Motivic complex es and relative cycles 11.1.17 The advantage of this deﬁnition is that an y separated S -scheme X of ﬁnite type deﬁnes a generalized motivic comple x, given b y the sheaf with transfers Λ tr S ( X ) seen as a complex concentrated in deg ree 0 (see Deﬁnition 10.4.2 ). The categor y DM eﬀ Λ , as a generalized premotivic category , admits the f ollowing operations: • Giv en any mor phism f : T / / S in S , there e xists an adjunction of the f or m: L f ∗ : DM eﬀ ( S , Λ ) / / o o DM eﬀ ( T , Λ ) : R f ∗ . • Giv en a separated morphism f : T / / S of ﬁnite type in S (non necessarily smooth), there e xists an adjunction of the f or m: L f ] : DM eﬀ ( S , Λ ) / / o o DM eﬀ ( T , Λ ) : f ∗ = L f ∗ . • Giv en any noether ian ﬁnite dimensional scheme S , the categor y DM eﬀ ( S , Λ ) is symmetric closed monoidal. These operations satisfy the proper ties of a generalized premotivic categor y f or which we ref er the reader to section 1.4 . As in the non g eneralized case, we g et from the general constr uction (see ( 5.3.23.2 )) an adjunction of tr iangulated generalized premotivic categor ies (11.1.17.1) Σ ∞ : DM eﬀ Λ / / o o DM Λ : Ω ∞ . T o an y separated S -scheme X of ﬁnite type, we associate a generalized motiv e as: M S ( X ) : = Σ ∞ Λ tr S ( X ) . By construction, the generalized premotivic categor y DM eﬀ Λ (resp. DM Λ ) satisﬁes the homotop y proper ty , Nisnevic h descent proper ty (resp. and stability property). 11.1.18 By vir tue of Remark 10.3.18 , the Nisnevich topology is bounded in Sh tr (− , Λ ) . Theref ore, as a corollary of Proposition 5.2.38 (resp. Corollar y 5.2.39 ), we obtain in particular that DM eﬀ ( S , Λ ) (resp. DM ( S , Λ ) ) is compactly generated, with the essen- tially small famil y of objects Λ tr S ( X ) (resp. M S ( X )( n ) ) for a separated S -scheme of ﬁnite type X (resp. and an integer n ∈ Z ) as compact generators. Recall that for any scheme S , the obvious restr iction functor ϕ ∗ : Sh tr ( S , Λ ) / / Sh tr ( S , Λ ) admits a left adjoint ϕ ! which is fully f aithful (Lemma 10.4.6 ). Moreov er , the adjoint pair ( ϕ ! , ϕ ∗ ) satisﬁes the assumption of Proposition 6.1.4 so that applying Corollary 6.1.9 giv es the f ollo wing proposition: 11 Motivic complex es 309 Proposition 11.1.19 Given any sc heme S , the adjoint pair ( ϕ ! , ϕ ∗ ) can be deriv ed and induces the f ollowing pair of adjoint functors ϕ ! : DM ( S , Λ ) / / o o DM ( S , Λ ) : ϕ ∗ , r esp. ϕ ! : DM eﬀ ( S , Λ ) / / o o DM eﬀ ( S , Λ ) : ϕ ∗ , (11.1.19.1) suc h that ϕ ! is fully fait hful. Mor e g enerally , the family of these adjunctions f or a noetherian ﬁnite dimensional sc heme S deﬁnes an enlarg ement of premo tivic categories (Deﬁnition 1.4.13 ). The abuse of notations is justiﬁed because of the follo wing essentially commutativ e diagram of functors: DM eﬀ Λ Σ ∞ / / ϕ !   DM Λ ϕ !   DM eﬀ Λ Σ ∞ / / DM Λ (11.1.19.2) Recall that, given a smooth separated S -scheme X , w e hav e the relation: ϕ ! ( M S ( X )) = M S ( X ) . Remar k 11.1.20 Bew are that the functor ϕ ∗ is f ar from being conservativ e. The f ol- lo wing e xample was sugges ted b y V . V ologodsky: let Z be a nowhere dense closed subscheme of S . Then ϕ ∗ ( M S ( Z )) = 0 . In fact, one can see that DM ( S , Λ ) is a localization of the categor y DM ( S , Λ ) with respect to the objects M such that ϕ ∗ ( M ) = 0 . 11.1.21 With rational coeﬃcients, the preceding proposition can be reﬁned. Recall that the qfh -sheaﬁﬁcation functor ( 10.5.9 ) induces by 5.3.28 a premotivic adjunction α ∗ : DM Q / / o o DM qfh , Q : α ∗ . Theorem 11.1.22 If S is a g eometrically unibranc h excellent noetherian scheme of ﬁnite dimension then the following composite functor α ∗ ϕ ! : DM ( S , Q ) / / DM qfh , Q ( S ) is fully fait hful. Proof Note that DM eﬀ ( S , Q ) and D eﬀ A 1 ( Sh qfh ( S , Q ) ) are compactly generated; see e xample 5.1.29 and Proposition 5.2.38 . Hence this corollary follo ws from Theorem 10.5.15 and Proposition 6.1.8 .  Remar k 11.1.23 Recall this theorem can be rephrased by saying that motives ov er S satisﬁes qfh -descent – see Remark 5.2.11 and more generall y Section 3 . In the ne xt section, w e will give applications of this fact to motivic cohomology . Theorem 11.1.24 The f ollowing assertions hold: 310 Motivic comple xes and relative cycles 1. The triangulated pr emotivic categories DM eﬀ Λ and DM Λ ar e w eakly continuous (Deﬁnition 4.3.2 ). 2. The g eneralized triangulated premo tivic categories DM eﬀ Λ and DM Λ ar e weakly continuous. Proof Note that Proposition 10.3.19 show s precisely that the generalized premotivic abelian categor y Sh tr (− , Λ ) satisﬁes Proper ty (wC) of Paragraph 5.1.35 . Theref ore, the asser tion (2) f ollow s from Propositions 5.2.41 and 5.3.44 . Moreo v er , the asser tion (1) f ollo ws from Corollar y 6.1.12 giv en the enlarg ement obtained in Proposition 11.1.19 .  Example 11.1.25 From the pre vious theorem and Proposition 4.3.4 , we obtain in particular that f or any pro-scheme ( S α ) α ∈ A with aﬃne and dominant transition map such that S = lim o o α ∈ A S α is noetherian ﬁnite dimensional, there e xists canonical equiv alences of categor ies: 2 - lim / / α  DM eﬀ g m , Λ ( S α )  / / DM eﬀ g m , Λ ( S ) , 2 - lim / / α  DM g m , Λ ( S α )  / / DM g m , Λ ( S ) . This result generalizes [ Ivo07 , 4.16]. 11.2 Motivic cohomology 11.2.a Deﬁnition and functoriality Deﬁnition 11.2.1 Let S be a scheme and ( n , m ) ∈ Z 2 be a couple of integers. W e deﬁne the motivic cohomology of S in degree n and twist m with coeﬃcients in Λ as the Λ -module H n , m M ( S , Λ ) = Hom DM ( S , Λ )  1 S , 1 S ( m )[ n ]  . Assuming m ≥ 0 , we deﬁne the eﬀectiv e motivic cohomology of S in deg ree n and twist m with coeﬃcients in Λ as the Λ -module H n , m M , eﬀ ( S , Λ ) = Hom DM eﬀ ( S , Λ )  Λ t r S , Λ t r S ( m )[ n ]  . Motivic cohomology (resp. eﬀective motivic cohomology) is contrav ar iant with respect to mor phisms of schemes and the monoidal structure on DM Λ (resp. DM eﬀ Λ ) deﬁnes a r ing structure compatible with pullbacks: giv en tw o cohomology classes: α : 1 S / / 1 S ( m )[ n ] , α 0 : 1 S / / 1 S ( m 0 )[ n 0 ] , one simply put: 11 Motivic complex es 311 α . α 0 = α ⊗ S α 0 . The link between motivic cohomology and eﬀectiv e motivic cohomology is provided b y Proposition 5.3.39 . Giv en any scheme S and an y couple of integers ( n , m ) ∈ Z 2 , one has a canonical identiﬁcation: H n , m M ( S , Λ ) = lim / / r > > 0 Hom DM eﬀ ( S , Λ )  Λ t r S ( r ) , Λ t r S ( m + r )[ n ]  . 11.2.2 Let Λ 0 be a localization of Λ . Then using the left adjoint of the premotivic adjunction ( 11.1.4.2 ), we get a canonical mor phism H n , m M ( S , Λ ) ⊗ Λ Λ 0 / / H n , m M ( S , Λ 0 ) . It is obviousl y compatible with pullbacks and the product s tructure. According to Proposition 11.1.5 , this map is an isomor phism. Example 11.2.3 Let k be a per f ect ﬁeld. Given any smooth separated k -scheme S of ﬁnite type, with structural mor phism f , and any pair of integers ( n , m ) ∈ Z 2 , motivic cohomology as deﬁned in the previous deﬁnition coincide with motivic cohomology as deﬁned by V oev odsky in [ V SF00 , chap. 5] according to the f ollowing computation and Remark 11.1.14 : H n , m M ( X , Z ) = Hom DM ( X , Z ) ( 1 X , 1 X ( m )[ n ]) = Hom DM ( X , Z ) ( 1 X , f ∗ ( 1 k )( m )[ n ]) = Hom DM ( k , Z ) ( L f ] ( 1 X ) , 1 k ( m )[ n ]) = Hom DM ( k , Z ) ( M k ( X ) , 1 k ( m )[ n ]) = Hom DM g m ( k , Z ) ( M k ( X ) , 1 k ( m )[ n ]) . In par ticular , it coincides with higher Cho w groups (cf [ V oe02a ]) according to the f ollowing formula: H n , m M ( X , Z ) = C H m ( X , 2 m − n ) . Recall in par ticular the f ollo wing computations: H n , m M ( X , Z ) =              Z π 0 ( X ) if n = m = 0 , G m ( X ) if n = m = 1 , C H m ( X ) if n = 2 m , 0 if m < 0 , n > min ( m + dim ( X ) , 2 m ) where C H m ( X ) is the usual Chow g roup of m -codimensional cycles in X . Note we will e xtend the identiﬁcation of motivic cohomology as deﬁned in the previous deﬁnition with the g eneral v ersion deﬁned by V oev odsky – [ V oe98 ] – in section 11.2.c . 11.2.4 Consider a separated mor phism p : X / / S of ﬁnite type. R ecall from the S f t -ﬁbred structure of DM Λ that M S ( X ) = L p ] p ∗ ( 1 S ) . Using the adjunction prop- erty of the pair ( L p ] , p ∗ ) , w e easily g et: 312 Motivic complex es and relative cycles H n , m M ( X , Λ ) = Hom DM ( X , Λ )  1 X , 1 X ( m )[ n ]  = Hom DM ( X , Λ )  1 X , 1 X ( m )[ n ]  = Hom DM ( S , Λ )  M S ( X ) , 1 S ( m )[ n ]  . (11.2.4.1) In par ticular , given any ﬁnite S -cor respondence α : X • / / Y betw een separated S -schemes of ﬁnite type, we obtain a pullback α ∗ : H n , m M ( Y , Λ ) / / H n , m M ( X , Λ ) which is, among other proper ties, functor ial with respect to composition of ﬁnite S -correspondences and e xtends the natural contra variant functoriality of motivic cohomology . In par ticular , given an y ﬁnite Λ -universal morphism f : Y / / X , we obtain a pushout f ∗ : H n , m M ( Y , Λ ) / / H n , m M ( X , Λ ) b y consider ing the transpose of the graph of f . Proposition 11.2.5 Let f : Y / / X be a ﬁnite Λ -univer sal morphism of schemes. Assume X is connected and let d > 0 be the degr ee of f (cf. 9.1.12 ). Then for any element x ∈ H n , m M ( X , Λ ) , f ∗ f ∗ ( x ) = d . x . This is a simple application of Proposition 9.1.13 . W e left to the reader the e xercise to state projection and base chang e f ormulas f or this pushout. Example 11.2.6 Let f : Y / / X be a ﬁnite mor phism. R ecall that f is Λ -univ ersal in the f ollo wing par ticular cases: • f is ﬂat (see Example 8.1.50 ); • X is regular and f sends the generic points of Y to generic points of X (see Corollary 8.3.28 ). In par ticular , motivic cohomology is cov ar iant with respect to this kind of ﬁnite morphisms. Another impor tant application of the g eneralized motives is obtained using the Corollary 10.5.8 : Proposition 11.2.7 Let f : X 0 / / X be a separat ed univer sal homeomor phism of ﬁnite type. Assume that c har ( X ) ⊂ Λ × . Then the pullbac k functor H n , m M ( X , Λ ) / / H n , m M ( X 0 , Λ ) is an isomorphism. Remar k 11.2.8 The preceding considerations hold similarly for the eﬀectiv e motivic cohomology . Example 11.2.9 In characteristic 0 , motivic cohomology (eﬀective and non-eﬀective) is in variant under semi-nor malization ([ Swa80 ]). 11 Motivic complex es 313 When restr icted to e xcellent geometricall y unibranc h sc heme X , motivic coho- mology (eﬀectiv e and non-eﬀectiv e) is in variant under normalization. Indeed, the normalization X 0 / / X of such a scheme is a universal homeomor phism ([ GD67 , I V 0 , 23.2.2]) of ﬁnite type. 11.2.b Eﬀectiv e motivic cohomology in w eight 0 and 1 11.2.10 Let S be a scheme and X a smooth S -scheme. For an y subscheme Y of X , we denote b y Λ t r S ( X / Y ) the cokernel of the canonical mor phism of sheaf with transfers Λ t r S ( Y ) / / Λ t r S ( X ) . As this morphism is a monomor phism, w e obtain a canonical distinguished tr iangle in DM eﬀ ( S , Λ ) Λ t r S ( Y ) / / Λ t r S ( X ) / / Λ t r S ( X / Y ) / / Λ t r S ( X )[ 1 ] . Using this notation and according to Deﬁnition 2.4.17 , the T ate motivic comple x is deﬁned as: Λ t r S ( 1 ) = Λ t r S ( P 1 S /{ ∞ } )[− 2 ] . The f ollo wing computation is classical: Λ t r S ( 1 ) = Λ t r S ( P 1 S / A 1 S )[− 2 ] = Λ t r S ( A 1 S / G m )[− 2 ] ; the ﬁrst identiﬁcation f ollow s from homotopy inv ar iance and the second one b y Nisnevic h descent (cf. Prop. 5.2.13 ). Proposition 11.2.11 Suppose S is a nor mal sc heme. Then the sheaf on Sm S r epresent ed by G m admits a canonical structure of a sheaf with tr ansfer s and ther e is a canonical isomor phism in DM eﬀ ( S , Λ ) : G m ⊗ Z Λ ' / / Λ t r S ( 1 )[ 1 ] . Proof Let U be an open subscheme of A 1 S and X be a smooth S -scheme. Note that X is nor mal according to [ GD67 , 18.10.7]. Consider a cycle α = Õ i n i . h Z i i of X × S U with n i ∈ Λ and Z i irreducible ﬁnite and dominant ov er an irreducible component of X . Then Z i is a divisor in X × S U and according to [ GD67 , 21.14.3], it is ﬂat ov er X . In other w ords, α is a Hilber t cycle which implies it is Λ -universal (Example 8.1.50 ). As a consequence, w e obtain the equality H i Γ ( X ; C ∗ Λ t r S ( U )) = H s i n g − i ( X × S U / X ) ⊗ Z Λ where the functor C ∗ is the associated Suslin singular comple x (see ( 5.2.32.1 )) and the right-hand side is the Suslin homology of X × S U / X ( cf. [ S V00b ]). Suppose in addition that X and U are aﬃne and let Z = P 1 S − U . According to a theorem of Suslin and V oev odsky ( cf. [ SV00b , th. 3.1]), 314 Motivic complex es and relativ e cycles H s i n g − i ( X × S U / X ) =  Pic ( X × S P 1 S , X × S Z ) if i = 0 0 otherwise; the group on the left-hand side is the r elativ e Picard gr oup . In particular , the comple x C ∗ Λ t r S ( U ) , seen as a complex of preshea ves with transfers, is con- centrated in cohomological degree 0 and its 0 -th cohomology is the presheaf X  / / Pic ( X × S P 1 S , X × S Z ) ⊗ Z Λ . Consider the f ollo wing e xact sequence of preshea v es with transfers: 0 / / Λ t r S ( G m ) / / Λ t r S ( A 1 S ) / / ˜ Λ t r S ( A 1 S / G m ) / / 0 . Applying the functor C ∗ to it, relativ ely to the category of complex es of presheav es with transf ers, we obtain a distinguished tr iangle in D ( PSh tr ( S , Λ )) : C ∗ Λ t r S ( G m ) / / C ∗ Λ t r S ( A 1 S ) / / C ∗ ˜ Λ t r S ( A 1 S / G m ) / / C ∗ Λ t r S ( G m )[ 1 ] . T aking the associated long e xact sequence of cohomology presheav es, we obtain that the comple x of presheav es with transfers C ∗ ˜ Λ t r S ( A 1 S / G m ) is concentrated in cohomological degree 0 and − 1 , and w e get an e xact sequence of preshea v es: 0 / / ˆ H − 1 [ C ∗ ˜ Λ t r S ( A 1 S / G m )] / / ˆ H 0 [ C ∗ Λ t r S ( G m )] / / ˆ H 0 [ C ∗ Λ t r S ( A 1 S )] / / ˆ H 0 [ C ∗ ˜ Λ t r S ( A 1 S / G m )] / / 0 . By deﬁnition of the relative Picard g roup, given any smooth (aﬃne) scheme X , we get an ex act sequence of abelian groups: (11.2.11.1) 0 / / G m ( X ) / / Pic ( X × S P 1 S , X 0 t X ∞ ) / / Pic ( X × S P 1 S , X 0 ) / / 0 . Thus w e deduce that: ˆ H 0 [ C ∗ ˜ Λ t r S ( A 1 S / G m )] = 0 , ˆ H − 1 [ C ∗ ˜ Λ t r S ( A 1 S / G m )] = G m ⊗ Z Λ . This giv es in par ticular a canonical isomor phism: C ∗ ˜ Λ t r S ( A 1 S / G m )[− 1 ] ' G m ⊗ Z Λ in D ( PSh tr ( S , Λ )) . T aking its image in DM eﬀ ( S , Λ ) we obtain a canonical isomor- phism which can be written as: C ∗ Λ t r S ( A 1 S / G m )[− 1 ] ' G m ⊗ Z Λ . Thus w e can conclude because, according to Lemma 5.2.35 , the canonical map Λ t r S ( A 1 S / G m ) / / C ∗ Λ t r S ( A 1 S / G m ) is an isomor phism in DM eﬀ ( S , Λ ) .  11 Motivic complex es 315 Remar k 11.2.12 In the course of the proof, a canonical structure of a sheaf with transf ers o v er S on G m has naturall y appeared – described by the e xact sequence ( 11.2.11.1 ). This structure is classical (see [ MVW06 , Ex. 2.4]). One can describe it as f ollo ws. Let X and Y be smooth S -schemes. Assume X is connected (thus ir reducible as it is nor mal). Let Z be a closed integ ral subscheme Z of X × S Y which is ﬁnite surjective o v er X . Then Z / X cor responds to an e xtension of function ﬁelds L / K . The norm map of L / K induces a mor phism of abelian g roups: N Z / X : G m ( Z ) / / G m ( X ) . Then we associate with Z , seen as a ﬁnite cor respondence from X to Y , the f ollowing morphism: G m ( Y ) p ∗ / / G m ( Z ) N Z / X / / G m ( X ) where p : Y / / Z is the natural projection. The f ollowing proposition is well-kno wn to the e xper t. W e include a proof f or completeness. Proposition 11.2.13 F or any regular scheme X and any integ er i ≥ 0 , H i Nis ( X , G m ) =          O X ( X ) × if i = 0 , Pic ( X ) if i = 1 , 0 o ther wise wher e Pic ( X ) is the Picard gr oup of X . Proof Let Y be an y étale scheme ov er X . W e let C 0 ( V ) be the abelian g roup made b y the in v ertible rational functions on V and C 1 ( V ) be the group of 1 -codimensional cy cles in V . Classically , one associates with an y rational function f on V its W eil divisor div ( f ) ∈ C 1 ( V ) . Recall, when V is integral with function ﬁeld K , f ∈ K , one puts: div V ( f ) = Õ x ∈ V ( 1 ) v x ( f ) . x ; the sum r uns o v er the points of codimension 1 in V and v x ( f ) is the valuation of f corresponding to the valuation r ing O X , x . A ccording to this deﬁnition, we g et a complex: 0 / / G m ( V ) / / C 0 ( V ) div V / / C 1 ( V ) . This seq uence is functor ial with respect to pullback of étale X -schemes. Thus w e ha v e deﬁned a mor phism of presheav es on X ´ e t : π : G m / / C ∗ . Giv en any Nisnevich distinguished square Q (Example 2.1.11 ), one can check easil y that the image of Q b y C 0 (resp. C 1 ) is cocar tesian. As a consequence C ∗ is a comple x of Nisnevic h sheav es which satisﬁes the Brown-Gers ten proper ty – i.e. it 316 Motivic complex es and relative cycles is Nisnevich ﬂasque in the sense of Deﬁnition 5.1.9 according to Proposition 5.2.13 applied to the der iv ed category of Nisnevic h sheav es o v er X . On the other hand, π is a quasi-isomor phism of Nisnevic h sheav es o ver S : indeed it is well-kno wn that f or any regular local ring A , the sequence 0 / / A × / / F rac ( A ) × div A / / Z 1 ( A ) / / 0 is ex act. This is an easy consequence of the fact A is a unique factorization domain – the classical Auslander -Buchsbaum theorem, (e.g. [ Mat70 , 20.3]). In par ticular , we g et H i ( X , G m ) = H i ( C ∗ ( X )) and this concludes.  The f ollo wing theorem is a g eneralization of a w ell-known computation of V o- ev odsky for smooth schemes o v er a per f ect ﬁeld. The second case is a corollar y of the tw o preceding propositions. Theorem 11.2.14 Let S be a sc heme and n ∈ Z an integ er . The f ollowing computation holds: 1. H n , 0 M , eﬀ ( S , Λ ) = Hom DM eﬀ ( S ) ( Λ t r S , Λ t r S [ n ]) =  Λ π 0 ( S ) if n = 0 0 otherwise; 2. if S is regular , H n , 1 M , eﬀ ( S , Λ ) = Hom DM eﬀ ( S ) ( Λ t r S , Λ t r S ( 1 )[ n ]) =        O S ( S ) × ⊗ Z Λ if n = 1 Pic ( S ) ⊗ Z Λ if n = 2 0 otherwise Proof For the ﬁrst case, according to Proposition 10.2.5 , the sheaf Λ t r S is Nisnevich local and A 1 -local as a comple x of shea v es. It is ob viously acy clic for the Nisnevich topology . Thus, we conclude using again 10.2.5 in the case n = 0 . Consider the second case. According to Proposition 11.2.13 , the sheaf G m on Sm S is A 1 -local. Thus according to Proposition 11.2.11 G m ⊗ Λ [− 1 ] is an A 1 -resolution of Λ t r S ( 1 ) . In par ticular , Hom DM eﬀ ( S ) ( Λ t r S , Λ t r S ( 1 )[ n ]) = Hom D ( Sh tr ( S , Λ )) ( Λ t r S , G m ⊗ Λ [ n − 1 ]) = H n − 1 Nis ( S , G m ) ⊗ Λ where the second identiﬁcation f ollo ws from R emark 10.4.4 . The conclusion follo ws from another application of Proposition 11.2.13 .  The f ollo wing corollar y is a (v ery) w eak cancellation result in DM eﬀ ( S ) : Corollary 11.2.15 Let S be a r egular scheme. Then R Hom ( Λ t r S ( 1 ) , Λ t r S ( 1 )) = Λ t r S . Mor eov er , if m = 0 or m = 1 , for any integ er n > m , 11 Motivic complex es 317 R Hom ( Λ t r S ( n ) , Λ t r S ( m )) = 0 . Proof W e consider the ﬁrst asser tion. An y smooth S -scheme is regular . Hence, it is suﬃcient to pro v e that for any connected regular scheme S , for any integer n ∈ Z , Hom DM eﬀ ( S ) ( Λ t r S ( 1 ) , Λ t r S ( 1 )[ n ]) =  Λ if n = 0 0 other wise. Using the e xact triangle (11.2.15.1) Λ t r S ( G m ) / / Λ t r S ( A 1 ) / / Λ t r S ( 1 )[ 2 ] + 1 / / and the second case of the previous theorem, we obtain the follo wing long e xact sequence · · · / / Hom ( Λ t r S ( A 1 ) , Λ t r S ( 1 )[ n ]) / / Hom ( Λ t r S ( G m ) , Λ t r S ( 1 )[ n ]) / / Hom ( Λ t r S ( 1 ) , Λ t r S ( 1 )[ n − 1 ]) / / Hom ( Λ t r S ( A 1 ) , Λ t r S ( 1 )[ n + 1 ]) / / · · · Then w e conclude using the previous theorem and the fact Pic ( A 1 × S ) = Pic ( G m × S ) whenev er S is regular . For the last assertion, we are reduced to prov e that if S is a regular sc heme, f or an y integers n > 0 and i , Hom DM eﬀ ( S ) ( Λ t r S ( n ) , Λ t r S [ i ]) = 0 . This is obviousl y implied b y the case n = 1 . Consider the distinguished tr iangle ( 11.2.15.1 ) again. Then the long ex act sequence attached to the cohomological functor Hom DM eﬀ ( S , Λ ) (− , Λ t r S ) and applied to this triangle together with the ﬁrst case of the previous theorem allo ws us to conclude.  11.2.c The motivic cohomology ring spectrum 11.2.16 A ccording to deﬁnition 10.4.2 and paragraph 10.4.3 , we hav e an adjunction of abelian premotivic categor ies γ ∗ : Sh ( − , Λ ) / / o o Sh tr (− , Λ ) : γ ∗ such that γ ∗ is conser v ativ e and e xact. According to Paragraph 5.3.28 , it induces an adjunction of tr iangulated premotivic categories (11.2.16.1) L γ ∗ : D A 1 , Λ / / o o DM Λ : R γ ∗ . 318 Motivic complex es and relativ e cycles Composing with the premotivic adjunction between the s table homotopy categor y and the A 1 -derived homotopy categor y ( 5.3.35.1 ), we ﬁnally get a canonical premo- tivic adjunction: (11.2.16.2) ϕ ∗ : SH / / o o DM Λ : ϕ ∗ . Recall that, because ϕ ∗ is monoidal, ϕ ∗ is weakl y monoidal. In particular , f or an y scheme S , one gets canonical morphisms 1 S / / ϕ ∗ ( 1 S ) , ϕ ∗ ( 1 S ) ∧ ϕ ∗ ( 1 S ) / / ϕ ∗ ( 1 S ) which giv es a structure of a commutative monoid to the spectr um ϕ ∗ ( 1 S ) i.e. a r ing spectrum. Deﬁnition 11.2.17 Giv en an y scheme S , one deﬁnes the motivic cohomology ring spectrum ov er S with coeﬃcients in Λ as the commutative r ing spectr um: H Λ M , S : = ϕ ∗ ( 1 S ) . The proper ties of the functor ϕ ∗ immediately implies that the ring spectr um H Λ M , S represents motivic cohomology . One no w easily checks that this ring spectrum coincides with the or iginal one of V oev odsky (see [ V oe98 , sec. 6.1]) in the case Λ = Z . Theref ore, our deﬁnition of motivic cohomology (with Z -coeﬃcients) ag rees with that giv en by V oev odsky in loc. cit. 11.2.18 Consider a localization Λ 0 of Λ . Then one gets an essentially commutative diagram of r ight adjoints of premotivic adjunctions: D A 1 ( S , Λ ) ⊗ Λ Λ 0 t t DM ( S , Λ ) ⊗ Λ Λ 0 o o SH ( S ) D A 1 ( S , Λ 0 ) j j ( 1 ) O O DM ( S , Λ 0 ) ( 2 ) O O o o where the map ( 1 ) is the canonical equiv alence (see Proposition 5.3.37 ) and the map (2) is the equiv alence from ( 11.1.4.2 ). Note in par ticular that ( 2 ) is monoidal (as its reciprocal equivalence is monoidal as the left adjoint of a premotivic adjunction). Thus this essentially commutativ e diagram deﬁnes a canonical mor phism of r ing spectra: (11.2.18.1) H Λ M , S ⊗ Λ Λ 0 / / H Λ 0 M , S . As a corollar y of Proposition 11.1.5 , we get the f ollowing result: Proposition 11.2.19 The map ( 11.2.18.1 ) is an isomor phism of ring spectra. Remar k 11.2.20 In a previous v ersion of this te xt, we only g et the abov e result in particular cases. The main argument for the general case obtained abov e can be traced back to Proposition 8.1.53 . 