The 0-th stable A^1-homotopy sheaf and quadratic zero cycles

The 0 -th stable A 1 -homotop y sheaf and quadratic zero cycles Aravind Asok ∗ Department of Mathematics Univ ersity of Southern California Los Angeles, CA 90089 -2532 asok@usc.edu Christian Haesemeyer † Department of Mathematics Univ ersity of California, Los Angeles Los Angeles, CA 90095 -1555 chh@math.ucla .edu Abstract W e study the 0 -th stable A 1 -homo topy sheaf of a smo oth proper variety over a ﬁeld k assumed to be inﬁnite, perf ect and to h av e characteristic unequal to 2 . W e provide an explicit description of this sheaf in ter ms o f the theory of (twisted) Chow-W itt grou ps as d eﬁned b y Barge-Morel and d ev elop ed by Fasel. W e study the notion of ratio nal po int up to stable A 1 - homoto py , deﬁned in terms of the stable A 1 -homo topy sheaf of gro ups mentioned above. W e show that, for a sm ooth proper k -variety X , existence of a r ational po int up to stable A 1 - homoto py is equiv alent to existence of a 0 -cycle of degree 1 . Contents 1 Introduction 2 2 Preliminaries and r eductions 5 2.1 Moti vic categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.2 Suslin homolog y and 0 -cycles of de gree 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.3 Abelianization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 3 Duality and A 1 -homology 20 3.1 Properties of Thom spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 3.2 Enhanced moti vic complex es and cohomo logy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 3.3 Stable representability in top codimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 3.4 Projectivity an d Base chang e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 3.5 Atiyah duality in the stable A 1 -deriv ed catego ry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 4 T wisted Chow-Witt groups a nd Thom isomorphisms 31 4.1 T wist ed Cho w-W itt groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 4.2 The Thom isomorphism theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 4.3 The zeroth stable A 1 -homotop y sheaf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 4.4 Rational points up to stable A 1 -homotop y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ∗ Aravind Asok was p artially supported by National Science Fou ndation A wards DMS-0900813 and DMS-0966589. † Christian Haesemeyer was partially supp orted by National Science Foun dation A ward DMS-0966821. 1 2 1 Introduction 1 Introd uction Assume k is a ﬁ eld and X is a s mooth proper v ariety ov er k . One say s t hat X has a 0 - cycle of degree 1 if there e xist ﬁnitely many ﬁnite e xtensio ns L i ov er k of coprime de grees such that X ( L i ) is non- empty for each i . Existen ce of a 0 -cycle of degree 1 is inherentl y a motiv ic homological condition. Indeed , by deﬁnition, X has a 0 -c ycle o f d egree 1 if and only if the degree map d eg : C H 0 ( X ) → Z is split surject iv e, and a 0 -cyc le of deg ree 1 is a choice of splittin g, or equiv alent ly a lift of 1 ∈ Z . Friedland er-V oe vo dsky duality implies that the group C H 0 ( X ) is a motivi c homology g roup (we make this much more pre cise in § 2.2 using V oe vo dsky’ s deri ved cate gory of motiv es [MVW06]). If X ( k ) is non-empty , then the degree map deg : C H 0 ( X ) → Z is split su rjecti ve: sending 1 ∈ Z to x determine s a splittin g. Howe ver , the con verse is not true in general (see, e.g., [CTC79, § 5] for a countere xample). Thus, if X has no 0 -cyc le of degree 1 , X cannot hav e a k -rational point; loo sely speaking, we will sa y that e xistence of a 0 - cycle of degree 1 is a motiv ic homolog ical obstru ction to ex istence of a k -rational point. Just as inte gral sing ular homolo gy is the tar get of the Hure wicz homomorphism fro m (sta ble) homotop y groups, moti vic homolog y is the targe t of a Hurewicz -style map from motivi c stable homotop y groups [V oe98, Mor04a]. Furthermore, the degree homomorphis m C H 0 ( X ) → Z is the pushfo rward map in moti vic homology for the m ap X → Sp ec k . Bein g cov arian tly functorial, moti vic stable homotop y groups are also furnish ed with an analog of the de gree m ap. Question. W hat kind of obstruction to e xistence of a rati onal point is pr ovid ed by motivic sta ble homotop y gr oups of a smooth pr oper variety? T o answer the questio n, we stud y the notion of “ratio nal point up to stab le A 1 -homoto py . ” T o be more precise, let π s 0 (Σ ∞ P 1 X + ) be the zeroth moti vic stable homotopy group of X with a disjoint base-p oint attache d (see § 2.1). The str ucture morp hism X → Sp ec k induc es a pushforwa rd map π s 0 (Σ ∞ P 1 X + ) → π s 0 (Σ ∞ P 1 Sp ec k + ) . A rational poin t up to stable A 1 -homoto py is a cho ice of splitting of thi s map. In Sectio n 4.4 we us e a slight ly differe nt thou gh equi va lent deﬁniti on; there is e ven a more-o r-les s elementary