11 Motivic complex es 319 11.2.21 Let f : T / / S be a mor phism of schemes. Recall from the str ucture of the premotivic adjunction ( ϕ ∗ , ϕ ∗ ) deﬁned abo v e that we g et an ex chang e mor phism: f ∗ ϕ ∗ / / ϕ ∗ f ∗ Applying this natural transf or mation to the unit object 1 S of DM ( S , Λ ) , one gets a canonical mor phism of ring spectra: τ f : f ∗ ( H Λ M , S ) / / H Λ M , T . Remark that this show s the collection ( H Λ M , S ) is a section of the ﬁbred categor y SH . Recall also the follo wing conjecture of V oev odsky ([ V oe02b , conj. 17]): Conjecture 11.2.22 (V oev odsky) F or any morphism f as abov e, the map τ f is an isomorphism. Remar k 11.2.23 At least, V oev odsky f ormulated this conjecture in the case where Λ = Z . According to the preceding proposition, this implies the case of any coeﬃcients ring Λ ⊂ Q . W e will solv e aﬃr mativ ely a par ticular case of this conjecture in 16.1.7 when Λ = Q . W e will see belo w that this conjecture of V oev odsky is strongl y related to the beha viour of the six operations in DM Λ ; see Proposition 11.4.7 . R ef erences f or other kno wn cases of variants of the conjecture may be found in Remark 11.4.8 . 11.3 Orientation and purity 11.3.1 For any scheme S , we let P ∞ S be the ind-scheme S / / P 1 S / / · · · / / P n S / / P n + 1 S / / made of the obvious closed immersions. This ind-scheme has a comultiplication giv en by the Seg re embedding P ∞ S × S P ∞ S / / P ∞ S Deﬁne Λ t r S ( P ∞ ) = lim / / Λ t r S ( P n ) . Applying Theorem 11.2.14 in the case S = Sp ec ( Z ) , w e obtain a canonical isomor phism: Hom DM eﬀ ( Spec ( Z ) , Λ ) ( Λ t r ( P ∞ ) , Λ t r ( 1 )[ 2 ]) = Pic ( P ∞ ) ⊗ Z Λ , whose aim is a free Λ -algebra of po wer ser ies in one variable. Considering the canonical dual in v er tible sheaf on P ∞ , w e obtain a canonical f or mal g enerator of this Λ -algebra and thus a mor phism DM eﬀ ( Sp ec ( Z ) , Λ ) : c 1 : Λ t r ( P ∞ ) / / Λ t r ( 1 )[ 2 ] . 320 Motivic complex es and relative cycles For any scheme S , considering the canonical projection f : S / / Sp ec ( Z ) , w e obtain b y pullback along f a morphism of DM eﬀ ( S , Λ ) c 1 , S : Λ t r S ( P ∞ S ) / / Λ t r S ( 1 )[ 2 ] . Consider G m as a sheaf of groups o v er Sm S . Follo wing [ MV99 , par t 4], we introduce its classifying space B G m as a simplicial sheaf o v er Sm S . From proposition 1.16 of loc. cit. , we get Hom H s • ( S ) ( S + , B G m ) = Pic ( S ) . Moreo v er , in the homotopy categor y of pointed simplicial shea v es H • ( S ) , we hav e a canonical isomor phism B G m ' P ∞ S ( cf. loc. cit. , prop. 3.7). Thus, ﬁnally , w e obtain a canonical map of pointed sets Pic ( S ) = Hom H s • ( S ) ( S + , B G m ) / / Hom H • ( S ) ( S + , P ∞ ) / / Hom DM eﬀ ( S , Λ ) ( Λ t r S , Λ t r S ( P ∞ /∗)) / / Hom DM eﬀ ( S , Λ ) ( Λ t r S , Λ t r S ( P ∞ )) . Deﬁnition 11.3.2 Consider the abo v e notations. W e deﬁne the ﬁrst motivic Cher n class as the f ollo wing composite mor phism c 1 : Pic ( S ) / / Hom DM eﬀ ( S , Λ ) ( Λ t r S , Λ t r S ( P ∞ S )) ( c 1 , S ) ∗ / / Hom DM eﬀ ( S , Λ ) ( Λ t r S , Λ t r S ( 1 )[ 2 ]) / / Hom DM ( S , Λ ) ( 1 S , 1 S ( 1 )[ 2 ]) = H 2 , 1 M ( S , Λ ) The ﬁrst motivic Chern class is evidentl y compatible with pullback. Remar k 11.3.3 Bew are that the map Pic ( S ) / / Hom DM eﬀ ( S , Λ ) ( Λ t r S , Λ t r S ( P ∞ S )) deﬁned abov e is not necessar il y a morphism of abelian groups. How ev er, the com- posite: Pic ( S ) / / Hom DM eﬀ ( S , Λ ) ( Λ t r S , Λ t r S ( P ∞ S )) ( c 1 , S ) ∗ / / Hom DM eﬀ ( S , Λ ) ( Λ t r S , Λ t r S ( 1 )[ 2 ]) is the isomorphism of Theorem 11.2.14 when S is normal. In par ticular , it is a morphism of abelian groups in this case. W e will give an ar gument belo w f or the general case. 11.3.4 The tr iangulated categor y DM ( S , Λ ) thus satisﬁes all the axioms of [ Dég08 , §2.1] (see also Paragraph 2.3.1 of loc. cit. in the regular case). In par ticular , we der iv e from the main results of loc. cit. the f ollo wing facts: 1. Let p : P / / S be a projectiv e bundle of rank n . Let c : 1 S / / 1 S ( 1 )[ 2 ] be the ﬁrst Cher n class of the canonical line bundle on P . Then the map M S ( P ) Í i p ⊗ c i / / n Ê i = 0 1 S ( i )[ 2 i ] is an isomor phism. This giv es the projectiv e bundle theorem in motivic coho- mology f or an y base scheme. 11 Motivic complex es 321 One deduces using the method of Grothendieck that motivic cohomology pos- sesses Cher n classes of v ector bundles which satisﬁes all the usual properties (see remark belo w f or additivity). 2. Let i : Z / / X be a closed immersion between smooth separated S -schemes of ﬁnite type. Assume i has pure codimension c and let j be the complementar y open immersion. Then there is a canonical purity isomorphism : p X , Z : M S ( X / X − Z ) / / M S ( Z )( c )[ 2 c ] . This deﬁnes in par ticular the Gysin triang le M S ( X − Z ) j ∗ / / M S ( X ) i ∗ / / M S ( Z )( c )[ 2 c ] ∂ X , Z / / M S ( X − Z )[ 1 ] . 3. Let f : Y / / X be a projective mor phism between smooth separated S -schemes of ﬁnite type. Assume f has pure relative dimension d . Then there is an associ- ated Gysin mor phism f ∗ : M S ( X ) / / M S ( Y )( d )[ 2 d ] functorial in f . W e ref er the reader to loc. cit f or v arious f or mulas inv olving the Gysin mor phism (projection f or mula, e xcess intersection,...) Note in par ticular that we deduce from that Gysin mor phism the f ollowing map in motivic cohomology: f ∗ : H n , i M ( Y , Λ ) / / H n + 2 d , i + d M ( X , Λ ) . 4. For any smooth projectiv e S -scheme X , the premotive M S ( X ) admits a strong dual . If X has pure relativ e dimension d o ver S , the strong dual of M S ( X ) is M S ( X )(− d )[− 2 d ] . Remar k 11.3.5 According to loc. cit. , there exis ts f or any scheme S a f or mal group law F S ( x , y ) with coeﬃcients in the g raded ring H 2 ∗ , ∗ M ( S , Λ ) . If one consider the Seg re embedding Σ : P ∞ S / / P ∞ S × S P ∞ S one has: F S ( x , y ) = σ ∗ ( 1 ) through the isomor phism: H 2 ∗ , ∗ M ( P ∞ S × S P ∞ S , Λ ) ' H 2 ∗ , ∗ M ( S , Λ )[[ x , y ]] which results from the projectiv e bundle formula in motivic cohomology . A ccording to Remark 11.3.3 , whenev er S is normal, one gets F S ( x , y ) = x + y . In par ticular , F Spec ( Z ) ( x , y ) = x + y . On the other hand, according to the abov e deﬁnition of F S ( x , y ) , F S ( x , y ) is compatible with pullback. Thus one deduces that F S ( x , y ) = x + y f or any scheme S . 11.3.6 A ccording to the proper ties that w e hav e previousl y pro v ed, motivic cohomol- ogy , and in par ticular the big raded par t H 2 n , n M ( X , Z ) , possesses many of the desired property of a generalized Chow theory for regular schemes (see [ BGI71 , XIV , §8]). 322 Motivic complex es and relative cycles Note in par ticular that the e xistence of Chern classes allo ws to deﬁne a Cher n character: K 0 ( X ) ⊗ Z Q c h / / H 2 ∗ , ∗ M ( X , Z ) ⊗ Q ' H 2 ∗ , ∗ M ( X , Q ) where the ﬁnal isomorphism f ollow s from Paragraph 11.2.2 . In particular, we will pro v e in the ne xt section (Corollar y 16.1.7 ) that, when X is regular , this map is an isomorphism as e xpected. Remar k 11.3.7 Among the good proper ties of motivic cohomology is the f act it is deﬁned, with its r ing str ucture and natural functor iality , other arbitrar y schemes. On the other hand, ev en when X is regular , one cannot describe at the moment the cohomology group H 2 n , n M ( X , Z ) in ter ms of classes of n -codimensional cy cles in X modulo an appropr iate equiv alence relation. Let us ho we ver mention the tw o f ollo wing interesting f acts: 1. Let X be a scheme of ﬁnite type o v er Sp ec ( Z ) and X p be its ﬁber o ver a pr imer p . Then one has a pullback map: H 2 n , n M ( X , Z ) / / H 2 n , n M ( X p , Z ) , σ  / / σ p . When X is an ar ithmetic scheme (regular and ﬂat o v er Z ) with good reduction at p , the targ et is the Cho w group of n -codimensional cycles (see Example 11.2.3 ). Then σ p should be thought as the specialization of its generic ﬁber (which lies in H 2 n , n M ( X Q , Z ) = C H n ( X Q ) according to the Example 11.2.3 ). This construction should coincide with other specialization maps in the ar ithmetic case (see f or e xample [ Ful98 , §20.3]). 2. Let X be a smooth S -scheme. Then any n -codimensional closed subscheme Z of X which is smooth ov er S deﬁnes using the Gysin mor phism an element [ Z ] = i ∗ ( 1 ) ∈ H 2 n , n M ( X , Z ) which should be called the fundamental class of X . One can e xtract from [ Dég08 ] some e xpected proper ties of these fundamental classes (relation to Cher n classes, pullback proper ties such as compatibility with base change). In par ticular , an y S -point of X deﬁnes an element of H 2 d , d M ( X , Z ) where d is the dimension of X (assumed of pure dimension). In par ticular , the group H 2 d , d M ( X , Z ) is close to a g roup of cy cles in X of relativ e dimension 0 o ver S . 11.3.8 W e end up this series of remarks on motivic cohomology with the follo wing construction that the reader might enjo y . Let S be any scheme and P S be the categor y of smooth projectiv e S -schemes. Giv en any scheme X and Y in P S , one can use the g roup H 2 d , d M ( X × S Y , Λ ) where d is the relativ e dimension of Y as a group of cor respondences using the properties obtained so far from motivic cohomology . In par ticular , one can mimic the construction of the category of Cho w motives ov er a ﬁeld k using the category P S 11 Motivic complex es 323 and these cor respondences. One obtains an additive monoidal categor y Chow 0 ( S , Λ ) of str ong Chow motiv es . A ccording to the duality proper ty of motiv es (Paragraph 11.3.4 , point 4) one also obtains a canonical isomor phism Hom DM ( S , Λ ) ( M S ( X ) , M S ( Y )) = H 2 d , d M ( X × S Y , Λ ) . Thus one deduces a canonical full embedding of monoidal categor ies: Cho w 0 ( S , Λ ) / / DM g m ( S , Λ ) which extends the w ell-kno wn case when S is a per f ect ﬁeld. Remar k 11.3.9 Bew are that, with rational coeﬃcients, a sharper notion of Cho w motiv es – in more precise terms, these are motives of w eight zero – ha ve been introduced recently (see [ Héb11 ], [ Bon14 ]). 11.4 The six functors 11.4.1 Recall that according to Deﬁnition 10.4.2 and Paragraph 10.4.3 , w e ha v e an adjunction of abelian premotivic categor ies γ ∗ : Sh ( − , Λ ) / / o o Sh tr (− , Λ ) : γ ∗ such that γ ∗ is e xact and conservativ e. Moreov er , f or an y scheme S , any smooth S -schemes X , Y and an y open immersion j : U / / X , the canonical map: j ∗ : c S ( Y , U ) / / c S ( Y , X ) is obviousl y a monomor phism. Thus the abelian premotivic category Sh tr (− , Λ ) satisﬁes the assumptions (i)-(iv) of Paragraph 6.3.1 . In par ticular , w e deduce from Corollaries 6.3.12 and 6.3.15 the follo wing theorem: Proposition 11.4.2 The premo tivic triangulated categor y DM Λ satisﬁes the support property . Mor eov er , f or any sc heme S and any closed immersion i : Z / / X betw een smooth S -schemes, DM Λ satisﬁes the localization property with respect to i , (Loc i ). An important corollary of this proposition is that giv en any separated mor phism f : Y / / X of ﬁnite type, one can construct an adjunction of tr iangulated categor ies: f ! : DM ( Y , Λ ) / / o o DM ( X , Λ ) : f ! such that f ! = f ∗ when f is proper (see Section 2.2 ). W e will elaborate on this fact at the end of this section. 11.4.3 Note that in par ticular , the premotivic categor y DM Λ satisﬁes the w eak localization property (wLoc). According to the premotivic adjunction ( 11.2.16.2 ) 324 Motivic comple xes and relative cycles and the e xistence of the ﬁrst Cher n class in motivic cohomology (Deﬁnition 11.3.2 ), one can apply Example 2.4.40 to the premotivic tr iangulated categor y DM Λ (which satisﬁes the Nisnevich separation property by construction). This implies in particular that DM Λ is oriented as a premotivic tr iangulated category (Deﬁnition 2.4.38 ). In par ticular , one can appl y Corollar y 2.4.43 to DM Λ and get the f ollo wing result: Proposition 11.4.4 Any smoot h projectiv e mor phism f is DM Λ -pur e: the canonical purity map ( 2.4.39.3 ) f ] / / f ! ( d )[ 2 d ] , is an isomorphism wher e d is the relativ e dimension of f . In par ticular , DM Λ is weakl y pure. The only property of the premotivic triangu- lated category DM Λ that w e cannot pro v e is the localization proper ty f or general closed immersions. Ho w ev er , the proper ties we ha v e seen so far allo ws to construct the 6 operations and establish some of its proper ties that are already of interest. Let us summarize this f ormalism, from Theorem 2.2.14 together with Lemma 2.4.23 : Theorem 11.4.5 F or any separat ed mor phism of ﬁnite type f : Y / / X , ther e exists an essentially unique pair of adjoint functors f ! : DM ( Y , Λ ) / / o o DM ( X , Λ ) : f ! suc h that : 1. Ther e exists a structur e of a cov ariant (resp. contrav ariant) 2 -functor on f  / / f ! (r esp. f  / / f ! ). 2. Ther e exists a natur al transf ormation α f : f ! / / f ∗ whic h is an isomorphism when f is pr oper . Mor eov er , α is a mor phism of 2 -functor s. 3. F or any smooth pr ojective mor phism f : X / / S of relativ e dimension d , ther e ar e canonical natural isomorphisms p t f : f ] / / f ! ( d )[ 2 d ] p 0 t f : f ∗ / / f ! (− d )[− 2 d ] whic h are dual to each other . 4. F or any car tesian squar e: Y 0 f 0 / / g 0   ∆ X 0 g   Y f / / X , suc h that f is separ ated of ﬁnite type, ther e exist natur al transf or mations g ∗ f ! ∼ / / f 0 ! g 0 ∗ , g 0 ∗ f 0 ! ∼ / / f ! g ∗ , whic h are isomor phisms in the f ollowing cases: 11 Motivic complex es 325 • g is smooth; • f is projectiv e and smooth. 5. F or any smooth projectiv e morphism f : Y / / X , ther e exist natural isomor - phisms E x ( f ∗ ! , ⊗ ) : ( f ! K ) ⊗ X L ∼ / / f ! ( K ⊗ Y f ∗ L ) , Hom X ( f ! ( L ) , K ) ∼ / / f ∗ Hom Y ( L , f ! ( K )) , f ! Hom X ( L , M ) ∼ / / Hom Y ( f ∗ ( L ) , f ! ( M )) . Remar k 11.4.6 As an example of application, let us recall the construction of the general trace map (from [ A GV73 ]) in the case of a smooth projective mor phism f : Y / / X of relative dimension d . It is the f ollo wing composite map: f ∗ f ∗ α − 1 f / / f ! f ∗ p 0 t f / / f ! f ! ( d )[ 2 d ] a d 0 ( f ! , f ! ) / / 1 ( d )[ 2 d ] . This allow s one to recov er the Gysin map associated with f , already constr ucted in Paragraph 11.3.4 , as w ell as the duality proper ty for the motive M X ( Y ) . W e will ref ormulate V oev odsky’ s conjecture 11.2.22 in terms of the six operations as f ollo ws. Proposition 11.4.7 W e ﬁx a base sc heme S as w ell as a ring of coeﬃcients Λ . The f ollowing assertions are equivalent : (i) f or any S -schemes X and Y and any morphism of ﬁnite type f : X / / Y , the canonical map τ f : f ∗ ( H Λ M , X ) / / H Λ M , Y is inv ertible; (ii) f or any S -scheme X , the canonical functor Ho ( H Λ M , X - mo d ) / / DM ( X , Λ ) is an equivalence of categories, and DM (− , Λ ) is a motivic category ov er S - sc hemes; (iii) the premo tivic category DM (− , Λ ) has the localization property for S -schemes; (iv) DM (− , Λ ) is a motivic category ov er S -sc hemes. Proof The fact that properties (iii) and (iv) are equivalent is ob vious, since the only missing property that is not kno wn f or DM Λ to be a motivic category is the localization proper ty . Condition (iv) is obviousl y a consequence of condition (ii). Keeping track of notations introduced in paragraph 11.2.16 , we shall obser v e that the f org etful functor ϕ ∗ : DM Λ / / SH commutes with the operator j ] , f or any open immersion j , as f ollow s. Since the f orgetful functor from D A 1 , Λ to SH is conservativ e and commutes with j ] f or an y open immersion j , it is suﬃcient to prov e that the functor R γ ∗ : DM Λ / / D A 1 , Λ has the same proper ty , which is precisel y Proposition 6.3.11 . 326 Motivic complex es and relativ e cycles Let us check that condition (i) (i.e. V oev odsky’ s conjecture 11.2.22 ) is a con- sequence of condition (iv). Let us assume that (iv) holds true, and that w e ha ve a morphism of ﬁnite type f : X / / Y . The proper ty that the canonical map τ f : f ∗ ( H Λ M , X ) / / H Λ M , Y is in vertible is local f or the Zar iski topology on X and on Y , so that we may assume that f is aﬃne. Since the map τ f is in v er tible for f smooth, w e observe from there that it is suﬃcient to pro v e that τ f is inv er tible when f is a closed immersion. Let j : U / / Y be the open immersion complement to f . Assuming (iv), there is a homotop y coﬁber sequence of the form j ] 1 U / / 1 Y / / f ∗ 1 X in DM ( Y , Λ ) , the image of which is isomor phic to the homotopy coﬁber sequence j ] H Λ M , U / / H Λ M , Y / / f ∗ H Λ M , X in SH ( Y ) , since the functor ϕ ∗ commutes with j ] (as recalled abo v e) and with f ∗ (f or obvious reasons). But the localization property in SH implies that the homo- top y coﬁber of the map j ] H Λ M , U / / H Λ M , Y is f ∗ f ∗ H Λ M , Y . Since the functor f ∗ is conservativ e in SH (being fully f aithful), this show s that the map τ f is in vertible. Let us assume that condition (i) is true. Since the f org etful functor ϕ ∗ is conser va- tiv e and commutes with i ∗ f or any closed immersion i , in order to prov e that condition (iv) holds, i.e. that DM Λ has the localization proper ty , it is suﬃcient to prov e that condition (ii) of Corollar y 2.3.18 is v er iﬁed in DM Λ . W e observe fur thermore that, f or an y smooth and projectiv e mor phism of S -schemes p : X / / Y ev er y where of relativ e dimension d , the functor ϕ ∗ commutes with p ] . Indeed, f or an y object M in DM ( X , Λ ) , w e hav e: ϕ ∗ p ] ( M ) ' ϕ ∗ p ∗ ( M )( d )[ 2 d ] ' p ∗ ϕ ∗ ( M )( d )[ 2 d ] ' p ! ( T h X ( T f ) ⊗ ϕ ∗ ( M )) ' p ] ϕ ∗ ( M ) (where the identiﬁcation T h X ( T f ) ⊗ ϕ ∗ ( M ) ' ϕ ∗ ( M )( d )[ 2 d ] comes from the or i- entation on ϕ ∗ ( M ) induced by its H Λ M , X -module str ucture). This implies that the functor ϕ ∗ commutes with f ] f or any smooth mor phism of S -schemes f . Indeed, this is a local condition with respect to the Zar iski topology both on the source and on the targ et of f , so that it is suﬃcient to check the case where f is quasi-projectiv e. Since the case where f is an open immersion is already known, and since w e just discussed the case where f is a smooth and projectiv e, this pro v es our claim. Finall y , w e obser v e that, giv en a closed immersion i : Z / / X as well as a smooth mor phism f : Y / / X , the diag ram g ] H Λ M , f − 1 ( X − Z ) / / f ] H Λ M , Y / / i ∗ i ∗ H Λ M , Y 11 Motivic complex es 327 is a homotop y coﬁber sequence, where g : f − 1 ( X − Z ) / / X is the restriction of f . Since the functor ϕ ∗ is conservativ e and commutes with f ] , g ] and i ∗ , this pro v es that DM Λ has the localization proper ty , b y Corollar y 2.3.18 . If condition (i) is tr ue, then, b y vir tue of Proposition 7.2.13 , there is a mor phism of premotivic categor ies α ∗ : Ho ( H Λ M - mo d ) / / o o DM Λ : α ∗ . Further more, Proposition 7.2.18 implies that, under condition (i), Ho ( H Λ M - mo d ) is a motivic categor y (in particular, has the localization property). W e just saw that DM Λ is a motivic category as well. T o pro ve that the functor α ∗ is an equivalence of categories, b y virtue of Corollar y 1.3.20 , it is suﬃcient to pro v e that, f or any smooth morphism f : X / / Y , the unit map f ] H Λ M , X / / α ∗ α ∗ f ] H Λ M , X ' α ∗ f ] α ∗ H Λ M , X is in v er tible. Since the operators α ∗ and f ] commute (when w e f orget the H Λ M , X - module structure, α ∗ is just ϕ ∗ ), it is suﬃcient to check this proper ty when f is the identity . But the map H Λ M , X / / α ∗ α ∗ H Λ M , X is in vertible (in fact the identity), b y deﬁnition.  Remar k 11.4.8 A v ar iant of V oev odsky’ s conjecture would be that the map τ f : f ∗ ( H Λ M , X ) / / H Λ M , Y is inv er tible f or regular S -schemes. W e invite the reader to chec k that this v ersion of the conjecture may be ref ormulated as in Proposition 11.4.7 (restricting ourselv es to regular S -schemes, obviousl y), essentially with the same proof. Evidence for this weak er f orm of the conjecture is given by the fact that ov er an y ﬁeld of exponent characteristic p , it is true with Λ = Z [ 1 / p ] ; see [ CD15 ]. A variant consists in replacing DM Λ b y its c dh -local version. In equal characteristic zero, this is pro v ed f or possibly singular scheme in [ CD15 ] (in characteristic p > 0 , this also holds up to p -torsion). The c dh -local version of H Λ M should be isomor phic to Spitzw eck’ s motivic cohomology spectr um [ Spi18 ]. P art IV Beilinson motiv es and alg ebraic K -theory 12 Stable homotopy theor y of schemes 331 11.4.9 In all this par t, S is assumed to be the category of noether ian schemes of ﬁnite dimension. 12 Stable homotop y theory of schemes 12.1 Ring spectra Consider a base scheme S . Recall that a ring spectrum E ov er S is a monoid object in the monoidal category SH ( S ) . W e sa y that E is commutativ e if it is commutative as a monoid in the symmetric monoidal category SH ( S ) . In what follo ws, we will assume that all our ring spectra are commutativ e without mentioning it. The premotivic categor y is Z 2 -graded where the ﬁrst inde x ref ers to the simplicial sphere and the second one to the T ate twis t. According to our general conv ention, a cohomology theor y representable in SH is Z 2 -graded accordingl y: given such a ring spectrum E , for any smooth S -scheme X , and any integer ( i , n ) ∈ Z 2 , we get a bigraded r ing: E n , i ( X ) = Hom SH ( S )  Σ ∞ X + , E ( i )[ n ]  . When X is a pointed smooth S -scheme, it deﬁnes a pointed sheaf of sets still denoted b y X and w e denote by ˜ E n , i ( X ) f or the cor responding cohomology ring. The coeﬃcient ring associated with E is the cohomology of the base E ∗∗ : = E ∗∗ ( S ) . The ring E ∗∗ ( X ) (resp. ˜ E ∗∗ ( X ) ) is in fact an E ∗∗ -algebra. 12.1.1 W e sa y E is a strict ring spectrum if there exis ts a monoid in the categor y of symmetric T ate spectra E 0 and an isomor phism of ring spectra E ' E 0 in SH ( S ) . In this case, a module M ov er the monoid E in the monoidal category SH ( S ) will be said to be strict if there exis ts an E 0 -module M 0 in the categor y of symmetric T ate spectra, as well as an isomor phism of E -modules M ' M 0 in SH ( S ) . 12.2 Orientation 12.2.1 Consider the inﬁnite tow er P 1 S / / P 2 S / / · · · / / P n S / / · · · of schemes pointed by the inﬁnity . W e denote by P ∞ S the limit of this tow er as a pointed Nisnevich sheaf of sets and by ι : P 1 S / / P ∞ S the induced inclusion. Recall the f ollo wing deﬁnition, classically translated from topology: Deﬁnition 12.2.2 Let E be a r ing spectr um o ver S . An orientation of E is a coho- mology class c in ˜ E 2 , 1 ( P ∞ S ) such that ι ∗ ( c ) is sent to the unit of the coeﬃcient r ing of E b y the canonical isomor phism ˜ E 2 , 1 ( P 1 S ) = E 0 , 0 . 332 Beilinson motives and algebraic K -theory W e then say that ( E , c ) is an oriented ring spectr um . W e shall say also that E is orientable if there exis ts an or ientation c . A ccording to [ MV99 , 1.16 and 3.7], we get a canonical map f or any smooth S -scheme X Pic ( X ) = H 1 ( X , G m ) / / Hom H • ( S ) ( X + , P ∞ ) / / Hom SH ( S ) ( Σ ∞ X + , Σ ∞ P ∞ ) (the ﬁrs t map is an isomor phism whene v er S is regular (or ev en geometrically unibranch)). Given this map, an orientation c of a r ing spectrum E deﬁnes a map of sets c 1 , X : Pic ( X ) / / E 2 , 1 ( X ) which is natural in X (and from its construction in [ MV99 ], one can check that c = c 1 , P ∞ S ( O ( 1 )) ). Usually , we put c 1 = c 1 , X . Example 12.2.3 1. The or iginal e xample of an oriented r ing spectr um is the alge- braic cobordism spectr um MGL introduced b y V oev odsky ( cf. [ V oe98 ]). 2. A ccording to Deﬁnition 11.3.2 , the motivic cohomology ring spectrum H Λ M , S deﬁned in 11.2.17 is an or iented r ing spectrum. 3. Consider a triangulated premotivic categor y T which satisﬁes the weak local- ization proper ty (wLoc) and such that there exis ts an adjunction of tr iangulated premotivic categories: ϕ ∗ : SH / / o o T : ϕ ∗ . Recall that ϕ ∗ is symmetr ic monoidal. Thus, its r ight adjoint is w eakly symmetr ic monoidal and f or any the spectr um H T , S : = ϕ ∗ ( 1 S ) admits a (commutativ e) r ing structure. Then T is or iented in the sense of Deﬁnition 2.4.38 if and only if the ring spectrum H T , S is oriented in the sense of Deﬁnition 12.2.2 – see Example 2.4.40 . Moreov er , an orientation of T is equiv alent to the data of or ientations H T , S f or an y scheme S which are stable by pullbac ks (on cohomology). Remar k 12.2.4 When E is a strict r ing spectr um, the categor y E - mo d satisﬁes the axioms of [ Dég08 , 2.1] (see example 2.12 of loc.cit. ). Recall the follo wing result, which ﬁrst appeared in [ V ez01 ]: Proposition 12.2.5 (Morel) Let ( E , c ) be an oriented ring spectrum. Then: E ∗∗ ( P ∞ S ) = E ∗∗ [[ c ]] and E ∗∗ ( P ∞ S × P ∞ S ) = E ∗∗ [[ x , y ]] , wher e x (resp. y ) is the pullbac k of c along the ﬁrs t (resp. second) projection. Remar k 12.2.6 When E is a strict r ing spectr um, this is [ Dég08 , 3.2] according to remark 12.2.4 . The proof f ollo ws an argument of Morel ([ Dég08 , lemma 3.3]) and the arguments of op.cit. , p. 634, can be easily used to obtain the proposition arguing directly f or the cohomology functor X  / / E ∗ , ∗ ( X ) . 12 Stable homotopy theor y of schemes 333 12.2.7 Recall that the Seg re embedding P n S × P m S / / P n + m + nm S deﬁne a map σ : P ∞ S × P ∞ S / / P ∞ S . It giv es the structure of an H -group to P ∞ S in the homotopy categor y H ( S ) . Con- sider the h ypothesis of the pre vious proposition. Then the pullback along σ in E -cohomology induces a map E ∗∗ [[ c ]] σ ∗ / / E ∗∗ [[ x , y ]] and follo wing Quillen, we check that the f or mal pow er series σ ∗ ( c ) deﬁnes a f or mal group law ov er the r ing E ∗∗ . Deﬁnition 12.2.8 Let ( E , c ) be an or iented r ing spectrum and consider the previous notation. The f ormal group law F E ( x , y ) : = σ ∗ ( c ) will be called the f or mal g roup la w associated to ( E , c ) . Recall the formal group law has the f orm: F E ( x , y ) = x + y + Õ i + j > 0 a i j . x i y j with a i j = a j i in E − 2 i − 2 j , − i − j . The coeﬃcients a i j describe the failure of additivity of the ﬁrst Cher n class c 1 . Indeed, assuming the previous deﬁnition, w e get the f ollowing result: Proposition 12.2.9 Let X be a smooth S -sc heme. 1. F or any line bundle L / X , the class c 1 ( L ) is nilpotent in E ∗∗ ( X ) . 2. Suppose X admits an ample line bundle. F or any line bundles L , L 0 ov er X , c 1 ( L 1 ⊗ L 2 ) = F E ( c 1 ( L 1 ) , c 1 ( L 2 )) ∈ E 2 , 1 ( X ) . W e refer to [ Dég08 , 3.8] in the case where E is strict; the proof is the same in the general case. Recall the follo wing theorem of V ezzosi ( cf. [ V ez01 , 4.3]): Theorem 12.2.10 (V ezzosi) Let ( E , c ) be an oriented spectra ov er S , with f or mal gr oup law F E . Then ther e exists a bijection betw een the f ollowing sets: (i) Orientation classes c 0 of E . (ii) Morphisms of ring spectra MGL / / E in SH ( S ) . (iii) Couples ( F , ϕ ) where F is a formal g roup law ov er E ∗∗ and ϕ is a pow er series ov er E ∗∗ whic h deﬁnes an isomor phism of formal gr oup law : ϕ is invertible as a pow er series and F E ( ϕ ( x ) , ϕ ( y )) = F ( x , y ) . 334 Beilinson motives and algebraic K -theor y 12.3 Rational category In what f ollow s, we shall use frequently the equivalence of premotivic categor ies (see 5.3.35 ) SH Q / / o o D A 1 , Q , and shall identify freely an y rational spectrum ov er a scheme S with an object of D A 1 ( S , Q ) . 13 Algebraic K -theory 13.1 The K -theory spectrum W e consider the spectrum KGL S which represents homotopy in variant K -theor y in SH ( S ) according to V oev odsky (see [ Cis13 ], [ V oe98 , 6.2], [ Rio10 , 5.2] and [ PPR09 ]). It is characterized by the f ollo wing proper ties: (K1) For any morphism of schemes f : T / / S , there is an isomor phism f ∗ K GL S ' K GL T in SH ( T ) . (K2) For any regular scheme S and an y integer n , there exis ts an isomor phism Hom SH ( S ) ( 1 S [ n ] , KGL S ) / / K n ( S ) (where the r ight-hand side is Quillen algebraic K -theory as deﬁned by Thomason and T robaugh, [ TT90 ], in the case where S does not admit an ample famil y) such that, for any mor phism f : T / / S of regular schemes, the f ollo wing diagram is commutativ e: Hom ( 1 S [ n ] , KGL S ) / /   Hom ( f ∗ 1 S [ n ] , f ∗ K GL S ) Hom ( 1 T [ n ] , KGL T )   K n ( S ) f ∗ / / K n ( T ) (where the lo w er horizontal map is the pullback in Quillen algebraic K -theor y along the mor phism f and the upper hor izontal map is obtained by using the functor f ∗ : SH ( S ) / / SH ( T ) and the identiﬁcation (K1)). (K3) For any scheme S , there e xists a unique structure of a commutativ e monoid on K GL S which is compatible with base change – using the identiﬁcation (K1) – and induces the canonical r ing structure on K 0 ( S ) . Thus, according to (K1) and (K3), the collection of the r ing spectr um K GL S f or any scheme S f or m an absolute r ing spectr um. As usual, when no confusion can ar ise, w e will not indicate the base in the notation KGL . Note that (K1) implies f or mall y that the isomor phism of (K2) can be e xtended f or an y smooth S -scheme X (with S regular), giving a natural isomor phism: 13 Algebraic K -theor y 335 Hom SH ( S ) ( Σ ∞ X + [ n ] , KGL ) / / K n ( X ) . 13.2 P eriodicity 13.2.1 Recall from the constr uction the f ollowing proper ty of the spectr um KGL : (K4) the spectr um KGL is a P 1 -periodic spectrum in the sense that there exis ts a canonical isomor phism K GL ∼ / / R Hom  Σ ∞ P 1 S , KGL  = KGL (− 1 )[− 2 ] . As usual, P 1 S is pointed by the inﬁnite point. This isomor phism, classically called the Bott isomorphism, is characterized uniquel y b y the fact that its adjoint isomor phism (obtained by tensor ing with 1 S ( 1 )[ 2 ] ) is equal to the composite (13.2.1.1) γ u : K GL ( 1 )[ 2 ] 1 ⊗ u / / K GL ∧ KGL µ / / K GL . where u : Σ ∞ P 1 / / K GL corresponds to the class [ O ( 1 )] − 1 in ˜ K 0 ( P 1 ) through the isomor phism (K2) and µ is the structural map of monoid from (K3). Using the isomor phism of (K4), the proper ty (K1) can be e xtended as follo ws: f or any smooth S -scheme X and an y integers ( i , n ) ∈ Z 2 , there is a canonical isomorphism: (13.2.1.2) K GL n , i ( X ) ∼ / / K 2 i − n ( X ) . Remar k 13.2.2 The element u is in vertible in the r ing KGL ∗ , ∗ ( S ) . Its in v erse is the Bott element β ∈ KGL 2 , 1 ( S ) . If we chose as an or ientation of the r ing spectr um K GL ( cf. 12.2.2 ) the class β . ([ O ( 1 )] − 1 ) ∈ KGL 2 , 1 ( P ∞ ) , the cor responding f or mal group law is the multiplicativ e f or mal group law : F ( x , y ) = x + y + β − 1 . x y . 13.3 Modules o ver algebraic K -theory Theorem 13.3.1 (Østvær , Röndigs, Spitzwec k) The spectrum KGL can be repr e- sented canonically by a cartesian monoid KGL 0 , as well as by a homotopy cartesian commutativ e monoid KGL β in the ﬁbred model categor y of symmetric P 1 -spectra, 336 Beilinson motiv es and algebraic K -theor y in such a way that ther e exists a morphism of monoids KGL 0 / / K GL β whic h is a termwise stable A 1 -equiv alence. Proof For any noether ian scheme of ﬁnite dimension S , one has a str ict commutative ring spectrum KGL β S which is canonicall y isomorphic to KGL S in SH ( S ) as r ing spectra; see [ RSØ10 ]. One can check that the objects KGL β S do form a commuta- tiv e monoid ov er the diagram of all noetherian sc hemes of ﬁnite dimension (i.e. a commutativ e monoid in the categor y of sections of the ﬁbred categor y of P 1 -spectra o v er the categor y of noetherian schemes of ﬁnite dimension), either by hand, by f ollowing the e xplicit construction of loc. cit. , either b y modifying its construction v ery slightly as f ollow s: one can perform mutatis mutandis the constr uction of loc. cit. in the P 1 -stabilization of the A 1 -localization of the model ca tegory of Nisnevic h simplicial sheav es o v er (any essentially small adequate subcategory of ) the categor y of all noether ian schemes of ﬁnite dimension, and get an object KGL β , whose re- striction to each of the categories Sm / S is the object KGL β S . From this point of view , w e clearl y hav e canonical maps f ∗ ( K GL β S ) / / K GL β T f or an y mor phism of schemes f : T / / S . The object KGL β is homotop y car tesian, as the composed map L f ∗ ( K GL S ) ' L f ∗ ( K GL β S ) / / f ∗ ( K GL β S ) / / K GL β T ' KGL T is an isomor phism in SH ( T ) . Consider now a coﬁbrant resolution K GL 0 Spec ( Z ) / / K GL β Spec ( Z ) in the model category of monoids of the categor y of symmetr ic P 1 -spectra o v er Sp ec ( Z ) ; see Theorem 7.1.3 . Then, we deﬁne, f or each noether ian scheme of ﬁnite dimension S , the P 1 -spectrum KGL 0 S as the pullback of KGL 0 Spec ( Z ) along the map f : S / / Sp ec ( Z ) . As the functor f ∗ is a left Quillen functor , the object KGL 0 S is coﬁbrant (both as a monoid and as a P 1 -spectrum), so that we get, b y constr uction, a ter m wise coﬁbrant car tesian strict P 1 -ring spectrum KGL 0 , as well as a mor phism K GL 0 / / K GL β which is a termwise stable A 1 -equiv alence.  13.3.2 For each noether ian scheme of ﬁnite dimension S , one can consider the model categories of modules o v er K GL 0 S and KGL β S respectiv ely ; see 7.2.2 . The change of scalars functor along the stable A 1 -equiv alence K GL 0 S / / K GL β S deﬁnes a left Quillen equiv alence, whence an equivalence of homotopy categories: Ho ( K GL 0 S - mo d ) ' Ho ( KGL β S - mo d ) . Deﬁnition 13.3.3 W e deﬁne the premotivic tr iangulated category of K GL -modules o v er S as the ﬁbred tr iangulated categor y whose ﬁber o v er a scheme S in S is deﬁned as: K GL - mo d ( S ) : = Ho ( KGL β S - mo d ) . 13.3.4 By deﬁnition, for any smooth S -scheme X , we ha v e a canonical isomor phism Hom SH ( S ) ( Σ ∞ ( X + ) , KGL [ n ]) ' Hom KGL ( K GL S ( X ) , KGL [ n ]) 13 Algebraic K -theor y 337 (where KGL S ( X ) = KGL S ∧ L S Σ ∞ ( X + ) , while Hom KGL stands f or Hom KGL - mod ( S ) ). A ccording to (K1) and (K3), f or an y regular scheme X , we thus get a canonical isomorphism: (13.3.4.1)  S : Hom KGL ( K GL S [ n ] , KGL S ) ∼ / / K n ( S ) . Using Bott periodicity (K4), and the compatibility with base chang e, this isomor- phism can be extended f or any smooth S -scheme X and an y pair ( n , m ) ∈ Z 2 : (13.3.4.2)  X / S : Hom KGL ( K GL S ( X ) , KGL S ( m )[ n ]) ∼ / / K 2 m − n ( X ) . Corollary 13.3.5 The premo tivic triangulated category K GL - mo d ) f orm a motivic category, and the functor s SH ( S ) / / K GL - mo d ( S ) , M  / / K GL S ∧ L S M f or a scheme S in S deﬁne a morphism of motivic categories SH / / K GL - mo d ov er the categor y of noet herian schemes of ﬁnite dimension. Proof This f ollo ws from the preceding theorem and from 7.2.13 and 7.2.18 .  13.4 K -theory with support 13.4.1 Consider a closed immersion i : Z / / S with complementary open immer - sion j : U / / S . Assume S is regular . W e use the deﬁnition of [ Gil81 , 2.13] f or the K -theor y of S with suppor t in Z denoted b y K Z ∗ ( S ) . In other w ords, we deﬁne K Z ( S ) as the homotopy ﬁber of the restriction map R Γ ( S , KGL S ) = K ( S ) / / K ( U ) = R Γ ( U , KGL U ) , and put: K Z n ( S ) = π n ( K Z ( S )) . Applying the der iv ed global section functor R Γ ( S , −) to the homotop y ﬁber sequence (13.4.1.1) i ! i ! K GL S / / K GL S / / j ∗ j ∗ K GL S , w e get a homotop y ﬁber sequence (13.4.1.2) R Γ ( S , i ! i ! K GL S ) / / R Γ ( S , KGL S ) / / R Γ ( U , K GL S ) from which w e deduce an isomor phism in the stable homotop y categor y of S 1 - spectra: 338 Beilinson motiv es and algebraic K -theor y (13.4.1.3) R Γ ( Z , i ! K GL S ) = R Γ ( S , i ! i ! K GL S ) ' K Z ( S ) . W e thus get the follo wing proper ty : (K6) There is a canonical isomor phism Hom SH ( S )  1 S [ n ] , i ! i ! K GL S  / / K Z n ( S ) which satisﬁes the f ollo wing compatiblities: (K6a) the f ollo wing diagram is commutativ e: Hom ( 1 [ n + 1 ] , j ∗ j ∗ K GL S ) / /   Hom  1 [ n ] , i ! i ! K GL S  / /   Hom ( 1 [ n ] , KGL S )   K n + 1 ( U ) / / K Z n ( S ) / / K n ( S ) where the upper hor izontal arrow s are induced b y the localization sequence ( 13.4.1.1 ), and the lo wer one is the canonical sequence of K -theor y with suppor t. The e xtreme left and r ight v er tical maps are the isomor phisms of (K2); (K6b) f or an y mor phism f : Y / / S of regular schemes, k : T / / Y the pullback of i along f , the follo wing diag ram is commutativ e: Hom ( 1 [ n ] , i ! i ! K GL S ) / /   Hom ( f ∗ 1 [ n ] , f ∗ i ! i ! K GL S ) / / Hom ( 1 [ n ] , k ! k ! K GL Y )   K Z n ( S ) f ∗ / / K T n ( Y ) where the lo w er horizontal map is given by the functor iality of relative K -theor y (induced b y the funtoriality of K -theor y) and the upper one is obtained using the functor f ∗ of SH , the canonical e x chang e mor phism f ∗ i ! i ! / / k ! k ! f ∗ and the identiﬁcation (K1). This proper ty can be extended to the motivic categor y Ho ( KGL - mo d ) and we get a canonical isomor phism (13.4.1.4)  i : Hom KGL ( K GL S [ n ] , i ! i ! K GL S ) ∼ / / K Z n ( S ) satisfying the analog of (K6a) and (K6b). 13.5 Fundamental class 13.5.1 Consider a car tesian square of regular schemes 13 Algebraic K -theor y 339 Z 0 k / / g   S 0 f   Z i / / S with i a closed immersion. W e will say that this square is T or -independant if Z and S 0 are T or -independent o ver S in the sense of [ BGI71 , III, 1.5]: f or an y i > 0 , T or S i ( O Z , O S 0 ) = 0 . 91 In this case, when we assume in addition that all the schemes in the previous square are regular and that i is a closed immersion we get from [ TT90 , 3.18] 92 the f or mula f ∗ i ∗ = k ∗ g ∗ : K ∗ ( Z ) / / K ∗ ( S 0 ) in Quillen K -theory . An impor tant point f or us is that there exis ts a canonical homo- topy between these mor phisms at the lev el of the W aldhausen spectra. 93 According to the localization theorem of Quillen [ Qui73 , 3.1], we get: Theorem 13.5.2 (Quillen) F or any closed immer sion i : Z / / S betw een r egular sc hemes, ther e exists a canonical isomor phism q i : K Z n ( S ) / / K n ( Z ) . Mor eov er , this isomorphism is functorial with respect to the T or -independent squar es as abov e, with i a closed immersion and all the schemes regular . Remar k 13.5.3 In the condition of this theorem, the follo wing diag ram is commutativ e b y construction: K Z n ( S ) + + q i   K n ( S ) K n ( Z ) i ∗ 3 3 where the non labeled map is the canonical one. Deﬁnition 13.5.4 Let i : Z / / S be a closed immersion betw een regular schemes. W e deﬁne the fundamental class associated with i as the mor phism of KGL - modules: η i : i ∗ K GL Z / / K GL S deﬁned b y the image of the unit element 1 through the f ollowing morphism: K 0 ( Z ) q − 1 i / / K Z 0 ( S )  − 1 i / / Hom ( K GL S , i ! i ! K GL S ) = Hom ( i ∗ K GL Z , KGL S ) . 91 For example, when i is a regular closed immersion of codimension 1 , this happens if and only if the abo ve square is transv ersal. 92 When all the schemes in the square admit ample line bundles, w e can refer to [ Qui73 , 2.11]. 93 In the proof of Quillen, one can also trace back a canonical homotop y with the restriction mentioned in the preceding f ootnote. 340 Beilinson motiv es and algebraic K -theor y W e also denote by η 0 i : K GL Z / / i ! K GL S the mor phism obtained b y adjunction. Remar k 13.5.5 The fundamental class has the f ollo wing functor iality properties. (1) By deﬁnition, and applying remark 13.5.3 , the composite map K GL S / / i ∗ i ∗ ( K GL S ) = i ∗ K GL Z η i / / K GL S corresponds via the isomor phism  S to i ∗ ( 1 ) ∈ K 0 ( S ) . According to [ BGI71 , Exp. VII, 2.7], this class is equal to λ − 1 ( N i ) where N i is the conor mal sheaf of the regular immersion i . (2) In the situation of a T or -independent square as in 13.5.1 , remark that f ∗ η i = η k through the canonical e xc hange isomor phism f ∗ i ∗ = k ∗ g ∗ — appl y the functoriality of  i from (K6b) and the one of q i . (3) Using the identiﬁcation i ! i ∗ = 1 , we get η 0 i = i ! η i . Consider a car tesian square as in 13.5.1 and assume f is smooth. Then the square is T or -independent and we get g ∗ η 0 i = η 0 k using the e x chang e isomor phism g ∗ i ! = k ! f ∗ . 13.6 Absolute purity for K -theory Proposition 13.6.1 F or any closed immersion i : Z / / S between r egular sc hemes, the f ollowing diagram is commutative: Hom KGL ( K GL Z [ n ] , KGL Z ) η 0 i / /  Z   (∗) Hom KGL ( K GL Z [ n ] , i ! K GL S )  i   K n ( Z ) q − 1 i / / K Z n ( S ) Proof In this proof, we denote b y [− , −] the bifunctor Hom KGL (− , −) . St ep 1: W e assume that i : Z / / S admits a retraction p : S / / Z . Consider a K GL -linear map α : KGL Z [ n ] / / K GL Z . Then, η 0 i ( α ) cor responds by adjunction to the composition i ∗ K GL Z [ n ] i ∗ ( α ) / / i ∗ K GL Z η i / / K GL S . Applying the projection formula f or the motivic categor y Ho ( K GL - mo d ) , we g et: i ∗ ( α ) = i ∗ ( 1 ⊗ i ∗ p ∗ ( α )) = i ∗ ( 1 ) ⊗ p ∗ ( α ) . Here 1 stands f or the identity mor phism of the KGL -module K GL Z . This sho ws that η 0 i ( α ) corresponds b y adjunction to the composite map: η i ⊗ p ∗ ( α ) : i ∗ K GL Z [ n ] = i ∗ K GL Z [ n ] ⊗ KGL S / / K GL S ⊗ KGL S = KGL S 13 Algebraic K -theor y 341 (the tensor product is the KGL -linear one). By assumption, i ∗ : K ∗ ( Z ) / / K ∗ ( S ) admits a retraction which implies the canonical map O i : K Z ∗ ( S ) / / K ∗ ( S ) admits a retraction ( cf. remark 13.5.3 ). T o check that the diag ram (∗) is commutativ e, we can thus compose with O i . Recall the ﬁrst point of remark 13.5.5 : applying proper ty (K6a) and the fact the iso- morphism  S : [ KGL S [ n ] , KGL S ] / / K n ( S ) is compatible with the alg ebra struc- tures, w e are ﬁnally reduced to pro v e that i ∗ ( α ) = i ∗ ( 1 ) . p ∗ ( α ) ∈ K n ( S ) . This f ollow s from the projection formula in K -theor y (see [ Qui73 , 2.10] and [ TT90 , 3.17]). St ep 2: W e shall reduce the general case to Step 1. W e consider the f ollowing def or mation to the nor mal cone diag ram: let D be the blo w-up of A 1 S in the closed subscheme { 0 } × Z , P be the projective completion of the nor mal bundle of Z in S and s be the canonical section of P / Z ; w e get the f ollowing diag ram of regular schemes: Z s 1 / / i   A 1 Z   Z s 0 o o s   S / / D P o o (13.6.1.1) where s 0 (resp. s 1 ) is the zero (resp. unit) section of A 1 Z o v er Z . These squares are cartesian and T or -independent in the sense of 13.5.1 . The maps s 0 and s 1 induce isomorphisms in K -theor y because Z is regular . Thus, the second point of remark 13.5.5 allo ws reducing to the case of the immersion s which was done in Step 1.  13.6.2 Consider a car tesian square T k / / g   X f   Z i / / S such that S and Z are regular, i is a closed immersion and f is smooth. In this case, the f ollo wing diagram is commutative Hom KGL ( K GL Z ( T )[ n ] , KGL Z ) η 0 i / / Hom KGL ( K GL Z ( T )[ n ] , i ! K GL S ) Hom KGL ( K GL T [ n ] , KGL T ) η 0 k / / Hom KGL ( K GL T [ n ] , k ! K GL X ) using the adjunction ( g ] , g ∗ ) , the e xc hange isomor phism g ∗ i ! ' k ! f ∗ (which uses relativ e purity for smooth mor phisms) and the third point of remark 13.5.5 . In particular, the preceding proposition has the follo wing consequences: 342 Beilinson motives and algebraic K -theor y Theorem 13.6.3 (Absolute purity) F or any closed immersion i : Z / / S betw een r egular schemes, the map η 0 i : K GL Z / / i ! K GL S is an isomorphism in the category Ho ( KGL - mo d )( Z ) (or in SH ( Z ) ). Corollary 13.6.4 Given a cartesian squar e as abov e, for any pair ( n , m ) ∈ Z 2 , the f ollowing diagr am is commutative: Hom ( K GL S ( X ) , i ∗ K GL Z ( m )[ n ]) η i / / Hom ( K GL S ( X ) , KGL S ( m )[ n ])  X / S ∼   Hom ( K GL Z ( T ) , KGL Z ( m )[ n ])  T / Z ∼   K 2 m − n ( T ) k ∗ / / K 2 m − n ( X ) wher e the v ertical maps ar e the isomorphisms ( 13.3.4.2 ) . 13.7 T race maps 13.7.1 Let S be a regular scheme. Let Y be a smooth S -scheme. The obvious map Pic ( Y ) / / K 0 ( Y ) together with the canonical maps K 0 ( Y ) ∼ / / Hom KGL ( K GL S ( Y ) , KGL S ) β ∗ / / Hom KGL ( K GL S ( Y ) , KGL S ( 1 )[ 2 ]) deﬁnes Cher n classes in the categor y Ho ( KGL - mo d )( S ) ; the y cor responds to the orientation deﬁned in remark 13.2.2 . Let p : P / / S be a projective bundle of rank n . Let v = [ O ( 1 )] − 1 in K 0 ( P ) . It corresponds to a map v : KGL S ( P ) / / K GL S . A ccording to [ Dég08 , 3.2] and our choice of Chern classes, the f ollo wing map K GL S ( P ) Í i β i . v i  p ∗ / / Ê 0 ≤ i ≤ n K GL S ( i )[ 2 i ] is an isomor phism. As β is inv er tible, it f ollow s that the map (13.7.1.1) ϕ P / S : K GL S ( P ) Í i v i  p ∗ / / Ê 0 ≤ i ≤ n K GL S is an isomor phism as well. Using this f ormula, the map Hom ( ϕ P / S , KGL S ) is equal to the isomor phism of Quillen ’ s projective bundle theorem in K -theor y ( cf. [ Qui73 , 4.3]): 13 Algebraic K -theor y 343 f P / S : n Ê i = 0 K ∗ ( S ) / / K ∗ ( P ) , ( S 0 , . . . , S n )  / / Õ i p ∗ ( S i ) . v i . Let p ∗ : K ∗ ( P ) / / K ∗ ( S ) be the pushout by the projective mor phism p . According to the projection formula, it is K ∗ ( S ) -linear . In par ticular , it is deter mined by the n + 1 -uple ( a 0 , . . . , a n ) where a i = p ∗ ( v i ) ∈ K 0 ( S ) through the isomor phism f P / S . Let a i : K GL S / / K GL S be the map cor responding to a i . Deﬁnition 13.7.2 Consider the previous notations. W e deﬁne the tr ace map as- sociated with the projection p : P / / S as the morphism of KGL -modules T r KGL p : p ∗ ( K GL P ) / / K GL S determined as the composite p ∗ ( K GL P ) = R Hom ( KGL S ( P ) , KGL S ) ( ϕ ∗ P / S ) − 1 / / n Ê i = 0 K GL S ( a 0 , . . . , a n ) / / K GL S . From this deﬁnition, it follo ws that T r p represents the push-forward b y p in K -theor y : Hom KGL ( K GL S [ n ] , p ∗ K GL P ) T r KGL p ∗ / / Hom KGL ( K GL S [ n ] , KGL S )  S   Hom KGL ( K GL P [ n ] , KGL P )  P   K n ( P ) p ∗ / / K n ( S ) Consider moreo v er a car tesian square: Q q / / g   P p   Y f / / S such that f is smooth. From the projectiv e base change theorem, w e get f ∗ p ∗ p ∗ = q ∗ q ∗ g ∗ . Using this identiﬁcation, we easily obtain that f ∗ T r KGL p = T r KGL q . Thus, w e conclude that the map Hom KGL ( K GL S ( Y )[ n ] , p ∗ K GL P ) T r KGL p / / Hom KGL ( K GL S ( Y )[ n ] , KGL S ) represents the usual pushout map q ∗ : K n ( Q ) / / K n ( Y ) through the canonical isomor phisms ( 13.3.4.2 ). 13.7.3 Consider a projectiv e morphism f : T / / S between regular schemes and choose a factorization 344 Beilinson motives and algebraic K -theory T i / / P p / / S where i is a closed immersion and p is the projection of a projectiv e bundle. Let us deﬁne a mor phism T r KGL ( p , i ) : f ∗ K GL T = p ∗ i ∗ K GL T p ∗ η i / / p ∗ K GL P T r KGL p / / K GL S . A ccording to 13.6.4 and the previous paragraph, f or an y car tesian square Y g / / b   X a   T f / / S such that a is smooth, the follo wing diagram is commutativ e. Hom KGL ( K GL S ( X ) , f ∗ K GL T ( m )[ n ]) T r KGL ( p , i )∗ / / Hom KGL ( K GL S ( X ) , KGL S ( m )[ n ])  X / S '   Hom KGL ( K GL T ( Y ) , KGL Z ( m )[ n ])  Y / T '   K 2 m − n ( Y ) g ∗ / / K 2 m − n ( X ) (13.7.3.1) Deﬁnition 13.7.4 Considering the abo v e notations, we deﬁne the trace map associ- ated to f as the mor phism T r KGL f = T r KGL ( p , i ) : f ∗ f ∗ K GL S / / K GL S . Remar k 13.7.5 By deﬁnition, the trace map T r KGL f is a mor phism of K GL -modules. As a consequence, the map obtained by adjunction η 0 f : K GL T ' f ∗ K GL S / / f ! K GL S is also a mor phism of KGL -module. This implies that the mor phism η 0 f (and thus also T r KGL f ) is completely deter mined by the element η 0 f ∈ Hom KGL ( K GL T , f ! K GL S ) ' Hom SH ( T ) ( 1 T , f ! K GL S ) . Moreo v er , as p is smooth, there is a canonical isomor phism p ! K GL S ' KGL P (b y relativ e pur ity f or p and b y per iodicity ; see [ Rio10 , lemma 6.1.3.3]). From there, we deduce from Theorem 13.6.3 that we ha v e a canonical isomor phism f ! K GL S ' i ! K GL P ' KGL T . 14 Beilinson motives 345 This implies that we hav e an isomor phism: Hom SH ( T ) ( 1 T , f ! K GL S ) ' K 0 ( T ) . Hence, the map η 0 f is completely deter mined by a class in K 0 ( T ) . The problem of the functoriality of trace maps in the motivic categor y Ho ( KGL - mo d ) is thus a matter of functoriality of this element η 0 f in K 0 , whic h can be translated f aithfully to the problem of the functor iality of pushf orwards f or K 0 . Ho w ev er , the only property of trace maps we shall use here is the f ollo wing. Proposition 13.7.6 Let f : T / / S be a ﬁnite ﬂat morphism of regular schemes such that the O S -module f ∗ O T is (g lobally) free of rank d . Then the follo wing composite map K GL S / / f ∗ f ∗ K GL S T r KGL f / / K GL S is equal to d . 1 KGL S in Ho ( KGL - mo d )( S ) (whence in SH ( S ) as well). Proof Let ϕ be the composite map of Ho ( KGL - mo d )( S ) K GL S / / f ∗ f ∗ K GL S T r f / / K GL S . As ϕ is K GL S -linear b y construction, it cor responds to an element ϕ ∈ Hom KGL ( K GL S , KGL S ) ' Hom SH ( S ) ( 1 S , KGL S ) ' K 0 ( S ) . A ccording to the commutative diagram ( 13.7.3.1 ), if we appl y the global sections functor Hom SH ( S ) ( 1 S , −) to ϕ , we obtain through the evident canonical isomor - phisms the composition of the usual pullback and pushf or w ard by f in K -theor y : K 0 ( S ) f ∗ / / K 0 ( T ) f ∗ / / K 0 ( S ) . With these notations, the element of K 0 ( S ) corresponding to ϕ is the pushf or w ard of 1 T = f ∗ ( 1 S ) b y f , while the element cor responding to the identity of KGL S is of course 1 S . Under our assumptions on f , it is ob vious that we hav e the identity f ∗ ( 1 T ) = d . 1 S ∈ K 0 ( S ) . This means that ϕ is d times the identity of KGL S .  14 Beilinson motiv es 14.1 The γ -ﬁltration 14.1.1 W e denote by KGL Q the Q -localization of the absolute r ing spectrum KGL , considered as a car tesian section of D A 1 , Q . From [ Rio10 , 5.3.10], this spectr um has the f ollo wing proper ty : 346 Beilinson motives and algebraic K -theory (K5) For any scheme S , there e xists a canonical decomposition, called the Adams decomposition K GL Q , S ' Ê i ∈ Z K GL ( i ) S compatible with base chang e and such that f or an y regular scheme S , the isomorphism of (K2) induces an isomor phism: Hom D A 1 ( S , Q )  Q S ( X )[ n ] , KGL ( i ) S  ' K ( i ) n ( X ) : = Gr i γ K n ( X ) Q where the right-hand side is the i -th graded piece of the γ -ﬁltration on K -theor y groups. W e will denote by π i : K GL Q , S / / K GL ( i ) S , resp. ι i : K GL ( i ) S / / K GL Q , S the projection (resp. inclusion) deﬁned by the decomposition (K3) and we put p i = ι i π i f or the cor responding projector on KGL Q , S . Deﬁnition 14.1.2 (Riou) W e deﬁne the Beilinson motivic cohomology spectrum as the rational T ate spectrum H B , S = KGL ( 0 ) S . Remar k 14.1.3 Note that, b y deﬁnition, f or an y mor phism of schemes f : T / / S , w e hav e f ∗ H B , S ' H B , T . Lemma 14.1.4 The isomorphism γ u of ( 13.2.1.1 ) is homog eneous of degr ee + 1 with r espect to the gr aduation (K5). In other w ords, for any integ er i ∈ Z , the f ollowing composite map is an isomorphism K GL ( i ) ( 1 )[ 2 ] ι i / / K GL Q ( 1 )[ 2 ] γ u / / K GL Q π i / / K GL ( i + 1 ) . For any integer i ∈ Z , we thus get a canonical isomor phism (14.1.4.1) H B ( i )[ 2 i ] ∼ / / K GL ( i ) . Proof It is suﬃcient to check that, f or j , i + 1 , ( p j ◦ γ u ◦ p i = 0 , p j ◦ γ − 1 u ◦ p i = 0 in Hom D A 1 ( S , Q ) ( K GL Q , KGL Q ) . But according to [ Rio10 , 5.3.1 and 5.3.6], we hav e only to chec k these equalities f or the induced endomor phism of K 0 (seen as a presheaf on the category of smooth schemes ov er Sp ec ( Z ) ). This f ollow s then from 14 Beilinson motives 347 the compatibility of the projective bundle isomor phism with the γ -ﬁltration; see [ BGI71 , Exp. VI, 5.6].  14.1.5 Recall from [ NSØ09 ] that KGL Q is canonically isomorphic (with respect to the or ientation 13.2.2 ) to the universal oriented rational r ing spectr um with multi- plicativ e formal g roup law introduced in [ NSØ09 ]. The isomor phism of the pre- ceding corollar y sho ws in par ticular that H B is obtained from KGL Q b y killing the elements β n f or n , 0 . In par ticular , this sho ws that H B is canonically isomorphic to the spectr um denoted by L Q in [ NSØ09 ], which cor responds to the universal additiv e f ormal g roup law ov er Q . This implies that H B has a natural structure of a (commutativ e) ring spectrum. Proposition 14.1.6 The multiplication map µ : H B ⊗ H B / / H B is an isomorphism. This trivially implies that the follo wing map is an isomor phism: (14.1.6.1) 1 ⊗ η : H B / / H B ⊗ H B . Proof It is enough to treat the case S = Sp ec ( Z ) . W e will proo v e that the projector ψ : H B ⊗ H B µ / / H B 1 ⊗ η / / H B ⊗ H B is an isomor phism (in whic h case it is in fact the identity). W e do that f or the isomorphic ring spectrum L Q . Let H t o p Q be the topological spectr um representing rational singular cohomol- ogy . In the ter minology of [ NSØ09 ], L Q is a T ate spectr um representing the Landwe- ber e xact cohomology which cor responds to the Adams graded M U ∗ -algebra Q obtained by killing e v ery generator of the Lazard r ing M U ∗ . The cor responding topological spectrum is of course H t o p Q . A ccording to [ NSØ09 , 9.2], the spectrum E = L Q ⊗ L Q is a Landw eber e xact spectrum cor responding to the M U ∗ -algebra Q ⊗ M U ∗ Q = Q . In par ticular , the cor - responding topological spectr um is simpl y H t o p Q . Thus, according to [ NSØ09 , 9.7], applied with F = E = L Q ⊗ L Q , we get an isomorphism of Q -v ector spaces End ( L Q ⊗ L Q ) = Hom Q ( Q , E ∗∗ ) = Q . Thus ψ = λ . I d f or λ ∈ Q . But λ = 0 is ex cluded because ψ is a projector on a non-trivial factor , so that we can conclude.  348 Beilinson motives and algebraic K -theor y 14.2 Deﬁnition Deﬁnition 14.2.1 Let S be any scheme. W e say that an object E of D A 1 ( S , Q ) is H B -acyclic if H B ⊗ E = 0 in D A 1 ( S , Q ) . A mor phism of D A 1 ( S , Q ) is an H B -equiv alence if its cone is H B -acy clic (or, equiv - alently , if its tensor product with H B is an isomor phism). An object M of D A 1 ( S , Q ) is H B -local if, for an y H B -acy clic object E , the group Hom ( E , M ) vanishes. W e denote by DM B ( S ) the V erdier quotient of D A 1 ( S , Q ) by the localizing sub- category made of H B -acy clic objects (i.e. the localization of D A 1 ( S , Q ) by the class of H B -equiv alences). The objects of DM B ( S ) are called the Beilinson motiv es . Proposition 14.2.2 An object E of D A 1 ( S , Q ) is H B -acyclic if and only if we have K GL Q ⊗ E = 0 . Proof This follo ws immediately from proper ty (K5) (see 14.1.1 ) and Lemma 14.1.4 .  Proposition 14.2.3 The localization functor D A 1 ( S , Q ) / / DM B ( S ) admits a fully fait hful right adjoint whose essential imag e in D A 1 ( S , Q ) is the full subcategory spanned by H B -local objects. More precisely , ther e is a left Bousﬁeld localization of the stable model category of symmetric T ate spectra Sp ( S , Q ) by a small set of maps whose homotopy categor y is pr ecisely DM B ( S ) . Proof For each smooth S -scheme X and any integers n , i ∈ Z , we hav e a functor with values in the categor y of Q -vector spaces F X , n , i = Hom D A 1 ( S , Q ) ( Σ ∞ Q S ( X ) , H B ⊗ (−)( i )[ n ]) : Sp ( S , Q ) / / Q - mo d which preser v es ﬁltered colimits. W e deﬁne the class of H B -w eak equiv alences as the class of maps of Sp ( S , Q ) whose image by F X , n , i is an isomor phism f or all X and n , i . By vir tue of [ Bek00 , Prop. 1.15 and 1.18], w e can apply Smith’ s theorem [ Bek00 , Theorem 1.7] (with the class of coﬁbrations of Sp ( S , Q ) ), which implies the proposition.  Remar k 14.2.4 W e shall often make the abuse of considering DM B ( S ) as a full subcategory in D A 1 , Q ( S ) , with an implicit ref erence to the preceding proposition. Note that H B -acy clic objects are stable by the operations f ∗ , f ] and ⊗ , so that applying Corollary 5.2.5 , we obtain a premotivic categor y DM B together with a premotivic adjunction: (14.2.4.1) β ∗ : D A 1 , Q / / o o DM B : β ∗ . Proposition 14.2.5 The spectr um H B , S is H B -local and the unit map η H B : 1 / / H B , S is an H B -equiv alence in D A 1 ( S , Q ) . 14 Beilinson motives 349 Proof The unit map η : 1 S / / H B , S is an H B -equiv alence by 14.1.6 . Consider a rational spectr um E o v er S such that E ⊗ H B = 0 and a map f : E / / H B . It f ollo ws tr ivially from the commutative diag ram E f / / 1 ⊗ η   H B , S 1 ⊗ η   E ⊗ H B , S f ⊗ 1 / / H B , S ⊗ H B , S µ / / H B , S that f = 0 , which show s that H B , S is H B -local.  Corollary 14.2.6 The family of ring spectra H B , S comes from a coﬁbrant cartesian commutativ e monoid ( 7.2.10 ) of the symme tric monoidal ﬁbred model category of T ate spectr a ov er the category of schemes. Proof By virtue of Proposition 14.2.5 and of Corollar y 7.1.9 , there e xists a coﬁbrant commutativ e monoid in the model categor y of symmetr ic T ate spectra o v er Sp ec ( Z ) which is canonicall y isomorphic to H B , Z in D A 1 ( Sp ec ( Z ) , Q ) (as commutativ e ring spectrum). For a mor phism of schemes f : S / / Sp ec ( Z ) , we can then deﬁne H B , S as the pullback of H B , Z (at the lev el of the model categor ies); using Proposition 7.1.11 , we see that this deﬁnes a coﬁbrant car tesian commutative monoid on the ﬁbred category of spectra which is isomor phic to H B , S as commutativ e ring spectra in D A 1 ( S , Q ) .  14.2.7 From now on, w e shall assume that H B is given by a coﬁbrant car tesian commutativ e monoid of the symmetric monoidal ﬁbred model category of T ate spectra ov er the categor y of schemes. By vir tue of propositions 7.2.11 and 7.2.18 ), w e get the motivic categor y Ho ( H B - mo d ) of H B -modules, together with a commutativ e diagram of mor phisms of premotivic categor ies D A 1 , Q β ' ' H B ⊗(−) / / Ho ( H B - mo d ) DM B ϕ 5 5 (an y H B -acy clic object becomes null in the homotopy categor y of H B -modules by deﬁnition, so that H B ⊗ (−) f actors uniquely through DM B b y the universal proper ty of localization). Proposition 14.2.8 The f org etful funct or U : Ho ( H B - mo d )( S ) / / D A 1 ( S , Q ) is fully fait hful. Proof W e ha v e to prov e that, for any H B , S -module M , the map H B , S ⊗ M / / M is an isomor phism in D A 1 , Q ( S ) . As this is a natural transf or mation betw een e xact functors which commute with small sums, and as D A 1 , Q is a compactly g enerated 350 Beilinson motiv es and algebraic K -theor y triangulated category , it is suﬃcient to chec k this f or M = H B , S ⊗ E , with E a (compact) object of D A 1 , Q ( S ) (see 7.2.7 ). In this case, this follo ws immediately from the isomor phism ( 14.1.6.1 ).  Theorem 14.2.9 The functor DM B ( S ) / / Ho ( H B , S - mo d ) is an equivalence of tri- angulated monoidal categories. Proof This f ollo ws formall y from the preceding proposition b y deﬁnition of DM B (see f or instance [ GZ67 , Chap. I, Prop. 1.3]).  Remar k 14.2.10 The preceding theorem show s that the premotivic categor y of H B - modules Ho ( H B - mo d ) as w ell as the mor phism D A 1 , Q / / Ho ( H B - mo d ) are com- pletely independent of the choice of the strictiﬁcation of the (commutative) monoid structure on H B giv en by Corollar y 14.2.6 . Corollary 14.2.11 The premotivic category DM B ' Ho ( H B - mo d ) is a Q -linear motivic categor y. Proof It f ollow s from Proposition 7.2.18 and Theorem 14.2.9 that DM B satisﬁes the homotop y , stability and localization proper ties (because this is tr ue f or D A 1 , Q b y 6.2.2 ). It is also well generated because it is a localization of D A 1 , Q . Thus we can apply Remark 2.4.47 to conclude.  Remar k 14.2.12 One can also prov e that DM B is motivic much more directly : this f ollow s from the f act that D A 1 , Q is motivic and that the six Grothendieck operations preserve H B -acy clic objects, so that all the proper ties of D A 1 , Q induce their analogs on DM B b y the 2 -univ ersal proper ty of localization (w e leav e this as an easy e x ercise f or the reader). Deﬁnition 14.2.13 For a scheme X , w e deﬁne its Beilinson motivic cohomology b y the f ormula: H q B ( X , Q ( p )) = Hom DM B ( X ) ( 1 X , 1 X ( p )[ q ]) . In fact, according to the preceding corollar y , the cohomology theor y deﬁned abo v e is represented by the r ing spectrum H B . In par ticular , we can no w justify the ter mi- nology of Beilinson motives: Corollary 14.2.14 F or any regular scheme X , we have a canonical isomorphism H q B ( X , Q ( p )) ' Gr p γ K 2 p − q ( X ) Q . 14.2.15 Recall from P aragraph 14.1.5 that H B , S is canonically oriented for an y scheme S . Moreov er , these or ientations are compatible with pullbacks with respect to S . This means in par ticular that the motivic triangulated categor y DM B is oriented (see Example 12.2.3 ). In par ticular , the ﬁbred categor y DM B satisﬁes the usual Grothendieck 6 functors f or malism. W e ref er the reader to Theorem 2.4.50 f or the precise statement. 14 Beilinson motives 351 It was remarked in Paragraph 14.1.5 that H B , S is the univ ersal oriented ring spectrum with additive formal group law ov er S . This proper ty can be e xpressed by the f ollo wing nice descr iption of Beilinson motiv es: Corollary 14.2.16 Let E be a rational spectrum ov er S . The f ollowing conditions ar e equivalent : (i) E is a Beilinson motiv e (i.e. is in the essential imag e of the right adjoint of the localization functor D A 1 , Q / / DM B ); (ii) E is H B -local; (iii) the map η ⊗ 1 E : E / / H B ⊗ E is an isomorphism; (iv) E is an H B -module in D A 1 , Q ; (v) E is admits a strict H B -module structur e. If, in addition, E is a commutative ring spectrum, these conditions ar e equiv alent to the f ollowing ones: (Ri) E is orientable; (Rii) E is an H B -alg ebra; (Riii) E admits a unique structur e of H B -alg ebra; And, if E is a strict commutative ring spectr um, these conditions ar e equiv alent to the f ollowing conditions: (Riv) ther e exists a mor phism of commutativ e monoids H B / / E in the stable model category of T ate spectr a; (Rv) ther e exists a unique morphism H B / / E in the homotopy category of commu- tativ e monoids of the categor y of T ate spectr a. Proof The equivalence betw een statements (i)–(v) f ollow s immediately from 14.2.9 . If E is a r ing spectr um, the equivalence with (Ri), (Rii) and R(iii) is a consequence of 12.2.10 and of the fact that MGL Q is H B -local; see [ NSØ09 , Cor . 10.6]. It remains to pro v e the equivalence with (Riv) and (Rv). Then, E is H B -local if and only if the map E / / H B ⊗ E is an isomor phism. But this map can be seen as a morphism of strict commutative r ing spectra (using the model structure of 7.1.8 applied to the model category of T ate spectra) whose target is clearl y an H B -algebra, so that (Riv) is equiv alent to (ii). It remains to check that there is at mos t one strict H B -algebra structure on E (up to homotop y), which follo ws from the fact that H B is the initial object in the homotop y categor y of commutative monoids of the model category giv en by Theorem 7.1.8 applied to the model structure of Proposition 14.2.3 .  Corollary 14.2.17 One has the f ollowing pr operties. 1. The ring structure on the spectrum H B is given by the f ollowing structur al maps (with the notations of 14.1.1 ). H B ⊗ H B ι 0 ⊗ ι 0 / / K GL Q ⊗ KGL Q µ KGL / / K GL Q π 0 / / H B , Q η KGL / / K GL Q π 0 / / H B . 352 Beilinson motiv es and algebraic K -theory 2. The map ı 0 : H B / / K GL Q is compatible with the monoid structur es. 3. Let H B [ t , t − 1 ] = É i ∈ Z H B ( i )[ 2 i ] be the free H B -alg ebra g enerat ed by one inv ertible gener ator t of bidegr ee ( 2 , 1 ) . Then the section u : Q ( 1 )[ 2 ] / / K GL Q induces an isomorphism of H B -alg ebras: γ 0 u : H B [ t , t − 1 ] / / K GL Q . Proof Proper ty (1) f ollow s from properties (2) and (3). Proper ty (2) is a tr ivial consequence of the previous corollar y . Using the isomor phisms ( 14.1.4.1 ) of Lemma 14.1.4 , w e get a canonical isomor phism H B , S [ t , t − 1 ] ∼ / / Ê i ∈ Z K GL ( i ) . Through this isomorphism, the map γ 0 u corresponds to the Adams decomposition (i.e. to the isomor phism (K5) of 14.1.1 ) from which we deduce proper ty (3).  Remar k 14.2.18 One deduces easily , from the preceding proposition and from 14.1.6 , another proof of the fact that K GL Q is a strict commutative r ing spectr um. The isomor phism (3) is in fact compatible with the grading of each ter m: the factor H B . t i is sent to the factor KGL ( i ) . Recall also the parameter t cor responds to the unit β − 1 in KGL ∗ , ∗ . Corollary 14.2.19 The Adams decomposition is compatible with the monoid struc- tur e on KGL Q : f or any integ er i , j , l such that l , i + j , the follo wing map is zero. K GL ( i ) ⊗ KGL ( j ) ι i ⊗ ι j / / K GL Q ⊗ KGL Q µ / / K GL Q π l / / K GL ( l ) 14.2.20 Let R be a Q -algebra with structural morphism ϕ . Recall from P aragraph 5.3.36 that we get an adjunction of premotivic tr iangulated categories: ϕ ∗ : D A 1 , Q / / D A 1 , R : ϕ ∗ . Moreo v er , f or an y object M and N of D A 1 , Q ( S ) , the canonical map (14.2.20.1) Hom ( M , N ) ⊗ Q R / / Hom ( ϕ ∗ ( M ) , ϕ ∗ ( N )) . is an isomor phism pro vided M is compact or R is a ﬁnite Q -vector space. In par ticular , the r ing spectr um KGL R : = ϕ ∗ ( K GL Q ) represents Quillen algebraic K -theor y with coeﬃcients in R ov er regular schemes. W e can repeat Deﬁnition 14.2.1 with R -coeﬃcients and this giv es the category DM B ( S , R ) of Beilinson motiv es with R -coeﬃcients together with an adjunction: ϕ ∗ : DM B / / DM B (− , R ) : ϕ ∗ . Moreo v er , using the canonical map ( 14.2.20.1 ) and the fact it is an isomor phism when M is a constructible Beilinson motives, w e immediately e xtend all the proper ties pro v ed so far from Q -coeﬃcients to R -coeﬃcients. 14 Beilinson motives 353 14.3 Motivic proper descent Recall from Deﬁnition 4.3.2 we hav e deﬁned the notion of continuity for a tr iangu- lated premotivic categor y whic h is the homotopy categor y of a premotivic model category , such as the triangulated motivic categor y DM B — in this case, the notion of continuity is relative to the T ate twist. Proposition 14.3.1 The motivic triangulated categor y DM B is continuous. Proof W e consider the adjunction ( 14.2.4.1 ). According to Theorem 14.2.9 , the func- tor β ∗ commutes with pullbacks b y arbitrary mor phisms. Thus the continuity property f or DM B f ollow s from the continuity property f or D A 1 , Q which was established in Example 6.1.13 .  W e will give the main applications of continuity in the section on constr uctible Beilinson motiv es. Recall from 4.3.9 the f ollowing corollar y of the continuity prop- erty of the motivic categor y DM B : Corollary 14.3.2 Let X be a scheme, and consider an X -scheme Y of ﬁnite type. Giv en a point x ∈ X , we denote by X h x the spectr um of the local henselian ring of X at the point x . Let a x : Y × X X h x / / Y be the canonical map. Then the family of functors DM B ( Y ) / / DM B ( Y × X X h x ) , E  / / a ∗ x ( E ) is conser v ative. As the reader might e xpect, this proposition is v er y useful to reduce global properties of the motivic categor y DM B to local proper ties. This is in par ticular illustrated by the f ollowing proposition. Theorem 14.3.3 The motivic category DM B is separat ed (on the category of noethe- rian sc hemes of ﬁnite dimension). Proof A ccording to Proposition 2.3.9 , it is suﬃcient to chec k that, f or an y ﬁnite surjective mor phism f : T / / S , the pullback functor f ∗ : DM B ( S ) / / DM B ( T ) is conservativ e. W e argue by induction on the dimension of S . Let us ﬁrst treat the case where dim ( S ) = 0 . Using the localization proper ty , we can assume that S and T are reduced ( cf. 2.3.6 ). Then S is a disjoint sum of spectra of ﬁelds. In particular, f is not only ﬁnite surjective but also ﬂat. Moreo ver , it is also globally free. It will be suﬃcient to pro v e that, for any Beilinson motiv e E o ver S , the adjunction map E / / f ∗ f ∗ ( E ) is a monomor phism in DM B . Using the projection formula in DM B applied to the ﬁnite mor phism f (point (5) of Theorem 2.4.50 ), this latter map is isomor phic to 354 Beilinson motives and algebraic K -theory  H B / / f ∗ f ∗ ( H B )  ⊗ 1 E . W e are ﬁnall y reduced to pro v e that the map H B , S / / f ∗ f ∗ H B , S is a monomor phism in DM B (an y monomor phism of a tr iangulated category splits). As H B , S is a direct factor of KGL Q , S , it is suﬃcient to ﬁnd a retraction of the adjunction map K GL Q , S / / f ∗ f ∗ K GL Q , S , and this f ollo ws from Proposition 13.7.6 . Let us ﬁnally sol ve the induction process. Applying the preceding proposition, w e can assume that S is local henselian. Let s be the closed point of S and U be the open complement. Let f s (resp. f U ) be the pullback of f abo v e s (resp. U ). Using the localization proper ty of DM B and the base change isomor phisms (point (4) of Theorem 2.4.50 ), it is suﬃcient to treat the case of the ﬁnite mor phisms f U and f s . The case of f U f ollow s b y the induction hypothesis while the case of f s f ollow s from the case treated previousl y . This ends up the induction process.  A ccording respectivel y to Proposition 3.3.33 and Theorem 3.3.37 , we deduce from the preceding proposition the follo wing result: Theorem 14.3.4 1. The motivic categor y DM B satisﬁes étale descent. 2. The motivic categor y DM B satisﬁes h -descent when res tricted to quasi-excellent sc hemes. Recall this means that for any étale hyperco ver (resp. h -h yperco v er of a q uasi- e x cellent scheme) p : X / / X and for any Beilinson motiv e E o ver X , the map p ∗ : R Γ ( X , E ) / / R Γ ( X , E ) = R lim o o n R Γ ( X n , E ) is an isomor phism in the derived categor y of the categor y of Q -v ector spaces (see Corollary 3.2.17 taking into account Deﬁnition 3.2.20 ). 14.4 Motivic absolute purity Theorem 14.4.1 (Absolute purity) Let i : Z / / S be a closed immersion betw een r egular schemes. Assume i has pure codimension n . Then, considering the notations of 14.1.1 , deﬁnition 13.5.4 , and the identiﬁcation ( 14.1.4.1 ) , the composed map H B , Z ι 0 / / K GL Q , Z η 0 i / / i ! K GL Q , S π n / / i ! H B , S ( n )[ 2 n ] is an isomorphism. This isomor phism, of equiv alently the map obtained b y adjunction: i ∗ ( H B , Z ) / / H B , S ( n )[ 2 n ] 14 Beilinson motives 355 is called the fundamental class associated with i . In fact, this is a canonical class in the Beilinson motivic cohomology of X with suppor t in Z of bidegree ( 2 n , n ) . Remar k 14.4.2 It f ollo ws from Remark 13.5.5 that the fundamental class in Beilinson motivic cohomology is compatible with pullback with respect to T or-independent square. Proof W e ha v e onl y to check that the abo ve composition induces an isomor phism after applying the functor Hom ( Q S ( X ) , −( a )[ b ]) f or a smooth S -scheme X and a cou- ple of integers ( a , b ) ∈ Z 2 . Using Remark 13.5.5 (3), this composition is compatible with smooth base chang e, and we can assume X = S . Let us consider the projector p a : K Z r ( S ) Q = K r ( S / S − Z ) Q / / K r ( S / S − Z ) Q induced by π a ◦ ι a : K GL Q / / K GL Q , and denote by K ( a ) r ( S / S − Z ) (with r = 2 a − b ) its image. By virtue of Proposition 13.6.1 , w e only ha v e to chec k that the f ollowing composite is an isomor phism: ρ i : K ( a ) r ( Z ) ι a / / K r ( Z ) Q q − 1 i / / K r ( S / S − Z ) Q π a / / K ( a + n ) r ( S / S − Z ) . From 13.5.2 , the mor phism ρ i is functor ial with respect to T or -independent car tesian squares of regular schemes ( cf. 13.5.1 ). Thus, using again the def or mation diagram ( 13.6.1.1 ), w e get a commutativ e diagram K ( a ) r ( Z ) / / ρ i   K ( a ) r ( A 1 Z )   K ( a ) r ( Z ) ρ s   o o K ( a + n ) r ( S / S − Z ) / / K ( a + n ) r ( D / D − A 1 Z ) K ( a + n ) r ( P / P − Z ) o o in which any of the horizontal maps is an isomor phism (as a direct f actor of an isomorphism). Thus, we are reduced to the case of the closed immersion s : Z / / P , canonical section of the projectivization of a v ector bundle E (where E is the normal bundle of the closed immersion i ). Moreov er , as the asser tion is local on Z , we may assume E is a tr ivial v ector bundle. Let p : P / / Z be the canonical projection, j : P − Z / / P the obvious open immersion. Considering the element v 0 : =  [ O ( 1 )] − 1  of K 0 ( P ) , we let v be its projection on the ﬁrst graded par t of the γ -ﬁltration, v ∈ K ( 1 ) 0 ( P ) . Recall that, according to the projectiv e bundle f or mula, the horizontal lines in the f ollowing commutative diagram are split shor t e xact sequences: 0 / / K r ( P / P − Z ) Q ν / /   K r ( P ) Q j ∗ / /   K r ( P − Z ) Q   / / 0 0 / / K ( a + n ) r ( P / P − Z ) ν 0 / / K ( a + n ) r ( P ) / / K ( a + n ) r ( P − Z ) / / 0 . 356 Beilinson motives and algebraic K -theory By assumption on E , v n lies in the kernel of j ∗ and the diagram allow s to identify the g raded piece K ( a + n ) r ( P / P − Z ) with the submodule of K ( a + n ) r ( P ) of the f orm K ( a ) r ( Z ) . v n . On the other hand, j ∗ s ∗ = 0 : there exis ts a unique element  ∈ K 0 ( Z ) such that s ∗ ( 1 ) = p ∗ (  ) . v n in K 0 ( P ) . From the relation p ∗ s ∗ ( 1 ) = 1 , w e obtain that  is a unit in K 0 ( Z ) , with inv erse the element p ∗ ( v n ) . By vir tue of [ BGI71 , Exp. VI, Cor . 5.8], p ∗ ( v n ) belongs to the 0 -th γ -graded par t of K 0 ( P ) Q so that the same holds f or its in v erse  . In the end, for any element z ∈ K r ( Z ) , w e get the f ollowing e xpression: s ∗ ( z ) = s ∗ ( 1 . s ∗ p ∗ ( z )) = s ∗ ( 1 ) . p ∗ ( z ) = p ∗ (  . z ) . v n . Thus, the commutativ e diagram K ( a ) r ( Z ) / / K r ( Z ) Q q − 1 s / / s ∗ ' ' K r ( P / P − Z ) Q ν   / / K ( a + n ) r ( P / P − Z ) ν 0   K r ( P ) Q / / K ( n ) r ( P ) implies that the isomor phism q − 1 s preserves the γ -ﬁltration (up to a shift by n ). Hence, it induces an isomor phism on the graded pieces by functor iality .  15 Constructible Beilinson motiv es 15.1 Deﬁnition and basic properties In this section, we appl y the general results of Section 4 to the tr iangulated motivic category DM B . Let us ﬁrst recall the deﬁnition of constructibility (Def. 4.2.1 ) which corresponds to the T ate twist. Deﬁnition 15.1.1 Giv en any sc heme S , we deﬁne the category DM B , c ( S ) of con- structible Beilinson motiv es ov er S as the thick tr iangulated subcategor y of DM B ( S ) generated by the motiv es of the f or m M S ( X )( i ) for a smooth S -scheme X and an integer i ∈ Z . Remar k 15.1.2 Constructible Beilinson motiv es play s tow ards Beilinson motives the same role as comple xes of étale sheav es with bounded cohomology and constructible cohomology sheav es pla ys against comple xes of étale sheav es (in the case of torsion coeﬃcients prime to the residue characteristics). This fact will be ev en more striking after Theorems 15.2.1 and 15.2.4 . 15.1.3 Recall from Corollary 6.2.2 that D A 1 , Q is compactly generated b y the T ate twist. A ccording to Theorem 14.2.9 , the same is tr ue f or the motivic categor y DM B . Thus Proposition 1.4.11 giv es the f ollo wing cr iterion of constr uctibility for Beilinson motiv es: 15 Constr uctible Beilinson motives 357 Proposition 15.1.4 Given any base scheme S , a Beilinson motiv e M ov er S is constructible if and only if it is compact. Remar k 15.1.5 In the sequel, we will give sev eral concrete descriptions of the categor y of constructible Beilinson motives (see Corollar ies 16.1.6 and 16.2.16 ). Recall from Proposition 14.3.1 that DM B is continuous (with respect to the T ate twist). Proposition 4.3.4 thus implies the follo wing proper ties of constructible Beilin- son motiv es: Proposition 15.1.6 Let ( S α ) α ∈ A be a pr o-object of noetherian ﬁnite dimensional sc hemes with aﬃne transition maps and such that the scheme S = lim o o α ∈ A S α is noetherian of ﬁnite dimension. Then the canonical functor : (15.1.6.1) 2 - lim / / α DM B , c ( S α ) / / DM B , c ( S ) is an equiv alence of monoidal triangulated categories. Example 15.1.7 U nder the assumptions of the abo ve proposition, for an y couple of integers ( p , q ) , the canonical map lim / / α H q B ( S α , Q ( p )) / / H q B ( S , Q ( p )) is an isomor phism. 94 15.2 Gro thendieck 6 functors formalism and duality The motivic tr iangulated categor y DM B is separated ( 14.3.3 ) and weakl y pure (see Deﬁnition 4.2.20 ; this f ollo ws directly from Theorem 14.4.1 ). Thus the abstract Theorem 4.2.29 giv es the ﬁniteness theorem, which we state here explicitl y to help the reader: Theorem 15.2.1 The triangulated subcategory DM B , c of DM B is stable by the f ollowing oper ations: 1. f ∗ f or any mor phism of sc hemes f . 2. f ∗ f or any morphism f : Y / / X of ﬁnite type suc h that X is quasi-excellent (r esp. any proper morphism f ). 3. f ! f or any separat ed mor phism of ﬁnite type f . 4. f ! f or any mor phism f : Y / / X of ﬁnite type such that X is quasi-excellent. 5. ⊗ X f or any scheme X . 6. Hom X f or any quasi-excellent scheme X . 94 This result is to be compared with [ Qui73 , Sec. 7, 2.2] — it concer ns homotop y inv ar iant K -theory rather than K -theory . 358 Beilinson motives and algebraic K -theory T o be more precise, point (1) and (5) are obvious, the non respe condition of point (2) is the hardest fact and f ollo ws from Theorem 4.2.24 , point (3) as well as the respe condition of point (2) is Corollar y 4.2.12 , point (4) is Corollary 4.2.28 and point (6) is Corollary 4.2.25 . 15.2.2 Let B be an ex cellent scheme such that dim ( B ) ≤ 2 . R ecall that B satisﬁes wide resolution of singularities up to quotient singular ities (see Def. 4.1.9 and the result of De Jong recalled in 4.1.11 ). Thus according to Corollar y 4.4.3 , we get the f ollowing descr iption of constructible Beilinson motiv es: Proposition 15.2.3 Let S be a separat ed B -scheme of ﬁnite type, and T ⊂ S a closed subsc heme. Then the triangulated categor y DM B , c ( S ) is the smalles t triangulat ed category of DM B ( S ) which contained motiv es of the f orm f ∗ ( 1 X )( n ) wher e n is an integ er and f : X / / S is a projectiv e morphism such that X is r egular connected and f − 1 ( T ) r e d is either empty, either X of the support of a strict nor mal crossing divisor . The main motivation to introd uce the notion of constructibility is Grothendieck duality . W e obtain this duality from the theoretical result on motivic tr iangulated categories, more precisely Corollar y 4.4.24 : Theorem 15.2.4 Let B be an excellent sc heme suc h that dim ( B ) ≤ 2 and S be a r egular separat ed B -scheme of ﬁnite type. Then for any separated mor phism f : X / / S of ﬁnite type, the premo tiv e f ! ( 1 S ) is a dualizing object of DM B , c ( X ) . In fact, if we put D X ( M ) : = Hom X ( M , f ! ( 1 S )) f or any constructible Beilinson motives M , the follo wing properties hold: (a) F or any separated S -scheme of ﬁnite type X , the functor D X pr eser v es con- structible objects. (b) F or any separated S -sc heme of ﬁnite type X , the natural map M / / D X ( D X ( M )) is an isomorphism f or any constructible Beilinson motive M . (c) F or any separat ed S -sc heme of ﬁnite type X , and for any Beilinson motiv e M and N ov er X , if N is constructible then we have a canonical isomorphism D X ( M ⊗ X D X ( N )) ' Hom X ( M , N ) . (d) F or any morphism betw een separ ated S -schemes of ﬁnite type f : Y / / X , we hav e natural isomor phisms 16 Compar ison theorems 359 D Y ( f ∗ ( M )) ' f ! ( D X ( M )) f ∗ ( D X ( M )) ' D Y ( f ! ( M )) D X ( f ! ( N )) ' f ∗ ( D Y ( N )) f ! ( D Y ( N )) ' D X ( f ∗ ( N )) wher e M (r esp. N ) is a constructible Beilinson motiv e ov er X (r esp. Y ). 15.2.5 Let R be a Q -algebra. 95 W e deﬁne the premotivic tr iangulated categor y of constructible Beilinson motives with coeﬃcients in R as the category of constructible objects of the category DM B (− , R ) deﬁned in Paragraph 14.2.20 . A ccording to loc. cit. , for any constructible Beilinson motiv es with coeﬃcients in Q , we get an isomorphism: Hom DM B , c ( S ) ( M , N ) ⊗ Q R / / Hom DM B , c ( S , R )  L ϕ ∗ ( M ) , L ϕ ∗ ( N )  . It is straightf or w ard to see that this isomor phism allow s to extend all the results pro v ed so far for Beilinson motives with coeﬃcient in Q to the case of R -coeﬃcients. 16 Comparison theorems 16.1 Comparison with V oe v odsky motiv es 16.1.1 W e consider the premotivic adjunction of 11.4.1 (16.1.1.1) γ ∗ : D A 1 , Q / / o o DM Q : γ ∗ . For a scheme S , γ ∗ ( 1 S ) is a (strict) commutative r ing spectr um, and, for any object M of DM Q ( S ) , γ ∗ ( M ) is naturally endo w ed with a structure of γ ∗ ( 1 S ) -module. On the other hand, as we ha v e the projective bundle formula in DM Q ( S ) ( 11.3.4 ), γ ∗ ( 1 S ) is or ientable ( 12.2.10 ), which implies that, for any object M of DM Q ( S ) , γ ∗ ( M ) is an H B , S -module, whence is H B -local ( 14.2.16 ). As consequence, w e get a canonical factorization of ( 16.1.1.1 ): (16.1.1.2) D A 1 , Q β ∗ / / DM B ϕ ∗ / / DM Q . Consider the commutativ e diagram of premotivic categor ies 95 The examples we hav e in mind are: R = E is a number ﬁeld, R = C , R = Q l , ¯ Q l f or a prime l . 360 Beilinson motives and algebraic K -theory D A 1 , Q γ ∗ / / ρ ]   DM Q ψ ]   D A 1 , Q γ ∗ / / DM Q (16.1.1.3) in whic h the two vertical maps are the canonical enlargements, and, in par ticular , are fully faithful (see 6.1.8 ). Let t denotes either the qfh -topology or the h -topology . W e also ha ve the follo wing commutativ e triangle D A 1 , Q γ ∗ / / a ∗ 5 5 DM Q α ∗ / / DM t , Q (16.1.1.4) in which both a ∗ and α ∗ are induced by the t -sheaﬁﬁcation functor; see 5.3.31 and 11.1.21 . W e obtain from ( 16.1.1.2 ), ( 16.1.1.3 ), and ( 16.1.1.4 ) the commutative diagram of premotivic categor ies belo w , in which χ ] = ϕ ∗ α ∗ ψ ] . D A 1 , Q β ∗ / / ρ ]   DM B χ ]   D A 1 , Q a ∗ / / DM t , Q (16.1.1.5) From no w on, we shall ﬁx an e x cellent noether ian scheme of ﬁnite dimension S . Theorem 16.1.2 W e hav e canonical equivalences of categories DM B ( S ) ' DM qfh , Q ( S ) ' DM h , Q ( S ) (r ecall that, for t = qfh , h , DM t , Q ( S ) stands for the localizing subcategory of DM t , Q ( S ) , spanned by the objects of shape Σ ∞ Q S ( X )( n ) , wher e X runs ov er the family of smooth S -schemes, and n ≤ 0 is an integ er; see 5.3.31 ). Proof Let t denote the qfh -topology or the h -topology . W e shall pro v e that the functor χ ] : DM B ( S ) / / DM t , Q ( S ) is fully faithful, and that its essential image is precisely DM t , Q . The functor β ∗ : DM B / / D A 1 , Q ( S ) is fully faithful, so that its composition with its left adjoint β ∗ is canonically isomor - phic to the identity . In par ticular , we g et isomor phisms of functors: χ ] ' χ ] β ∗ β ∗ ' a ∗ ρ ] β ∗ . 16 Compar ison theorems 361 The right adjoint of a ∗ is full y f aithful, and its essential image consis ts of the objects of D A 1 , Q ( S ) which satisfy t -descent ( 5.3.30 ). On the other hand, the functor ρ ] is full y f aithful, and an object of D A 1 , Q ( S ) satisﬁes t -descent if and onl y if its image by ρ ] satisﬁes t -descent ( 6.1.11 ). By vir tue of Theorem 14.3.4 , this implies immediately that χ ] is fully faithful. Let DM t , Q ( S ) be the localizing subcategor y of DM t , Q ( S ) spanned b y the objects of shape Σ ∞ Q ( X )( n ) , where X r uns o v er the famil y of smooth S -schemes, and n ≤ 0 is an integer ( 5.3.31 ). W e know that DM t , Q ( S ) is compactly generated (see 5.1.29 , 5.2.38 and 5.3.40 ), and that χ ] is a full y f aithful e xact functor which preserves small sums as w ell as compact objects from DM B ( S ) to DM t , Q ( S ) . As, b y construction, there e xists a generating famil y of compact objects of DM t , Q ( S ) in the essential image of χ ] , this implies that χ ] induces an equiv alence of triangulated categories DM B ( S ) ' DM t , Q ( S ) (see 1.3.20 ).  Let us underline the f ollowing result which completes Corollar y 14.2.16 : Theorem 16.1.3 Let E be an object of D A 1 ( S , Q ) . The f ollowing conditions are equiv alent : (i) E is a Beilinson motiv e; (ii) E satisﬁes h -descent; (iii) E satisﬁes qfh -descent; Proof W e already know that condition (i) implies condition (ii) (second point of Theorem 14.3.4 ), and condition (ii) implies ob viousl y condition (iii). It is thus suﬃcient to prov e that condition (iii) implies condition (i). If E satisﬁes qfh -descent, then ρ ] ( E ) satisﬁes qfh -descent in DM ( S , Q ) as well. The commutativity of ( 16.1.1.4 ) implies then that ρ ] ( E ) belongs to the essential imag e of γ ∗ (the r ight adjoint of γ ∗ ). As ρ ] is full y faithful, the commutativity of ( 16.1.1.3 ) thus implies that E itself belongs to the essential image of γ ∗ (the right adjoint to γ ∗ ). In par ticular , E is then a module ov er the r ing spectrum γ ∗ ( 1 S ) , which is itself an H B -algebra. W e conclude b y Corollar y 14.2.16 .  Theorem 16.1.4 If S is excellent and g eometrically unibranc h, then the comparison functor ϕ ∗ : DM B ( S ) / / DM Q ( S ) is an equiv alence of triangulated monoidal categories. Proof If S is geometrically unibranch, then we kno w that the composed functor DM Q ( S ) ψ ] / / DM Q ( S ) α ∗ / / DM qfh , Q ( S ) is fully faithful ( 11.1.22 ). The commutativ e diagram DM B ( S ) ϕ ∗ / / χ ] 3 3 DM Q ( S ) α ∗ ψ ] / / DM qfh , Q ( S ) 362 Beilinson motives and algebraic K -theory and Theorem 16.1.2 imply that ϕ ∗ is fully faithful. As ϕ ∗ is ex act, preser v es small sums as well as compact objects, and as DM Q ( S ) has a g enerating f amily of compact objects in the essential image of ϕ ∗ , the functor ϕ ∗ has to be an equiv alence of categories ( 1.3.20 ).  Remar k 16.1.5 Some version of the preceding theorem (the one obtained by replacing DM B b y Ho ( H B - mo d ) ) was already known in the case where S is the spectrum of a per f ect ﬁeld; see [ RØ08a , theorem 68]. The proof used de Jong’s resolution of singularities b y alterations and P oincaré duality in a crucial w ay . The proof of the preceding theorem we ga ve here relies on proper descent but does not use an y kind of resolution of singular ities. The preceding theorem allow s to give the follo wing descr iption of constr uctible Beilinson motiv es o ver geometrically unibranch schemes: Corollary 16.1.6 F or any geome trically unibranc h sc heme S , the functor ϕ ∗ induces an equiv alence of triangulated monoidal categories: DM B , c ( S ) ∼ / / DM g m ( S , Q ) wher e the right hand side is t he Q -linear version of the category of g eometric (V oevodsky) motiv es (Deﬁnition 11.1.10 ). Note that we also applied Proposition 11.1.5 to get this corollary . W e ﬁnally point out the follo wing impor tant fact about V oev odsky’s motivic cohomology spectrum H M , S = γ ∗ ( 1 S ) with rational coeﬃcients: Corollary 16.1.7 1. F or any g eometrically unibr anch excellent sc heme S , the canonical map H B , S / / H Q M , S is an isomorphism of ring spectra. 2. F or any mor phism f : T / / S of excellent geome trically unibranc h sc hemes, the canonical map f ∗ H Q M , S / / H Q M , T is an isomorphism of ring spectra. The second par t is the last conjecture of V oev odsky’ s paper [ V oe02b ] with rational coeﬃcients (and geometrically unibranch schemes) – see also Paragraph 11.2.21 . Proof The ﬁrst par t is a trivial consequence of the previous theorem, and the second f ollow s from the ﬁrst, as the Beilinson motivic cohomology spectr um is stable by pullbacks.  16.2 Comparison with Morel motiv es 16.2.1 Let S be a scheme. The per mutation isomorphism 16 Compar ison theorems 363 (16.2.1.1) τ : Q ( 1 )[ 1 ] ⊗ L Q Q ( 1 )[ 1 ] / / Q ( 1 )[ 1 ] ⊗ L Q Q ( 1 )[ 1 ] satisﬁes the equation τ 2 = 1 in D A 1 ( S , Q ) . Hence it deﬁnes an element  in End D A 1 ( S , Q ) ( Q ) which also satisﬁes the relation  2 = 1 . W e deﬁne tw o projec- tors (16.2.1.2) e + = 1 −  2 and e − = 1 +  2 . As the tr iangulated categor y D A 1 ( S , Q ) is pseudo abelian, we can deﬁne two objects b y the f or mulæ: (16.2.1.3) Q + = Im e + and Q − = Im e − . Then f or an object M of D A 1 ( S , Q ) , we set (16.2.1.4) M + = Q + ⊗ L Q M and M − = Q − ⊗ L Q M . It is obvious that for any objects M and N of D A 1 ( S , Q ) , one has (16.2.1.5) Hom D A 1 ( S , Q ) ( M i , N j ) = 0 f or i , j ∈ { + , − } with i , j . Denote b y D A 1 ( S , Q ) + (resp. D A 1 ( S , Q ) − ) the full subcategor y of D A 1 ( S , Q ) made of objects which are isomorphic to some M + (resp. some M − ) for an object M in D A 1 ( S , Q ) . Then ( 16.2.1.5 ) implies that the direct sum functor ( M + , M − )  / / M + ⊕ M − induces an equiv alence of tr iangulated categor ies (16.2.1.6) ( D A 1 ( S , Q ) + ) × ( D A 1 ( S , Q ) − ) ' D A 1 ( S , Q ) . W e shall call D A 1 ( S , Q ) + the category of Morel motives ov er S . The aim of this section is to compare this category with DM B ( S ) (see Theorem 16.2.13 ). This will consists essentially of pro ving that Q + is nothing else than Beilinson ’ s motivic spectr um H B (which w as announced by Morel in [ Mor06 ]). The main ing redients of the proof are the descr iption of DM B ( S ) as full subcategory of D A 1 ( S , Q ) , the homotopy t -str ucture on D A 1 ( S , Q ) , and Morel’ s computation of the endomor phism r ing of the motivic sphere spectr um in terms of Milnor- Witt K -theor y [ Mor03 , Mor04a , Mor04b , Mor12 ]. 16.2.2 For a little while, w e shall assume that S is the spectr um of a ﬁeld k . Recall that the algebr aic Hopf ﬁbration is the map A 2 − { 0 } / / P 1 , ( x , y )  / / [ x , y ] . This deﬁnes, by desuspension, a mor phism η : Q ( 1 )[ 1 ] / / Q in D A 1 ( S , Q ) ; see [ Mor03 , 6.2] (recall that w e identify D A 1 ( S , Q ) with SH Q ( S ) and that, under this identiﬁcation, Q ( 1 )[ 1 ] cor responds to Σ ∞ ( G m ) ). 364 Beilinson motives and algebraic K -theory Lemma 16.2.3 W e have η =  η in Hom D A 1 ( S , Q ) ( Q ( 1 )[ 1 ] , Q ) . Proof See [ Mor03 , 6.2.3].  16.2.4 Recall the homot opy t -structure on D A 1 ( S , Q ) ; see [ Mor03 , 5.2]. T o remain close to the conv entions of loc. cit. , w e shall adopt homological notations, so that, f or an y object M of D A 1 ( S , Q ) , we ha v e the f ollo wing tr uncation triangle τ > 0 M / / M / / τ ≤ 0 M / / τ > 0 M [ 1 ] . W e whall wr ite H 0 f or the zeroth homology functor in the sense of this t -structure. This t -structure can be descr ibed in terms of generators, as in [ A yo07a , deﬁnition 2.2.41]: the category D A 1 ( S , Q ) ≥ 0 is the smallest full subcategor y of D A 1 ( S , Q ) which contains the objects of shape Q S ( X )( m )[ m ] f or X smooth o v er S , m ∈ Z , and which satisﬁes the f ollo wing stability conditions: (a) D A 1 ( S , Q ) ≥ 0 is stable under suspension; i.e. for any object M in D A 1 ( S , Q ) ≥ 0 , M [ 1 ] is in D A 1 ( S , Q ) ≥ 0 ; (b) D A 1 ( S , Q ) ≥ 0 is closed under extensions: f or any distinguished tr iangle M 0 / / M / / M 00 / / M 0 [ 1 ] , if M 0 and M 00 are in D A 1 ( S , Q ) ≥ 0 , so is M ; (c) D A 1 ( S , Q ) ≥ 0 is closed under small sums. With this descr iption, it is easy to see that D A 1 ( S , Q ) ≥ 0 is also closed under tensor product (because the class of generators has this property). The categor y D A 1 ( S , Q ) ≤ 0 is the full subcategor y of D A 1 ( S , Q ) which consists of objects M suc h that Hom D A 1 ( S , Q ) ( Q S ( X )( m )[ m + n ] , M ) ' 0 f or X / S smooth, m ∈ Z , and n > 0 ; see [ A yo07a , 2.1.72]. Note that the hear t of the homotop y t -structure is symmetr ic monoidal, with tensor product ⊗ h deﬁned b y the formula: F ⊗ h G = H 0 ( F ⊗ L S G ) (the unit object is H 0 ( Q ) ). W e shall still wr ite η : H 0 ( Q ( 1 )[ 1 ]) / / H 0 ( Q ) f or the map induced b y the alge- braic Hopf ﬁbration. Proposition 16.2.5 T ensoring by Q ( n )[ n ] deﬁnes a t -exact endofunctor of D A 1 ( S , Q ) f or any integ er n . Proof As tensor ing by Q ( n )[ n ] is an equiv alence of categor ies, it is suﬃcient to prov e this f or n ≥ 0 . This is then a par ticular case of [ A yo07a , 2.2.51].  Proposition 16.2.6 F or any smooth S -sc heme X of dimension d , and for any object M of D A 1 ( S , Q ) , the map 16 Compar ison theorems 365 Hom ( Q S ( X ) , M ) / / Hom ( Q S ( X ) , M ≤ n ) is an isomorphism f or n > d . Proof Using [ Mor03 , lemma 5.2.5], it is suﬃcient to pro ve the analog f or the homo- top y t -structure on D eﬀ A 1 , Q ( S ) , which f ollo ws from [ Mor05 , lemma 3.3.3].  Proposition 16.2.7 The homotopy t -structure is non deg enerated. Even better , f or any object M of D A 1 ( S , Q ) , we hav e canonical isomorphisms L lim / / n τ > n M ' M and R lim o o n τ > n M ' 0 , as w ell as isomor phisms L lim / / n τ ≤ n M ' 0 and M ' R lim o o n τ ≤ n M . Proof The ﬁrs t assertion is a direct consequence of propositions 16.2.5 and 16.2.6 (because the objects of shape Q S ( X )( m )[ i ] , for X / S smooth, and m , i ∈ Z , form a generating f amily). As the objects Q S ( X )( m )[ m + n ] are compact in D A 1 ( S , Q ) , the category D A 1 ( S , Q ) ≤ 0 is closed under small sums. As D A 1 ( S , Q ) ≥ 0 is also closed under small sums, w e deduce easily that the truncation functors τ > 0 and τ ≤ 0 preserve small sums, which implies that the homology functor H 0 has the same proper ty . Moreo v er , if C 0 / / · · · / / C n / / C n + 1 / / · · · is a sequence of maps in D A 1 ( S , Q ) , then C = L lim / / n C n ﬁts in a distinguished triangle of shape Ê n C n 1 − s / / Ê n C n / / C / / Ê n C n [ 1 ] , where s is the map induced by the maps C n / / C n + 1 . This implies that, for an y integer i , w e hav e lim / / n H i ( C n ) ' H i ( C ) (where the colimit is taken in the hear t of the homotopy t -str ucture). As the homotopy t -str ucture is non degenerated, this prov es the two formulas L lim / / n τ > n M ' M and L lim / / n τ ≤ n M ' 0 . Let X be a smooth S -scheme of ﬁnite type, and p , q be some integ er . T o pro ve that the map Hom ( Q S ( X )( m )[ i ] , M ) / / Hom ( Q S ( X )( m )[ i ] , R lim o o n τ ≤ n M ) 366 Beilinson motives and algebraic K -theory is bijectiv e, w e ma y assume that m = 0 (replacing M b y M (− m )[− m ] and i by i − m , and using Proposition 16.2.5 ). Consider the Milnor shor t e xact sequence below , with A = Q S ( X )[ i ] (in which the ﬁrst map is injectiv e, but we will not use it): lim o o n 1 Hom ( A [ 1 ] , τ ≤ n M ) / / Hom ( A , R lim o o n τ ≤ n M ) / / lim o o n Hom ( A , τ ≤ n M ) / / 0 . Using Proposition 16.2.6 , as lim o o 1 of a constant functor vanishes, we get that the map Hom ( A , M ) / / Hom ( A , R lim o o n τ ≤ n M ) is an isomor phism. This giv es the isomor phism M ' R lim o o n τ ≤ n M . Using the previous isomor phism, and by contemplating the homotop y limit of the homotop y coﬁber sequences τ > n M / / M / / τ ≤ n M , w e deduce the isomor phism R lim o o n τ > n M ' 0 .  Lemma 16.2.8 W e have H B ∈ D A 1 ( S , Q ) ≥ 0 , so that w e have a canonical map H B / / H 0 ( H B ) in D A 1 ( S , Q ) . In par ticular , for any object M in the heart of the homot opy t -structur e, if M is endow ed with an action of the monoid H 0 ( H B ) , then M has a natural structure of H B -module in D A 1 ( S , Q ) . Proof As H B is isomor phic to the motivic cohomology spectr um in the sense of V oe v odsky ( 16.1.7 ), the ﬁrst assertion is the ﬁrst asser tion of [ Mor03 , theorem 5.3.2]. Theref ore, the tr uncation triangle for H B giv es a tr iangle τ > 0 H B / / H B / / H 0 ( H B ) / / τ > 0 H B [ 1 ] , which giv es the second asser tion. For the third asser tion, consider an object M in the heart of the homotopy t -str ucture, endow ed with an action of H 0 ( H B ) . Note that D A 1 ( S , Q ) ≥ 0 is closed under tensor product, so that H B ⊗ L S M is in D A 1 ( S , Q ) ≥ 0 . Hence w e hav e natural maps H B ⊗ L S M / / H 0 ( H B ⊗ L S M ) / / H 0 ( H 0 ( H B ) ⊗ L S M ) = H 0 ( H B ) ⊗ h M . Then the structural map H 0 ( H B ) ⊗ h M / / M deﬁnes a map H B ⊗ L S M / / M which giv es the expected action (observe that, as w e already know that H B -modules do f or m a thick subcategory of D A 1 ( S , Q ) ( 14.2.8 ), we don ’ t ev en need to chec k all the 16 Compar ison theorems 367 axioms of an internal module: it is suﬃcient to check that the unit Q / / H B induces a section M / / H B ⊗ L S M of the map constructed abov e).  Lemma 16.2.9 W e have the follo wing exact sequence in the heart of the homotopy t -structure. H 0 ( Q ( 1 )[ 1 ]) η / / H 0 ( Q ) / / H 0 ( H B ) / / 0 Proof Using the equiv alence of categor ies from the hear t of the homotop y t -str ucture to the categor y of homotop y modules in the sense of [ Mor03 , deﬁnition 5.2.4], b y vir tue of Corollar y 16.1.7 and [ Mor03 , theorem 5.3.2], w e kno w that H 0 ( H B ) corresponds to the homotopy module K M ∗ ⊗ Q associated with Milnor K -theory, while H 0 ( Q ) cor responds to the homotop y module K M W ∗ ⊗ Q associated with Milnor - Witt K -theor y (which follo ws easily from [ Mor12 , theorems 2.11, 6.13 and 6.40]). Considering K M ∗ and K M W ∗ as unramiﬁed shea ves in the sense of Morel [ Mor12 ], this lemma is then a reformulation of the isomor phism K M W ∗ ( F )/ η ' K M ∗ ( F ) f or an y ﬁeld F ; see [ Mor12 , remark 2.2].  Proposition 16.2.10 W e have H B + ' H B , and the induced map Q + / / H B giv es a canonical isomorphism H 0 ( Q + ) ' H 0 ( H B ) . Proof The map  ( 1 )[ 1 ] : Q ( 1 )[ 1 ] / / Q ( 1 )[ 1 ] can be descr ibed geometrically as the morphism associated with the pointed mor phism ı : G m / / G m , t  / / t − 1 (see the second asser tion of [ Mor03 , lemma 6.1.1]). In the decomposition K 1 ( G m ) ' k [ t , t − 1 ] × ' k × ⊕ Z , the map ı induces multiplication b y − 1 on Z . Using the per iodicity isomor phism K GL ( 1 )[ 2 ] ' KGL , w e get the identiﬁcations: K 1 ( G m ) ⊃ Hom SH ( k ) ( Σ ∞ ( G m )[ 1 ] , KGL ) ' Hom KGL ( K GL , KGL ) ' K 0 ( k ) ' Z . Theref ore,  acts as the multiplication by − 1 on the spectrum KGL Q , whence on H B as well. This means precisely that H B + ' H B . By Lemma 16.2.3 , the class 2 η vanishes in Q + , so that, appy ling the ( t -exact) functor M  / / M + to the exact sequence of Lemma 16.2.9 , we get an isomor phism H 0 ( Q + ) ' H 0 ( H B + ) ' H 0 ( H B ) .  Corollary 16.2.11 F or any object M in the heart of the homotopy t -structure, M + is a Beilinson motiv e. Proof The object M is an H 0 ( Q ) -module, so that M + is an H 0 ( Q + ) -module. By virtue of Proposition 16.2.10 , M + is then a module ov er H 0 ( H B ) , so that, by Lemma 16.2.8 , M + is naturally endow ed with an action of H B .  368 Beilinson motiv es and algebraic K -theor y Remar k 16.2.12 Until now , we did not reall y use the fact w e are in a Q -linear conte xt (replacing H B b y V oev odsky’s motivic spectr um, we just needed 2 to be inv er tible in the preceding corollar y). How ev er, the f ollo wing result really uses Q -linear ity (because, in the proof, w e see DM B ( S ) as a full subcategor y of D A 1 ( S , Q ) ; see Proposition 14.2.3 ). Theorem 16.2.13 F or any noetherian scheme of ﬁnite dimension S , the map Q + / / H B is an isomorphism in D A 1 ( S , Q ) . As a consequence, we have a canonical equiv alence of triangulated monoidal categories D A 1 ( S , Q ) + ' DM B ( S ) . This theorem has already been pro ved b y Morel when S is the spectr um of a per f ect ﬁeld – where the left hand side is the rational category of V oev odsky motiv es. Morel announced that the categor y D A 1 ( S , Q ) + should be the categor y of rational motives and this theorem conﬁr m his insight. Proof Observe that, if ev er Q + ' H B , we ha v e D A 1 ( S , Q ) + ' DM B ( S ) : this f ollow s from the fact that an object M of D A 1 ( S , Q ) belongs to D A 1 ( S , Q ) + (resp. to DM B ( S ) ) if and only if there exis ts an isomor phism M ' M + (resp. M ' H B ⊗ L S M ; see 14.2.16 ). It is thus suﬃcient to prov e the ﬁrst asser tion. As both Q + and H B are stable by pullback, it is suﬃcient to treat the case where S = Sp ec ( Z ) . Using Corollar y 14.3.2 , we may replace S by any of its henselisations, so that, by the localization proper ty , it is suﬃcient to treat the case where S is the spectrum of a (per f ect) ﬁeld k . W e shall pro v e directly that, f or an y object M of D A 1 ( S , Q ) , M + is an H B -module (or , equivalentl y , is H B -local). Note that DM B ( S ) is closed under homotop y limits and homotop y colimits in D A 1 ( S , Q ) : indeed the inclusion functor DM B / / D A 1 , Q has a left adjoint which preser v es a f amily of compact generators, whence it also has a left adjoint ( 1.3.19 ). By vir tue of Proposition 16.2.7 , w e ma y thus assume that M is bounded with respect to the homotopy t -str ucture. As DM B ( S ) is cer tainly closed under extensions in D A 1 ( S , Q ) , w e ma y ev en assume that M belongs to the hear t the homotop y t -structure. W e conclude with Corollar y 16.2.11 .  Corollary 16.2.14 F or any noetherian scheme of ﬁnite dimension S , if − 1 is a sum of squar es in all the residue ﬁelds of S , then Q − ' 0 in D A 1 ( S , Q ) , and w e have a canonical equiv alence of triangulated monoidal categories D A 1 ( S , Q ) ' DM B ( S ) . Proof It is suﬃcient to pro v e that, under this assumption, Q − ' 0 . As in the preceding proof, we may replace S by any of its henselisations ( 4.3.9 ), so that, b y the localization property (and by induction on the dimension), it is suﬃcient to treat the case where S is the spectr um of a ﬁeld k . W e ha v e to chec k that, if − 1 is a sum of squares in k , then we hav e  = − 1 . Using [ Mor03 , remark 6.3.5 and lemma 6.3.7], we see that, if k is of character istic 2 , we alw ay s hav e  = − 1 , while, if the c haracteristic of k is distinct from 2 , we hav e a mor phism of r ings 16 Compar ison theorems 369 GW ( k ) / / Hom D A 1 , Q ( Spec ( k ) ) ( Q , Q ) , where GW ( k ) denotes the Grothendieck - Witt r ing 96 o v er k . This morphism sends the class of the quadratic form − X 2 to −  and this pro ves the result. (F or a more precise v ersion of this, with integ ral coeﬃcients, see [ Mor12 , proposition 2.13].)  16.2.15 Recall from Example 5.3.43 that we can descr ibe the categor y D A 1 , c ( S , Q ) of compact objects of D A 1 ( S , Q ) as the tr iangulated monoidal categor y obtained from  K b ( Q ( Sm / S ) ) /( BG S ∪ T A 1 S )  \ b y formally in verting the T ate twis t. The operation  still acts on this categor y and the decomposition in + and − part of a motiv e respects constructibility as this is a decomposition b y direct f actors. The preceding theorem gives the follo wing description of constructible Beilinson motives: Corollary 16.2.16 F or any noet herian scheme of ﬁnite dimension S , ther e is a canon- ical equiv alence of triangulated monoidal categories DM B , c ( S ) ' D A 1 , c ( S , Q ) + When − 1 is a sum of sq uare in all the r esidue ﬁelds of S , this eq uivalence can be written: DM B , c ( S ) ' D A 1 , c ( S , Q ) . 16.2.17 Consider the Q -linear étale motivic categor y D A 1 , ´ e t (− , Q ) , deﬁned by D A 1 , ´ e t ( S , Q ) = D A 1 ( Sh ´ e t ( Sm / S , Q ) ) (see 5.3.31 ). The étale sheaﬁﬁcation functor induces a mor phism of motivic cate- gories (16.2.17.1) D A 1 ( S , Q ) / / D A 1 , ´ e t ( S , Q ) . W e shall pro v e the f ollo wing result, as an application of Theorem 16.2.13 . Theorem 16.2.18 F or any noetherian sc heme of ﬁnite dimension S , there is a canon- ical equiv alence of categories DM B ( S ) ' D A 1 , ´ e t ( S , Q ) . As f or Theorem 16.2.13 , the idea of this result is from F . Morel who already pro v ed it at least in the case of a base ﬁeld. In order prov e the abo v e Theorem, we shall study the behaviour of the decompo- sition ( 16.2.1.3 ) in D A 1 , ´ e t ( S , Q ) : Lemma 16.2.19 W e have Q − ' 0 in D A 1 , ´ e t ( S , Q ) . 96 i.e. the Grothendieck group of quadratic f orms 370 Beilinson motives and algebraic K -theor y Proof Proceeding as in the proof of Theorem 16.2.13 , w e ma y assume that S is the spectrum of a perfect ﬁeld k . By étale descent, we see that w e ma y replace k b y an y of its ﬁnite extension. In par ticular , we ma y assume that − 1 is a sum of squares in k . But then, by vir tue of Corollar y 16.2.14 , Q − ' 0 in D A 1 ( S , Q ) , so that, by functoriality , Q − ' 0 in D A 1 , ´ e t ( S , Q ) .  Proof (Proof of Theor em 16.2.18 ) Note that the functor ( 16.2.17.1 ) has a fully faithful right adjoint, whose essential image consists of objects of D A 1 ( S , Q ) which satisfy étale descent. As any Beilinson motive satisﬁes étale descent (ﬁrst point of 14.3.4 ), DM B ( S ) can be seen naturally as a full subcategory of D A 1 , ´ e t ( S , Q ) . On the other hand, by vir tue of the preceding lemma, any object of D A 1 ( S , Q ) which satisﬁes étale descent belongs to D A 1 ( S , Q ) + . Hence, b y Theorem 16.2.13 , an y object of D A 1 ( S , Q ) which satisﬁes étale descent is a Beilinson motive. This achie ves the proof.  Remar k 16.2.20 If S is e xcellent, and if all the residue ﬁelds of S are of character istic zero, one can pro ve Theorem 16.2.18 independently of Morel’ s theorem: this follo ws then directly from a descent argument, namely from Corollar y 3.3.38 and from Theorem 16.1.3 . Corollary 16.2.21 F or any regular noetherian scheme of ﬁnite dimension S , w e have canonical isomorphisms Hom D A 1 , ´ e t ( S , Q ) ( Q S , Q S ( p )[ q ]) ' Gr p γ K 2 p − q ( S ) Q . Proof This follo ws immediatel y from Theorem 16.2.18 , by deﬁnition of DM B ( 14.2.14 ).  Corollary 16.2.22 F or any g eometrically unibr anch excellent noetherian sc heme of ﬁnite dimension S , ther e is a canonical equiv alence of symmetric monoidal triangu- lated categories D A 1 , ´ e t ( S , Q ) ' DM ( S , Q ) . Proof This f ollo ws from theorems 16.1.4 and 16.2.18 .  Remar k 16.2.23 The preceding corollar y e xtends immediately to the case of coef- ﬁcients in a Q -algebra R (cf. Example 5.3.36 f or the left hand side and P aragraph 14.2.20 f or the r ight hand side). Corollary 16.2.24 Let S be an excellent noetherian scheme of ﬁnite dimension. An object of D A 1 ( S , Q ) satisﬁes h -descent if and only if it satisﬁes étale descent. Proof This f ollo ws from theorems 16.1.3 and 16.2.18 .  17 Realizations 371 17 Realizations 17.1 Tilting 17.1.1 Let M be a s table per f ect symmetr ic monoidal Sm -ﬁbred combinator ial model category ov er an adequate categor y of S -schemes S , such that Ho ( M ) is motivic, with generating set of twists τ . Consider a homotopy car tesian commutative monoid E in M . Then E - mo d is an Sm -ﬁbred model categor y , such that Ho ( E - mo d ) is motivic, and w e hav e a morphism of motivic categor ies (see 7.2.13 and 7.2.18 ) Ho ( M ) / / Ho ( E - mo d ) , M  / / E ⊗ L M . In practice, all the realization functors are obtained in this wa y (at least ov er ﬁelds), which can be formulated as f ollow s (for simplicity , we shall w ork here in a Q -linear conte xt, but, if w e are ready to consider higher categor ical constructions, there is no reason to make such an assumption). 17.1.2 Consider a quasi-ex cellent noetherian scheme S of ﬁnite dimension, as well as tw o stable symmetr ic monoidal Sm -ﬁbred combinatorial model categor ies M and M 0 o v er the categor y of S -schemes of ﬁnite type such that Ho ( M ) and Ho ( M 0 ) are motivic (as tr iangulated premotivic categories). W e also assume that both Ho ( M ) and Ho ( M 0 ) are Q -linear and separated. Consider a Quillen adjunction (17.1.2.1) ϕ ∗ : M / / o o M 0 : ϕ ∗ , inducing a mor phism of Sm -ﬁbred categor ies (17.1.2.2) L ϕ ∗ : Ho ( M ) / / Ho ( M 0 ) . W e consider both Ho ( M ) and Ho ( M 0 ) as endo wed with their T ate twists, which deﬁnes two motivic subcategor ies of constructible objects Ho ( M ) c and Ho ( M 0 ) c , respectiv ely . The functor L ϕ ∗ preserves constructible objects, and thus deﬁnes a morphism of premotivic categor ies (17.1.2.3) L ϕ ∗ : Ho ( M ) c / / Ho ( M 0 ) c . Proposition 17.1.3 Under the assumptions of 17.1.2 , if, f or any r egular S -scheme of ﬁnite type X , and f or any integ ers p and q , the map Hom Ho ( M )( X ) ( 1 X , 1 X ( p )[ q ]) / / Hom Ho ( M 0 )( X ) ( 1 X , 1 X ( p )[ q ]) is bi jectiv e, then the morphism ( 17.1.2.3 ) is an equiv alence of premo tivic categories. Mor eov er , if both Ho ( M ) and Ho ( M 0 ) are compactly g enerated by their T ate twists, then the mor phism ( 17.1.2.2 ) is an equivalence of motivic categories. 372 Beilinson motives and algebraic K -theory Proof Note ﬁrst that, f or any separated S -scheme of ﬁnite type X , and f or an y integers p and q , the map Hom Ho ( M )( X ) ( 1 X , 1 X ( p )[ q ]) / / Hom Ho ( M 0 )( X ) ( 1 X , 1 X ( p )[ q ]) is bi jectiv e. Indeed, it is equivalent to pro v e that the maps R Γ ( X , 1 X ( p )) / / R Γ ( X , ϕ ∗ ( 1 X )( p )) are isomor phisms in the der iv ed category of Q -vector spaces: by h -descent ( 3.3.37 ), and by vir tue of Gabber’ s weak unif or mization Theorem 4.1.2 , it is suﬃcient to treat the case where X is regular , which is done b y assumption. Let T be an S -scheme of ﬁnite type. T o prov e that the functor L ϕ ∗ : Ho ( M ) c ( T ) / / Ho ( M 0 ) c ( T ) is fully faithful, it is suﬃcient to choose tw o small f amilies A and B of objects of Ho ( M )( T ) such that the thick subcategor y g enerated by A (by B , respectivel y) contains Ho ( M )( T ) , and to check that the map Hom Ho ( M )( T ) ( A , B ) / / Hom Ho ( M 0 )( T ) ( ϕ ∗ ( A ) , ϕ ∗ ( B )) are bijectiv e, where A and B run o ver A and B , respectivel y . By vir tue of Proposition 4.2.13 , it is thus suﬃcient to prov e that, for an y separated smooth mor phism f : X / / T , f or any projective morphism g : Y / / T , and f or any integ ers p and q , the map Hom Ho ( M ) ( L f ] ( 1 X ) , R g ∗ ( 1 Y )( p )[ q ]) / / Hom Ho ( M 0 ) ( L f ] ( 1 X ) , R g ∗ ( 1 Y )( p )[ q ]) is an isomor phism. Consider the pullback square X × T Y pr 2 / / pr 1   Y g   X f / / T From Proposition 2.4.53 , the functor ϕ ∗ commutes with f ! when f is a separated mor - phism of ﬁnite type. One then easil y concludes using this fact and the isomor phisms (obtained b y adjunction and smooth (or proper) base chang e) Hom ( L f ] ( 1 X ) , R g ∗ ( 1 Y )( p )[ q ]) ' Hom ( 1 X , L f ∗ R g ∗ ( 1 X )( p )[ q ]) ' Hom ( 1 X , R pr 1 , ∗ L pr ∗ 2 ( 1 X )( p )[ q ]) ' Hom ( 1 X , R pr 1 , ∗ ( 1 X × T Y )( p )[ q ]) , ' Hom ( 1 X × T Y , 1 X × S Y ( p )[ q ]) , 17 Realizations 373 that ( 17.1.2.3 ) is fully faithful and that Ho ( M 0 ) c ( T ) is the thick subcategor y generated b y the imag e by L ϕ ∗ of constructible objects of Ho ( M )( T ) . In other w ords, the functor ( 17.1.2.3 ) is an equivalence of categor ies. If, moreo ver , both Ho ( M ) and Ho ( M 0 ) are compactly generated b y their T ate twists, then the sum preser ving e xact functor L ϕ ∗ : Ho ( M )( T ) / / Ho ( M 0 )( T ) is an equiv alence at the lev el of compact objects, hence is an equivalence of categor ies ( 1.3.20 ).  17.1.4 U nder the assumptions of 17.1.2 , assume that M and M 0 are strongl y Q - linear ( 7.1.4 ), left proper , tractable, satisfy the monoid axiom, and hav e coﬁbrant unit objects. Let E 0 be a ﬁbrant resolution of 1 in M 0 ( Sp ec ( k ) ) . By vir tue of Theorem 7.1.8 , we may assume that E 0 is a ﬁbrant and coﬁbrant commutative monoid in M 0 . Then R ϕ ∗ ( 1 ) = ϕ ∗ ( E 0 ) is a commutative monoid in M . Let E be a coﬁbrant resolution of ϕ ∗ ( E 0 ) in M ( Sp ec ( k ) ) . Using Theorem 7.1.8 , we may assume that E is a ﬁbrant and coﬁbrant commutative monoid, and that the map E / / R ϕ ∗ ( E 0 ) is a mor phism of commutativ e monoids (and a weak equivalence by construction). W e can see E and E 0 as car tesian commutative monoids in M and M 0 respectiv ely (b y consider ing their pullbacks along morphisms of ﬁnite type f : X / / Sp ec ( k ) ). W e obtain the essentially commutative diag ram of left Quillen functors belo w (in which the lo wer horizontal map is the functor induced by ϕ ∗ and b y the chang e of scalars functor along the map ϕ ∗ ( E ) / / E 0 ): M / /   M 0   E - mo d / / E 0 - mo d (17.1.4.1) where E - mo d and E 0 - mo d are respectiv ely the model premotivic categories of E -modules and E 0 -modules (see Proposition 7.2.11 ). Note fur thermore that the r ight hand v er tical left Quillen functor is a Quillen equiv alence by constr uction (identifying M 0 ( X ) with 1 X -modules, and using the fact that the mor phism of monoids 1 X / / E 0 X is a weak equivalence in M 0 ( X ) ). Theorem 17.1.5 Consider the assumptions of 17.1.4 , with S = Sp ec ( k ) the spectr um of a ﬁeld k . W e suppose furthermore that one of the follo wing conditions is v eriﬁed. (i) The ﬁeld k is per f ect. (ii) The motivic categories Ho ( M ) and Ho ( M 0 ) ar e continuous and semi-separated. Then the morphism Ho ( E - mo d ) c / / Ho ( E 0 - mo d ) c ' Ho ( M 0 ) c 374 Beilinson motives and algebraic K -theory is an equivalence of motivic categories. U nder these identiﬁcations, the morphism ( 17.1.2.3 ) corr esponds to the c hang e of scalar functor Ho ( M ) c / / Ho ( M 0 ) c ' Ho ( E - mo d ) c , M  / / E ⊗ L M . If moreo ver both Ho ( M ) and Ho ( M 0 ) are compactly g enerated by their T ate twists, then these identiﬁcations extend to non-constructible objects, so that, in par ticular , the morphism ( 17.1.2.2 ) corresponds to the chang e of scalar functor Ho ( M ) / / Ho ( M 0 ) ' Ho ( E - mo d ) , M  / / E ⊗ L M . Remar k 17.1.6 This theorem can be thought as (a par t of ) a tilting theor y f or motivic (homotop y) categories. Remark that the theorem abov e readily implies that the morphism of motivic categor ies ϕ ∗ : Ho ( M ) c / / Ho ( M 0 ) commutes with the six operations (because the, b y virtue of Theorem 4.4.25 , the functor M  / / E ⊗ L M has this proper ty , as well as the inclusion Ho ( M 0 ) c ⊂ Ho ( M 0 ) ). Proof For any regular k -scheme of ﬁnite type X , and f or an y integers p and q , the map Hom Ho ( M )( X ) ( 1 X , E X ( p )[ q ]) / / Hom Ho ( M 0 )( X ) ( 1 X , E 0 X ( p )[ q ]) is bi jective: this is easy to chec k whene v er X is smooth o ver k , whic h pro v es the assertion under condition (i), while, under condition (ii), we see immediately from Proposition 4.3.16 that we ma y assume condition (i). The ﬁrst assertion is then a special case of the ﬁrst asser tion of Proposition 17.1.3 . Similar ly , b y Proposition 7.2.7 , the second asser tion f ollo ws from the second asser tion of Proposition 17.1.3 .  Example 17.1.7 Let M be the stable Sm -ﬁbred model categor y of T ate spectra, so that Ho ( M ) = D A 1 , Q , and write M B f or the left Bousﬁeld localization of M by the class of H B -equiv alences (see 14.2.3 ), so that Ho ( M B ) = DM B . Let k be a ﬁeld of characteristic zero, endow ed with an embedding σ : k / / C . Giv en a comple x analytic manif old X , let M a n ( X ) be the categor y of comple x es of shea v es of Q -vector spaces on the smooth analytic site of X (i.e. on the categor y of smooth analytic X -manif olds, endow ed with the Grothendieck topology cor respond- ing to open cov er ings), endo w ed with its local model structure (see [ A y o07b , 4.4.16] and [ A y o10 ]). W e shall wr ite M eﬀ Betti ( X ) f or the stable left Bousﬁeld localization of M a n ( X ) by the maps of shape Q ( U × D 1 ) / / Q ( U ) f or any analytic smooth X ( C ) - manif old U (where D 1 denotes the closed unit disc). W e deﬁne at last M Betti ( X ) as the stable model categor y of analytic Q ( 1 )[ 1 ] -spectra in M eﬀ Betti ( X ) , where Q ( 1 )[ 1 ] stands f or the cokernel of the map Q / / Q ( A 1 , an − { 0 } ) induced by 1 ∈ C ; see [ A y o10 , section 1]. Giv en a k -scheme of ﬁnite type X , we shall write (17.1.7.1) D Betti ( X ) : = Ho ( M Betti ( X )) 17 Realizations 375 (where the topological space X ( C ) is endo wed with its canonical anal ytic structure). A ccording to [ A y o10 , 1.8 and 1.10], there e xists canonical equiv alences of categor ies (17.1.7.2) D Betti ( X ) ' Ho ( M eﬀ Betti ( X )) ' D ( X ( C ) , Q ) , where D ( X ( C ) , Q ) stands f or the (unbounded) derived category of the abelian cate- gory of shea v es of Q -vector spaces on the small site of X ( C ) . By vir tue of [ A yo10 , section 2], there e xists a symmetr ic monoidal left Quillen morphism of monoidal Sm -ﬁbred model categor ies o v er the categor y of k -schemes of ﬁnite type (17.1.7.3) An ∗ : M / / M Betti , which induces a mor phism of motivic categor ies o ver the category of k -schemes of ﬁnite type. Hence R An ∗ ( 1 ) is a r ing spectr um in D A 1 , Q ( Sp ec ( k ) ) which represents Betti cohomology of smooth k -schemes. As D Betti satisﬁes étale descent, it f ollo ws from Corollar y 3.3.38 that it satisﬁes h -descent, from which, b y vir tue of Theorem 16.1.3 , the mor phism ( 17.1.7.3 ) deﬁnes a left Quillen functor (17.1.7.4) An ∗ : M B / / M Betti , hence giv es r ise to a mor phism of motivic categor ies (17.1.7.5) DM B / / D Betti , the Betti r ealization functor of Beilinson motiv es. App yling Theorem 17.1.5 to ( 17.1.7.4 ), w e obtain a commutativ e ring spectrum E Betti = R An ∗ ( 1 ) which represents Betti cohomology of smooth k -schemes, such that the restriction of the functor ( 17.1.7.5 ) to constr uctible objects corresponds to the chang e of scalars functors M  / / E Betti ⊗ L M : (17.1.7.6) DM B , c ( X ) / / Ho ( E Betti - mo d ) c ( X ) ' D b c ( X ( C ) , Q ) . It should be pointed out that, here, D b c ( X ( C ) , Q ) means the der iv ed categor y of shea v es which are constructible of g eometric origin (i.e. constructible in the alg ebraic sense, and not in the analytic sense). In other words, once Betti cohomology of smooth k -schemes is kno wn, one can reconstruct canonically the bounded der iv ed categor ies of constr uctible sheav es of geometric or igin on X ( C ) f or any k -scheme of ﬁnite type X , from the theory of mix ed motiv es. W e expect all the realization functors to be of this shape (which should f ollo w from (some variant of ) Theorem 17.1.5 ): the (absolute) cohomology of smooth k -schemes with constant coeﬃcients determines the deriv ed categories of constructible shea ves of geometric origin ov er an y k -scheme of ﬁnite type. For instance, the geometric par t of the theory of v ar iations of mixed Hodg e str uctures should be obtained from Deligne cohomology , seen as a r ing spectrum in DM B ( k ) (or , more precisely , in M B ( k ) ). W ork in prog ress of Brad Dre w [ Dre13 , Dre18 ] goes in this direction. 376 Beilinson motives and algebraic K -theor y 17.2 Mix ed W eil cohomologies Let S be an ex cellent (regular) noether ian scheme of ﬁnite dimension, and K a ﬁeld of characteristic zero, called the ﬁeld of coeﬃcients . 17.2.1 Let E be a Nisnevic h sheaf of commutativ e diﬀerential graded K -alg ebras (i.e. is a commutative monoid in the categor y of sheav es of complex es of K -v ector spaces). W e shall wr ite H n ( X , E ) = Hom D eﬀ A 1 , Q ( X ) ( Q X , E [ n ]) f or any smooth S -scheme of ﬁnite type X , and for an y integer n (note that, if E satisﬁes Nisne vich descent and is A 1 -homotop y in variant, which we can alwa ys assume, using 7.1.8 , then H n ( X , E ) = H n ( E ( X )) ). W e introduce the f ollo wing axioms : W1 Dimension .— H i ( S , E ) ' ( K if i = 0 , 0 other wise. W2 Stability .— dim K H i ( G m , E ) = ( 1 if i = 0 or i = 1 , 0 other wise. W3 Künneth formula .— For any smooth S -schemes X and Y , the exterior cup product induces an isomor phism Ê p + q = n H p ( X , E ) ⊗ K H q ( Y , E ) ∼ / / H n ( X × S Y , E ) . W3 0 W eak Künneth formula .— For any smooth S -scheme X , the exterior cup product induces an isomor phism Ê p + q = n H p ( X , E ) ⊗ K H q ( G m , E ) ∼ / / H n ( X × S G m , E ) . 17.2.2 U nder assumptions W1 and W2, w e will call any non-zero element c ∈ H 1 ( G m , E ) a stability class . Note that such a class cor responds to a non-trivial map c : Q S ( 1 ) / / E in D eﬀ A 1 , Q ( S ) (using the decomposition Q ( G m ) = Q ⊕ Q ( 1 )[ 1 ] ). In par ticular , possibly after replacing E b y a ﬁbrant resolution (so that E is homotop y inv ar iant and satisﬁes Nisnevic h descent), such a stability class can be lifted to an actual map of comple x es of preshea ves. Such a lift will be called a stability structur e on E . Deﬁnition 17.2.3 A sheaf of commutativ e diﬀerential graded K -algebras E as abov e is a mixed W eil cohomology (resp. a stable cohomology ) if it satisﬁes the proper ties W1, W2 and W3 (resp. W1, W2 and W3 0 ) stated abo ve. 17 Realizations 377 Proposition 17.2.4 Let E be a stable cohomology. There exists a (commutativ e) ring spectrum E in DM B ( S ) with the follo wing properties. (i) F or any smoot h S -sc heme X , and any integ er i , there is a canonical isomor phism of K -v ector spaces H i ( X , E ) ' Hom DM B ( S ) ( M S ( X ) , E [ i ]) . (ii) Any c hoice of a stability structure on E deﬁnes a map Q ( 1 ) / / E in DM B ( S ) , whic h induces an E -linear isomorphism E ( 1 ) ' E . Proof One deﬁnes explicitl y the commutativ e r ing spectr um E as f ollow s. Firs t, by virtue of Theorem 7.1.8 , we may assume that E is a Nisnevic h sheaf of commutativ e diﬀerential graded algebras and is ﬁbrant f or the A 1 -local projectiv e model structure: f or an y smooth S -scheme X , the two maps H n ( E ( X )) / / H n Nis ( X , E ) / / H n Nis ( X × A 1 , E ) are isomorphisms for any n ∈ Z . Let L be the constant Nisne vich sheaf of complex es of K -v ector spaces associated to the kernel of the map induced by S = { 1 } ⊂ G m : L = ker  E ( G m ) 1 ∗ / / E ( S )  . W e remark that L is coﬁbrant, and one deﬁnes E n = Hom ( L ⊗ n , E ) this sheaf being endow ed with an action of the symmetric g roup on n letters by permuting the factors on L ⊗ n . W e then hav e canonical pair ings Hom ( L ⊗ m , E ) ⊗ Q Hom ( L ⊗ n , E ) / / Hom ( L ⊗ m + n , E ⊗ Q E ) / / Hom ( L ⊗ m + n , E ) which tur n the collection E = { E n } n ≥ 0 into a commutative monoid in the categor y of symmetric sequences of sheav es of comple xes of Q -v ector spaces; see Deﬁnition 5.3.7 . On the other hand, w e remark that L is the constant sheaf associated to Γ ( S , Hom ( Q ( 1 )[ 1 ] , E )) , from which we deduce that there is a natural map L / / Hom ( Q ( 1 )[ 1 ] , E ) which can be transposed into a canonical map Q ( 1 )[ 1 ] / / Hom ( L , E ) = E 1 . This deﬁnes a canonical structure of commutativ e monoid in the categor y symmetric Q ( 1 )[ 1 ] -spectra on the symmetr ic sequence E (see Remark 5.3.10 ) 97 . 97 Here, we w ork with Q ( 1 )[ 1 ] -spectra. How ev er , the paper [ CD12 ] is wr itten in the language of symmetric Q ( 1 ) -spectra. W e leav e as an ex ercise to the reader the task of the translation, which 378 Beilinson motiv es and algebraic K -theor y By vir tue of [ CD12 , Proposition 2.1.6], for any smooth S -scheme X , and any integer i , there is a canonical isomor phism of K -vector spaces H i ( X , E ) ' Hom D A 1 , Q ( S ) ( M S ( X ) , E [ i ]) , and any choice of a stability str ucture on E deﬁnes an isomor phism E ( 1 ) ' E . Moreo v er , [ CD12 , corollar y 2.2.8] and Theorem 12.2.10 asser t that this r ing spectr um E is or iented, so that, by Corollar y 14.2.16 , E is an H B -module, i.e. belongs to DM B ( S ) .  17.2.5 Giv en a stable cohomology E and its associated r ing spectr um E , we can see E as a car tesian commutativ e monoid: we deﬁne, f or an S -scheme X , with structural map f : X / / S : E X = L f ∗ ( E ) (which means that we take a coﬁbrant replacement E 0 of E in the model categor y of commutativ e monoids of the categor y of T ate spectra, and deﬁne E X = f ∗ ( E 0 ) ), and put (17.2.5.1) D ( X , E ) : = Ho ( E - mo d )( X ) = Ho ( E X - mo d ) . W e thus ha v e realization functors (17.2.5.2) DM B ( X ) / / D ( X , E ) , M  / / E X ⊗ L X M which commute with the six operations of Grothendiec k if ev er S is the spectrum of a ﬁeld (Theorem 4.4.25 ). Further more, D (− , E ) is a motivic category which is Q -linear (in fact K -linear), separated, and continuous. For an S -scheme X , deﬁne H q ( X , E ( p )) = Hom DM B ( X ) ( Q X , E ( p )[ q ]) ' Hom D ( X , E ) ( E X , E X ( p )[ q ]) (this notation is compatible with 17.2.1 by virtue of Proposition 17.2.4 ). Corollary 17.2.6 Any s table cohomology (in par ticular , any mixed W eil cohomol- ogy) extends naturally to S -schemes of ﬁnite type, and this extension satisﬁes coho- mological h -descent (in par ticular , étale descent as well as proper descent). Proof This f ollo ws immediately from the constr uction abo ve and from Theorem 14.3.4 .  17.2.7 W e denote by D ∨ ( S , E ) the localizing subcategor y of D ( S , E ) generated by its rigid objects (i.e. by the objects which hav e strong duals). For instance, f or any smooth and proper S -scheme X , E ( X ) = E ⊗ L S M S ( X ) belongs to D ∨ ( S , E ) ; see 2.4.31 . consists in checking that the functor { E n } n ≥ 0  / / { E n [ n ] } n ≥ 0 is a symmetric monoidal left Quillen equivalence from symmetr ic Q ( 1 )[ 1 ] -spectra to symmetr ic Q ( 1 ) -spectra, which is also a right Quillen functor (and thus, in par ticular , preserves and detects stable A 1 -equiv alences). 17 Realizations 379 If we denote by D ( K ) the (unbounded) der iv ed category of the abelian categor y of K -v ector spaces, we get the f ollo wing inter pretation of the Künneth formula. Theorem 17.2.8 If E is a mixed W eil cohomology, then the functor R Hom E ( E , −) : D ∨ ( S , E ) / / D ( K ) is an equiv alence of symmetric monoidal triangulated categories. Proof This is [ CD12 , theorem 2.6.2].  Theorem 17.2.9 If S is the spectrum of a ﬁeld, then D ∨ ( S , E ) = D ( S , E ) . Proof This f ollo ws then from Corollary 4.4.17 .  Remar k 17.2.10 It is not reasonable to expect the analog of Theorem 17.2.9 to hold whenev er S is of dimension > 0 ; see (the proof of ) [ CD12 , corollar y 3.2.7]. Heur isti- cally , f or higher dimensional schemes X , the rigid objects of D ( X , E ) are e xtensions of some kind of locally constant shea v es (in the ` -adic setting, these cor respond to Q ` -faisceaux lisses ). Corollary 17.2.11 If E is a mixed W eil cohomology, and if S is t he spectr um of a ﬁeld, then the functor R Hom E ( E , −) : D ( S , E ) / / D ( K ) is an equiv alence of symmetric monoidal triangulated categories. Remar k 17.2.12 This result can be thought as a tilting theory f or the spectra associated with mix ed W eil cohomologies. 17.2.13 Assume that E is a mixed W eil cohomology , and that S is the spectr um of a ﬁeld k . For each k -scheme of ﬁnite type X , denote b y D c ( X , E ) the categor y of constructible objects of D ( X , E ) : by deﬁnition, this is the thick tr iangulated subcategory of D ( X , E ) g enerated b y objects of shape E ( Y ) = E ⊗ L X M X ( Y ) f or Y smooth o v er X (we can drop T ate twists because of 17.2.4 (ii)). The categor y D c ( X , E ) also coincides with the category of compact objects in D ( X , E ) ; see 1.4.11 . W rite D b ( K ) for the bounded derived categor y of the abelian category of ﬁnite dimensional K -v ector spaces. Note that D b ( K ) is canonically equiv alent to the homotopy category of per f ect comple xes of K -modules, i.e. to the category of compact objects of D ( K ) . Corollary 17.2.14 Under the assumptions of 17.2.13 , we hav e a canonical equiv a- lence of symmetric monoidal triangulated categories D c ( Sp ec ( k ) , E ) ' D b ( K ) . Proof This f ollo ws from 17.2.11 and from the fact that equiv alences of categor ies preserve compact objects.  380 Beilinson motives and algebraic K -theory Corollary 17.2.15 Under the assumptions of 17.2.13 , if E 0 is another K -linear stable cohomology with associated ring spectrum E 0 , any mor phism of presheav es of com- mutativ e diﬀerential K -alg ebras E / / E 0 inducing an isomorphism H 1 ( G m , E ) ' H 1 ( G m , E 0 ) giv es a canonical isomorphism E ' E 0 in the homo topy category of commutativ e ring spectra. In particular , w e g et canonical equivalences of categories D ( X , E ) ' D ( X , E 0 ) f or any k -scheme of ﬁnite type X (and these are compatible with the six operations of Gro thendiec k , as well as with the realization functor s). Proof This f ollo ws from Theorem 17.2.9 and from [ CD12 , Theorem 2.6.5].  The preceding corollary can be stated in the f ollo wing wa y: if E and E 0 are tw o (strict) commutativ e r ing spectra associated to K -linear mixed W eil cohomologies deﬁned on smooth k -schemes E and E 0 , respectivel y , then an y mor phism E / / E 0 in the homotop y category of (commutativ e) monoids in the model categor y of K - linear T ate spectra is inv er tible. Moreo v er , E is isomor phic to E 0 if and only if E is isomorphic to E 0 (in the appropr iate homotopy categor ies of commutativ e monoids). T o be more precise (and more g eneral), this last asser tion f ollow s immediately from Corollary 17.2.15 and from the follo wing result. Proposition 17.2.16 Let E be a commutativ e monoid in the A 1 -stable model cat- egor y of sheav es of complexes of symme tric Q ( 1 )[ 1 ] -spectra o ver the Nisnevich smooth site of k . Suppose that ther e exists an isomorphism E ( 1 ) ' E in the homotopy category of E -modules and that H n ( Sp ec ( k ) , E ) = ( K if n = 0 , 0 other wise. Then E = R Γ (− , E ) is a stable cohomology theor y , and the commutativ e ring spectrum E associated to E by Proposition 17.2.4 is canonically isomorphic to E in the homot opy category of (strict) commutativ e ring spectra. Proof By vir tue of Theorem 7.1.8 , w e may assume that E is (coﬁbrant and) ﬁbrant. The r ing spectr um E is deﬁned by a symmetr ic sequence of comple xes of Nisnevich shea v es of K -v ector spaces E n , n ≥ 0 , (endo wed with an action of the symmetric group on n -letters), together with maps σ n : E n ( 1 )[ 1 ] / / E n + 1 inducing quasi- isomorphisms E n ∼ / / Hom ( K ( 1 )[ 1 ] , E n + 1 ) as w ell as pair ings E m ⊗ K E n / / E m + n satisfying a f ew compatibilities. In par ticular , E = R Γ (− , E ) = E 0 17 Realizations 381 is naturally endow ed with a structure of Nisne vich sheaf of commutative diﬀeren- tial graded algebras which satisﬁes Nisnevich descent and A 1 -homotop y inv ar iance. Moreo v er , f or any integer n ≥ 0 , the Nisnevic h sheaf of complex es of K -v ector spaces E n also has the properties of Nisnevic h descent and of A 1 -homotop y inv ar iance, and is naturally endow ed with a structure of E -module. It is clear that E is a stable coho- mology theor y , so that (the proof of ) Proposition 17.2.4 provides a commutative r ing spectrum E associated to it. With the notations introduced in the proof of Proposition 17.2.4 , w e kno w that E is made of the symmetr ic sequence { E n = Hom ( L ⊗ n , E ) } n ≥ 0 , where L is the constant sheaf associated to Γ ( S , Hom ( K ( 1 )[ 1 ] , E )) . Let us deﬁne L = L ( 1 )[ 1 ] . W e deﬁne a new symmetric sequence E b y the f or mula E n = Hom ( L ⊗ n , E n ) , n ≥ 0 , where the symmetr ic group acts through the diagonal S n / / S n × S n b y per mutation of the factors on L ⊗ n and by the structural action on E n . W e see that E is a commutativ e monoid in the categor y of symmetric seq uences with the pair ings deﬁned b y the tensor product map Hom ( L ⊗ m , E m ) ⊗ K Hom ( L ⊗ n , E n ) / / Hom ( L ⊗ m + n , E m ⊗ K E n ) composed with the multiplication of E : Hom ( L ⊗ m + n , E m ⊗ K E n ) / / Hom ( L ⊗ m + n , E m + n ) . Finall y , we can compose the transposition of the map σ 1 : E ( 1 )[ 1 ] / / E 1 , with the structural map K ( 1 )[ 1 ] / / Hom ( L , E ) = E 1 , to obtain: K ( 1 )[ 1 ] / / Hom ( L , E ) / / Hom ( L , Hom ( K ( 1 )[ 1 ] , E 1 ) ' Hom ( L , E 1 ) = E 1 . This deﬁnes a structure of commutative r ing spectr um on E . Note that L is chain homotop y equivalent to K [− 1 ] , so that the functors Hom ( L ⊗ n , −) preserve quasi- isomorphisms (more precisely , L is concentrated in cohomological degree 1 , and its ﬁrst cohomology sheaf is the constant sheaf associated to the K -v ector space of dimension one H 1 ( G m , E ) ). Therefore, one has a quasi-isomor phism of commutative monoids of K -linear T ate spectra E / / E , deﬁned by the canonical maps Hom ( L ⊗ n , E ) / / Hom ( L ⊗ n , Hom ( K ( n )[ n ] , E n ) ' Hom ( L ⊗ n , E n ) . It remains to produce a quasi-isomorphism of commutative monoids of T ate spectra E / / E . W e ha v e a structural map K ( 1 )[ 1 ] / / Hom ( L , E ) which can be transposed into a map L = L ( 1 )[ 1 ] / / E = E 0 . As E is a commutativ e monoid and each E n an E -module, w e hav e natural maps L ⊗ n ⊗ K E n / / E ⊗ n ⊗ K E n / / E ⊗ K E n / / E n 382 Beilinson motives and algebraic K -theory which can be transposed into S n -equiv ariant maps E n / / Hom ( L ⊗ n , E n ) = E n . These deﬁne a mor phism of commutativ e monoids of K -linear T ate spectra E / / E . It remains to chec k that the maps E n / / E n are quasi-isomorphisms f or each n ≥ 0 . As Hom ( K ( n )[ n ] , E ) ' E n , we can replace E n b y Hom ( L ⊗ n , E ) . The case n = 1 is then a ref or mulation of Proposition 17.2.4 ( ii ), and the general case f ollow s b y an obvious induction.  Theorem 17.2.17 Under the assumptions of par agr aph 17.2.13 , the six operations of Gro thendieck preserve constructibility in the motivic categor y D (− , E ) , as deﬁned in P arag raph 17.2.5 . Proof Observe that the motivic category D (− , E ) is Q -linear and separated (because DM B is so, see 7.2.18 ), as w ell as τ -compatible (because by Proposition 4.4.16 , it is ev en τ -dualizable which is strong er than τ -compatible; see Deﬁnition 4.4.13 ). W e conclude with 4.2.29 .  17.2.18 As a consequence, we hav e, f or any k -scheme of ﬁnite type X , a realization functor DM B , c ( X ) / / D c ( X , E ) and we deduce from Theorem 4.4.25 that it preserves all of Grothendieck six op- erations. For X = Sp ec ( k ) , by vir tue of Corollar y 17.2.14 , this cor responds to a symmetric monoidal e xact realization functor R : DM B , c ( Sp ec ( k ) ) / / D b ( K ) . This leads to a ﬁniteness result: Corollary 17.2.19 Under the assumptions of 17.2.13 , f or any k -sc heme of ﬁnite type X , and for any objects M and N in D c ( X , E ) , Hom E ( M , N [ n ]) is a ﬁnite dimensional K -v ector space, and it is trivial for all but a ﬁnite number of values of n . Proof Let f : X / / Sp ec ( k ) be the structural map. By vir tue of 17.2.17 , as M and N are constructible, the object R f ∗ R Hom X ( M , N ) is constructible as well, i.e. is a compact object of D ( Sp ec ( k ) , E ) . But R Hom E ( M , N ) is nothing else than the image of R f ∗ R Hom X ( M , N ) b y the e quiv alence of categories giv en b y Corollary 17.2.11 . Hence R Hom E ( M , N ) is a compact object of D ( K ) , which means that it belongs to D b ( K ) .  17.2.20 For a K -vector space V and an integer n , deﬁne V ( n ) = ( V ⊗ K Hom K ( H 1 ( G m , E ) ⊗ n , K ) if n > 0 , V ⊗ K H 1 ( G m , E ) ⊗(− n ) if n ≤ 0 . An y choice of a generator in K (− 1 ) = H 1 ( G m , E ) ' H 2 ( P 1 k , E ) deﬁnes a natural isomorphism V ( n ) ' V f or any integer n . W e ha v e canonical isomor phisms 17 Realizations 383 H q ( X , E ( p )) ' H q ( X , E )( p ) (using the fact that the equiv alence of Corollary 17.2.14 is monoidal). The realization functors ( 17.2.5.2 ) induce in par ticular cy cle class maps cl X : H q B ( X , Q ( p )) / / H q ( X , E )( p ) (and similarly f or cohomology with compact suppor t, for homology , and for Borel- Moore homology). Example 17.2.21 Let k be a ﬁeld of characteristic zero. W e then ha v e a mix ed W eil cohomology E dR deﬁned b y the algebraic de Rham comple x E dR ( X ) = Ω ∗ A / k f or any smooth aﬃne k -scheme of ﬁnite type X = Spec ( A ) (algebraic de Rham cohomology of smooth k -schemes of ﬁnite type is obtained b y Zariski descent); see [ CD12 , 3.1.5]. W e obtain from 17.2.4 a commutative r ing spectr um E dR , and, for a k -scheme of ﬁnite type X , we deﬁne D dR ( X ) = D c ( X , E dR ) . W e thus g et a motivic categor y D dR , and we hav e a natural deﬁnition of algebraic de Rham cohomology of k -schemes of ﬁnite type, giv en by H n dR ( X ) = Hom D dR ( X ) ( E dR , X , E dR , X [ n ]) . This deﬁnition coincides with the usual one: this is tr ue by deﬁnition f or separated smooth k -schemes of ﬁnite type, while the g eneral case follo ws from h -descent ( 17.2.6 ) and from de Jong’s Theorem 4.1.11 (or resolution of singularities à la Hiron- aka). W e ha v e, by construction, a de Rham realization functor R dR : DM B , c ( X ) / / D dR ( X ) which preserves the six operations of Grothendieck (Theorem 4.4.25 ). In particular, w e hav e cycle class maps H q B ( X , Q ( p )) / / H q dR ( X )( p ) . Note that, for any ﬁeld e xtension k 0 / k , we hav e natural isomor phisms H n dR ( X ) ⊗ k k 0 ' H n dR ( X × Spec ( k ) Sp ec ( k 0 ) ) . Example 17.2.22 Let k be a ﬁeld of characteristic zero, which is algebraicall y closed and complete with respect to some valuation (archimedian or not). W e can then deﬁne a stable cohomology E dR , an as analytic de Rham cohomology of X an , for an y smooth k -scheme of ﬁnite type X ; see [ CD12 , 3.1.7]. As abo v e, w e g et a ring spectrum E dR , an , and f or any k -scheme of ﬁnite type, a categor y of coeﬃcients 384 Beilinson motives and algebraic K -theor y D dR , an ( X ) = D c ( X , E dR , an ) , which allo ws to deﬁne the analytic de Rham cohomology of any k -scheme of ﬁnite type X b y H n dR , an ( X ) = Hom D dR , an ( X ) ( E dR , an , X , E dR , an , X [ n ]) . W e also ha v e a realization functor R dR , an : DM B , c ( X ) / / D dR , an ( X ) which preser v es the six operations of Grothendieck. W e then ha v e a mor phism of stable cohomologies E dR / / E dR , an which happens to be a quasi-isomor phism locally f or the Nisnevic h topology (this is Grothendieck’ s theorem in the case where K is archimedian, and Kiehl’s theorem in the case where K is non-archimedian; an ywa y , one obtains this directly from Corollary 17.2.15 ). This induces a canonical isomor phism E dR ' E dR , an in the homotopy category of commutative ring spectra. In particular, E dR , an is a mixed W eil cohomology , and, for an y k -scheme of ﬁnite type, we ha v e natural equiv alences of categor ies D dR ( X ) / / D dR , an ( X ) , M  / / E dR , an ⊗ L E dR M which commute with the six operations of Grothendieck and are compatible with the realization functors. Note that, in the case k = C , E dR , an coincides with Betti cohomology (after tensorization by C ), so that we hav e canonical fully faithful functors D Betti , c ( X ) ⊗ Q C / / D dR , an ( X ) which are compatible with the realization functors. More precisely , it f ollo ws from Proposition 17.2.16 that the Betti spectrum E Betti , obtained by appy ling Theorem 17.1.5 to A y oub’s realization functor ( 17.1.7.4 ), is the spectr um associated to Q -linear Betti cohomology , seen as a mixed W eil cohomology , from Proposition 17.2.4 . There- f ore, the holomor phic P oincaré Lemma, together with Corollary 17.2.15 , pro vide an isomorphism E Betti ⊗ Q C ' E dR , an in the homotopy categor y of commutative monoids of the model categor y of C -linear T ate spectra. W e thus hav e tr iangulated equivalences of categor ies D b c ( X ( C ) , C ) ' Ho ( E Betti ⊗ Q C - mo d ) c ( X ) ' D dR , an ( X ) 17 Realizations 385 which commute with the six operations as w ell as with the realization functors. In par ticular , f or X smooth, by the Riemann-Hilber t cor respondence, D dR , an ( X ) is equivalent to the bounded derived categor y of analytic regular holonomic D - modules on X which are constructible of g eometr ic or igin. But we can go backw ard: as pro v ed by Brad Drew in his thesis [ Dre13 ], using Corollar y 17.2.15 , one can actually show directly (i.e. motivically) that, for X smooth, D dR , an ( X ) is equiv alent to the bounded derived categor y of analytic regular holonomic D -modules on X which are constructible of geometric or igin, and thus giv e a ne w algebraic proof of the Riemann-Hilber t correspondence. Example 17.2.23 Let V be a complete discrete valuation ring of mixed characteris- tic with per f ect residue ﬁeld k and ﬁeld of functions K . The Monsky- W ashnitzer comple x deﬁnes a stable cohomology E M W o v er smooth V -schemes of ﬁnite type, deﬁned b y E M W ( X ) = Ω ∗ A † / V ⊗ V K f or an y aﬃne smooth V -scheme X = Sp ec ( A ) (the case of a smooth V -scheme of ﬁnite type is obtained by Zariski descent); see [ CD12 , 3.2.3]. Let E M W be the cor - responding r ing spectr um in DM B ( Sp ec ( V ) ) , and wr ite j : Sp ec ( K ) / / Sp ec ( V ) and i : Sp ec ( k ) / / Sp ec ( V ) f or the canonical immersions. As we obviousl y hav e j ∗ E M W = 0 (the Monsky- W ashnitzer cohomology of a smooth V -scheme with empty special ﬁber vanishes), we ha v e a canonical isomor phism E M W ' R i ∗ L i ∗ E M W . W e deﬁne the r igid cohomology spectrum E rig in DM B ( Sp ec ( k ) ) b y the formula E rig = L i ∗ E M W . This is a r ing spectr um associated to a K -linear mixed W eil cohomology: coho- mology with coeﬃcients in E rig coincides with r igid cohomology in the sense of Berthelot, and the K ünneth f or mula for r igid cohomology holds f or smooth and projectiv e k -schemes (as r igid cohomology coincides then with cr ys talline coho- mology), from which w e deduce the Künneth f or mula f or smooth k -schemes of ﬁnite type; see [ CD12 , 3.2.10]. 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Geom. , 5(6):686–702, 2018. 394 Inde x acy clic, H B -acy clic, 348 adequate, category of schemes, 31 adjunction of P -ﬁbred categor ies, see mor - phism of of premotivic categor ies, see mor- phism of Quillen adjunction, 87 admissible topology , see topology admissible, class of mor phisms, 3 algebra E ∞ -algebra, 233 H B -algebra, 351 alteration, 130 Galois alteration, 130 Galois alteration., 152 A uslander -Buchsbaum theorem, 316 base chang e P -base change, 7 proper base chang e, 9 smooth base chang e, 9 biﬁbred category , 7 Bott isomor phism, 335 bounded (topology), 179 bounded generating famil y , 179 Bro wn representability theorem, 24 , 25 , 44 , 51 bundle normal, 69 , 70 tangent, 70 , 78 virtual v ector bundle, 60 cartesian mor phism, see mor phism cd-structure, 34 , 295 lo w er , 35 upper , 34 Cho w’ s lemma, 31 , 50 class Chern, 320 , 342 fundamental, 339 , 355 coeﬃcients, for Beilinson motives, 352 , 359 coﬁbration, 171 termwise, 84 coherence, 5 , 8 , 11 , 19 cohomology algebraic De Rham, 383 analytic De Rham, 383 Beilinson motivic, 350 Betti, 375 Cho w group, see g roup eﬀectiv e motivic, 310 higher Cho w group, see group K -theor y , see K -theor y Landw eber ex act, 347 mix ed W eil, 376 Monsky- W ashnitzer , 385 motivic, 310 representable, 331 rigid, 385 stable, 376 commute, see functor compact, 23 , 181 , 200 , 307 compatible with (a topology) t , 170 , 171 , 177 compatible with transf ers, see topol- ogy compatible with twists, 19 , 29 comple x algebraic De Rham, 383 Monsky- W ashnitzer , 385 conservativ e, 47 , 52 , 53 , 148 constructibility , see constructible constructible, see also τ -constructible 29 ( Z × τ ) -constructible, 215 τ -constructible, 29 , 131 – 160 , 181 , 200 Beilinson motiv e, 356 – 359 motiv e, 306 , 307 motivic comple x, 306 A 1 -contractible, 191 cotransv ersality proper ty , 9 co v er , 97 Galois co v er , 114 INDEX 395 h -co ver , 122 , 129 pseudo-Galois co v er , 114 qfh -co ver , 120 , 122 cy cle Λ -cy cle, 243 , see also cycle 244 Λ -univ ersal (mor phism of ), 257 associated, 244 Hilbert, 245 pre-special (mor phism of ), 248 pseudo-equidimensional, 272 pullback, 252 associativity , 260 commutativity , 259 of Hilber t cy cles, 246 projection f ormulas, 262 pushf or ward, 244 restriction, 245 Samuel specialization, 269 special (mor phism of ), 250 specialization, 249 standard form, 244 trivial, 264 decomposition, A dams, 346 , 352 def or mation space, 69 derivator , Grothendieck, 94 , 104 , 106 , 175 derived derived P -premotivic categor y , 186 descent c dh -descent, 111 , 112 , 123 cohomological h -descent, 378 cohomological t -descent, 170 , 175 étale, 120 , 122 , 126 , 354 , 370 Galois, 247 h -descent, 123 , 125 , 126 , 354 , 370 Nisnevic h, 109 , 110 qfh -descent, 120 , 123 , 125 , 309 t -descent, 99 , 175 , 187 , 210 dg-structure, 170 , 185 diagram S -diagram, 81 , 174 , 185 direct imag e with compact suppor t, see functor , left e xceptional divisor W eil, 315 domain (of a Λ -cycle), 244 dual, strong, 67 , 321 duality local duality , 158 duality , Grothendieck, 160 , 358 dualizable, strongl y , 67 , 73 dualizing τ -dualizing, 153 embedding, Segre, 333 enlarg ement, of premotivic categor ies, see premotivic equidimensional absolutely , 245 ﬂat mor phism, 245 equiv alence A 1 -equiv alence, 191 H B -equiv alence, 348 of motivic categor ies, 371 of tr iangulated monoidal cate- gories, 350 , 361 strong A 1 -equiv alence, 191 termwise w eak equiv alence, 84 w eak equiv alence of commuta- tiv e monoids, 228 w eak equiv alence of modules, 233 w eak equivalence of monoids, 227 W -equiv alence, 184 equiv alence, of categor ies, 33 e x ceptional functor , see functor e x chang e isomorphism, 7 , 13 , 49 , 223 – 226 morphism, see ex chang e trans- f or mation transf or mation, 5 , 8 , 11 , 12 , 17 , 18 , 42 ﬁbrant A 1 -ﬁbrant, 191 ﬁbration 396 t -ﬁbration, 171 algebraic Hopf, 363 of commutativ e monoids, 228 of modules, 233 of monoids, 227 termwise, 84 W -ﬁbration, 184 ﬁbred ﬁbred category , 3 monoidal pre- P -ﬁbred category , 10 monoidal P -ﬁbred categor y , 12 model (category), 26 of ﬁnite cor respondences, 284 P -ﬁbred categor y , 7 τ -generated, see generated abelian, 22 abelian monoidal, 22 canonical, 7 canonical monoidal, 12 complete, 7 complete monoidal, 12 ﬁnitely τ -presented , see ﬁnitely presented geometricall y generated, see generated Grothendieck abelian, 22 Grothendieck abelian monoidal, 22 homotop y , 27 homotop y monoidal, 27 model, 25 monoidal P -ﬁbred model cat- egory , 172 triangulated, 23 triangulated monoidal, 23 pre- P -ﬁbred categor y , 4 ﬁltration, γ -ﬁltration, 346 ﬁnite cor respondence, 275 composition, 277 ﬁnite S -cor respondence, see ﬁ- nite cor respondence graph functor , see functor tensor porduct, see tensor prod- uct transpose, see mor phism ﬁnitely presented ﬁnitely τ -presented, 23 , 181 , 200 , 214 object of a categor y , 22 ﬁniteness theorem, 143 , 357 ﬂasque, t -ﬂasque comple x, 170 f or malism, Grothendieck 6 functors, 350 f or malism, Grothendieck six functors, 77 functor commutes, 7 ev aluation, 82 , 89 , 203 e x ceptional, 43 , 78 graph, 278 inﬁnite suspension, 29 left e x ceptional, 38 Quillen, 90 t -exact endofunctor, 364 Galois group, see g roup generated τ -generated, 15 , 16 , 20 , 28 , 53 , 143 compactly ( Z × τ ) -g enerated, 214 compactly τ -generated, 28 triangulated P -ﬁbred, 24 , 44 , 54 , 181 , 200 compactly generated, 23 triangulated P -ﬁbred, 23 , 356 , 371 , 374 geometricall y generated, 15 w ell generated, 23 triangulated P -ﬁbred, 23 global section, see section group Cho w , 311 Galois group, 114 H -g roup, 333 higher Cho w , 311 Picard, 315 relativ e Picard, 314 henselization, 147 INDEX 397 homeomorphism, univ ersal, 33 , 269 , 312 homotopic, A 1 -homotopic, 191 homotop y colimit, 88 limit, 88 object of homotop y ﬁx ed points, 118 homotop y car tesian, 111 , 119 , 121 , 125 , 152 , 189 object o v er a diagram, 96 square, 110 homotop y categor y , 4 , 28 , 69 , 194 homotop y linear , 240 homotop y pullback, see homotop y carte- sian h yperco v er , 97 , 187 Čech t -hyperco v ers, 289 ind-constructible, 264 inﬁnite suspension, see also func- tor 29 K -theor y homotop y inv ar iant, 334 Milnor , 367 Milnor - Witt, 367 Quillen, 334 with suppor t, 337 la w , f ormal group, 333 linear Q -linear (stable model category), 105 strongl y Q -linear , 227 local, 170 W -local, 184 A 1 -local, 191 , 196 H B -local, 348 localization triangle, seetr iangle, 46 map, trace, 325 , 343 model structure t -descent, 171 injectiv e (diagrams), 86 positiv e stable model structure, 233 projectiv e (diagrams), 85 W -local, 184 module H B -module, 349 , 351 strict H B -module, 351 modules K GL -modules, 336 o v er a homotop y car tesian com- mutativ e monoid, 237 , 373 o v er a monoid, 233 monoid, 227 , 237 cartesian, 237 , 239 cartesian commutativ e monoid, 201 commutativ e monoid, 228 homotop y car tesian, 238 monoid axiom, 227 , 233 , 235 , 239 monoidal stable homotop y 2 -functor, 77 morphism T -pure, see also mor phism, pure 62 cartesian —- of S -diag rams, 91 cocontinuous, 104 degree, 279 faithfull y ﬂat, 49 ﬁnite Λ -univ ersal, 312 Gysin, 321 of P -ﬁbred categor ies, 17 of P -ﬁbred model categories, 25 of P -premotivic categor ies, 27 , 29 of Λ -cy cles, 244 of abelian P -ﬁbred categor ies, 22 of abelian P -premotivic cate- gories, 28 of abelian monoidal P -ﬁbred categories, 22 of complete P -ﬁbred categor ies, 17 of derivators, 104 398 of monoidal P -ﬁbred model cat- egory , 26 of S -diagrams, 83 of tr iangulated P -ﬁbred cate- gories, 23 of tr iangulated P -premotivic cat- egories, 28 of tr iangulated monoidal P -ﬁbred categories, 23 of triangulated premotivic cate- gories, 80 pseudo-dominant, 244 pure, 64 , 67 , 73 pure (proper case), 62 Quillen —- of P -ﬁbred model categories, 96 radicial, 33 , 49 separated, 38 , 243 transpose, 276 univ ersally T -pure (proper case), 62 motiv e, 304 Beilinson, 348 , 367 Cho w (strong), 323 constructible, see constructible eﬀectiv e h -motives, 190 eﬀectiv e qfh -motives, 190 generalized, 307 geometric, 29 , 306 , 362 geometric eﬀectiv e, 306 h -motiv e, 212 qfh -motiv e, 212 Morel, 362 – 370 motivic comple x, 304 constructible, see constructible generalized, 307 stable, see also motiv e 304 , 304 multiplicity geometric, 244 Samuel (of a cycle), 269 Samuel (of a module), 268 Suslin- V oev odsky , 256 nilpotent, 333 Nisnevic h distinguished square, see distin- guished topology , see topology orientable, 351 orientation, 73 of a r ing spectrum, 331 of a triangulated premotivic cat- egory , 70 , 72 perfect, 234 , 238 perfect pair ing, 67 Picard category , 60 point, 243 fat point (of a cycle), 248 generic (of a cy cle), 243 geometric, 243 of a cy cle, 248 pointed, smooth S -scheme, 56 prederivator , 103 premotiv e, 28 T ate premotive, 61 premotivic case, 31 category , 28 category of h -motives, 212 category of qfh -motives, 212 enlarg ement of —- categor y , 30 , 219 , 303 , 309 generalized —- categor y , 28 morphism, see mor phism of pre- motivic categories P -premotivic A 1 -derived categor y , 190 abelian category , 28 category , 27 derived categor y , 173 stable A 1 -derived categor y , 208 triangulated categor y , 28 stable A 1 -derived premotivic cat- egory , 211 presentation local presentation of a simplicial object, 98 presented, see also ﬁnitely presented 23 presheaf INDEX 399 Λ -presheaf, 168 with transf ers, 286 projection f ormula P -projection formula, 11 projectiv e sys tem, of schemes, xviii , 143 , 182 , 264 , 287 , 297 pseudo-Galois, see co ver or distin- guished pullback of fundamental class, 355 purity absolute, 342 , 354 isomorphism (relative), 62 , 70 , 71 , 321 quasi-e xcellent, 129 quotient Gabriel, 217 radicial, see mor phism realization functor (associated with a stable coho- mology), 378 Betti, 375 de Rham, 383 of construcible motives, 382 rigid, 385 resolution of singular ities, 129 canonical —- up to quotient sin- gularities, 131 canonical dominant —- up to quotient singularities, 130 wide —- up to quotient singular - ities, 131 , 358 Riemann-Hilbert, 385 ring Grothendieck - Witt, 369 schematic closure, 243 scheme e x cellent, 129 , 302 , 309 geometricall y unibranch, 270 , 275 , 302 , 309 , 361 quasi-e xcellent, 129 , 137 , 354 regular , 274 , 275 , 277 , 316 strictly local, 272 unibranch, 270 section absolute deriv ed global section, 108 cartesian, 201 geometric, 14 , 15 , 17 , 52 geometric der iv ed global section, 103 sequence symmetric sequence, 203 sheaf étale sheaf with transfers, 288 generalized sheaf with transfers, 298 h -sheaf, 190 , 212 qfh -sheaf, 190 , 212 , 300 sheaf with transf ers, 288 , 298 t -sheaf of Λ -modules, 168 t -sheaf with transf ers, 288 siev e, 32 , 34 singular Suslin singular comple xe, 198 specialization, 322 spectra, see spectr um spectrum abelian T ate spectr um, 205 absolute T ate spectr um, 205 algebraic cobordism, 332 Beilinson motivic cohomology spectrum, 346 motivic cohomology r ing spec- trum, 318 , 332 rational, 334 ring —-, 331 ring —- (associated with a stable cohomology), 377 strict r ing —-, 331 T ate spectr um, 206 T ate Ω -spectr um, 209 univ ersal or iented ring —- with additiv e f ormal group la w , 351 sphere simplicial, 331 square 400 P -distinguished, 34 c dh -distinguished, 111 , 179 Nisnevic h distinguished, 34 , 108 , 179 proper c dh -distinguished, 35 pseudo-Galois qfh -distinguished, 115 qfh -distinguished, 115 , 125 , 180 T or -independant, 339 , 355 stable homotopy categor y of schemes, 28 strict transf orm, 247 , 252 strictiﬁcation theorem, 233 strongl y dualizable, see dualizable T ate motivic comple x, 305 twist, see twist, 331 tensor product of ﬁnite cor respondences, 280 Thom adjoint transf ormation, 56 class, 71 isomorphism, 70 premotiv e, 59 transf or mation, 56 tilting, 374 , 379 topology admissible, 168 c dh -topology , 35 compatible with transf ers, 289 , 295 h -topology , 113 , 360 mildly compatible with transfers, 291 , 293 , 296 Nisnevic h, 34 P -admissible, 168 proper c dh , 35 qfh -topology , 113 , 360 w eakly compatible with trans- f ers, 289 tractable, 96 , 228 trait of a cy cle, 248 transf er , see presheaf or sheaf transv ersal M -transv ersal square, 9 transv ersality proper ty , 9 triangle Gysin, 321 localization triangle, 46 Ma y er - Vietoris tr iangle, 110 t -str ucture heart, 366 non degenerated, 365 t -str ucture, homotop y , 364 twist, 15 , 28 commutes with τ -twists ( or twists), 15 , 17 , 19 of a triangulated monoidal P - ﬁbred category , 23 T ate, 28 , 61 , 209 τ -twisted, 15 underl ying simplicial set of a simplicial object, 98 univ ersal, 188 w eak equivalence, see equivalence Notation 401 No tation α ⊗ [ S S 0 , 247 α ⊗ S S s , 269 α ⊗ t r S α 0 , 280 ˜ α , 247 A S , 203 β ◦ α , 277 β R , k , 249 β ⊗ α α 0 , 252 h Z i X , 244 c S ( X , Y ) Λ , 275 C ∗ , 198 c 0 ( X / S , Λ ) , 275 D A 1 ( A ) , 208 D A 1 , g m ( A S ) , 215 D A 1 , Λ , 211 D A 1 , Λ , 211 D A 1 ( S , Λ ) + , 363 D Betti ( X ) , 375 deg x ( f ) , 279 D eﬀ A 1 ( A ) , 190 D eﬀ A 1 , Λ , 190 DM B , c ( S ) , 356 DM B ( S ) , 348 DM eﬀ g m ( S , Λ ) , 306 DM g m ( S , Λ ) , 306 DM h , Λ , 212 DM Λ , 304 DM Λ , 308 DM eﬀ Λ , 304 DM eﬀ Λ , 308 DM qfh , Λ , 212 DM eﬀ h , Λ , 190 DM eﬀ qfh , Λ , 190 • / / , 275 D ( X , E ) , 378 e A q ( M ) , 268 H B , 349 H q B ( X , Q ( p )) , 350 H n , m M , eﬀ ( S , Λ ) , 310 H n , m M ( S , Λ ) , 310 Hom • (− , −) , 167 H • ( S ) , 4 K GL β , 335 K GL 0 , 335 K GL Q , 345 K GL S , 334 K GL ( i ) S , 346 Λ t S ( X ) , 169 Λ t r S ( X ) , 286 Λ tr S ( X ) , 288 M a n ( X ) , 374 M eﬀ Betti ( X ) , 374 M Betti ( X ) , 374 MGL , 332 m SV ( x ; β ⊗ α α 0 ) , 256 M S ( X ) , 305 M S ( X ) , 308 P c or Λ , S , 278 P c art , 93 PSh ( P / S , Λ ) , 168 PSh  P c or Λ , S  , 286 R A 1 , 195 SH ( S ) , 28 Sh t ( P / S , Λ ) , 168 Sh t  P c or Λ , S  , 288 Sh tr (− , Λ ) , 298 Sh tr (− , Λ ) , 298 Sym ( A ) , 204 ⊗ S , 203 T ot π , 167 t f , 276 T r KGL p , 343 Λ t S ( X ) , 191 402 Index of properties of P -ﬁbred triangulated categories Name Symbol Def. related Remark result additiv e 2.1.1 adjoint proper ty (A dj) 2.2.13 2.2.14 adjoint proper ty f or f (A dj f ) 2.2.13 f morphism of schemes cotransv ersality proper ty 1.1.17 deﬁned f or an y P -ﬁbred category homotop y proper ty (Htp) 2.1.3 localization proper ty (Loc) 2.3.2 2.4.26 6.3.15 localization proper ty f or i (Loc i ) § 2.3.1 i closed immersion motivic 2.4.45 2.4.50 f or premotivic tr iangulated categor ies, 14.2.11 means: (Htp), (Stab), (Loc), (Adj) oriented 2.4.38 2.4.43 f or tr iangulated premotivic categor ies satisfying (wLoc) projection f ormula (PF) 2.2.13 projection f ormula for f (PF f ) 2.2.13 2.4.26 f morphism of schemes proper base chang e proper ty (BC) 2.2.13 2.4.26 proper base chang e proper ty f or f (BC f ) 2.2.13 f mor phism of schemes purity proper ty (Pur) 2.4.21 2.4.26 separated (Sep) 2.1.7 4.2.24 4.4.21 14.3.3 semi-separated (sSep) 2.1.7 3.3.33 stability proper ty (Stab) 2.4.4 support proper ty (Supp) 2.2.5 2.2.12 2.2.14 11.4.2 τ -compatible 4.2.20 4.2.29 τ set of twists τ -continuous 4.3.2 6.1.13 for homotopy P -ﬁbred categor ies, 11.1.24 τ set of twists 14.3.1 τ -dualizable 4.4.13 4.4.21 τ set of twists t -descent proper ty 3.2.5 f or homotopy P -ﬁbred categor ies, t topology transv ersality proper ty 1.1.17 f or an y P -ﬁbred category t -separated (t-sep) 2.1.5 t topology w eak localization proper ty (wLoc) 2.4.7 11.4.2 w eak pur ity property (wPur) 2.4.21 2.4.26 2.4.43

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