deﬁnitio n along the lines of the ﬁ rst deﬁnition of 0 -c ycle of de gree 1 we gav e above. T o our kno wledge, the notion of rational point up to stable A 1 -homoto py was ﬁrst explicitly considered by R ¨ ondig s; see [R ¨ on10, Theorem 5 .1] and the disc ussion immediately preced ing his theorem statement. Theor em 1 (See Theorem 4.4.5) . I f X is a smooth pr oper variety over a ﬁeld k assumed to be inﬁnit e, perfec t and to ha ve cha racteri stic uneq ual to 2 , then X has a ra tional poin t up to s table A 1 -homotop y if and only if X has a 0 -cy cle of de gr ee 1 . The 0 -th stable A 1 -homoto py gro up of X as di scussed above is the set of sections over k of a (Nisne vich) sheaf of abelian groups. Theorem 1 is a consequenc e of a concrete descrip tion of this sheaf, which will be written π s 0 (Σ ∞ P 1 X + ) in the sequel, and detailed study of the functor iality . T o moti va te our descript ion, we go back to the Hurewicz- style homomo rphism from the 0 -t h stable A 1 -homoto py group of a smooth v ariety X to the zeroth moti vic homolog y group mentioned abo ve. The 0 -th Suslin homolog y sheaf of a smooth proper k -scheme X , denoted H S 0 ( X ) , is a sheaﬁﬁcatio n of the Chow group of 0 -cy cles on X (see § 2.3 and Lemma 2.2.1). The Hure wicz functo r giv es rise to a sheaﬁﬁed Hure wicz homomorph ism from the 0 -th stable A 1 -homoto py sheaf 3 1 Introduction of a smooth proper varie ty X to its zeroth Suslin homology sheaf. On the other hand, a fundamen- tal computation of Morel desc ribes the 0 -th sta ble A 1 -homoto py sheaf of a point (more pre cisely , the motivic sphere spectru m) in ter ms of so-called Milnor -W itt K-th eory sh eav es, which combine Milnor K-theory and po wers of the fundament al ideal in the W itt ring [Mor06]. A pr iori , it s eems rea sonable to search for a descri ption of π s 0 (Σ ∞ P 1 X + ) in vo lving the fe atures of 0 -c ycles together w ith additio nal “quadra tic” data. Our description of the 0 -th stable A 1 -homoto py sheaf of a smooth proper vari ety is exactly in these terms and uses w hat are called (twisted) C ho w- W itt groups. As the name sugges ts, Cho w-W itt groups combine aspects of the theory of algebraic cyc les (Cho w grou ps) with aspects of the theo ry of quadratic forms (W itt groups), and the twist refers to a choice of line b undle on X . The main ne w compl ication is th at the resultin g theo ry is unorie nted in the sense of cohomology theorie s, though co nfusingly Cho w-W itt group s hav e also been called orient ed Cho w groups. For a smooth p roper k -scheme X , one deﬁnes the de gr ee 0 Chow-W it t gr o up g C H 0 ( X ) . V ery rough ly speaki ng, this group is a quotient of the free abeli an group on pairs ( x, q ) where x is a closed point of X and q is a qua dratic form ove r the residue ﬁ eld κ x (for the precise deﬁnition , see Deﬁnition 4.1.13); this constructi on encapsulate s the sens e in which the term “quadrati c” is used in the title of the paper . If X = Sp ec k , then g C H 0 (Sp ec k ) coincides with the G rothen dieck- W itt group of symmetric bilinear forms over k (in agr eement w ith Morel’ s compu tation). The resulti ng theory of Chow-W it t grou ps admits for getful maps to Cho w groups and has reasonable functo riality properties (e.g., pushforwar ds for pr oper maps in the ap propriate situa tions). Our main computa tional result is summarize d in the follo wing theorem. Theor em 2 (See T heorem 4.3.1) . Suppos e k is an inﬁnite perfect ﬁeld having char acteristic unequal to 2 , and X is a smooth pr oper k -variety . F or any separa ble, ﬁnitely g ener ated e xtension L/k , ther e ar e isomorphi sms π s 0 (Σ ∞ P 1 X + )( L ) ∼ − → g C H 0 ( X L ) functo rial w ith r espect to ﬁeld e xtension s. Theorem 2 is prov en in three essentia lly ind ependent steps, which correspon d to the three sec- tions of the paper sub sequent to this intr oduction. Using Morel’ s stable A 1 -conne ctiv ity theorem and duality statements, much of the work can be viewed as a reduction to Morel’ s computati ons of some stable A 1 -homoto py sheav es of the sphere spectrum. First, we reduce to a correspo nding homological statement: we sho w that in-so-fa r as our com- putati on is conce rned, we may replace the stable A 1 -homoto py catego ry by a correspond ing stabl e A 1 -deri ved catego ry . The latter is a v ariant of V oe vods ky’ s triangulate d categor y of motiv ic com- ple xes (with the T ate moti v e in vert ed). The stab le A 1 -deri ved cate gory was origi nally concei v ed by Morel (see [Mor04b, § 5.2]), and a detailed construc tion appears in the work of Cisinski-D ´ eglise [CD09]. W e refer to the pass age from the stable A 1 -homoto py catego ry to the stable A 1 -deri ved cate gory as abelianiza tion since it ca n be obtaine d by taking the deri ved func tors of the free abelia n group functor . The main result of Section 2 is T heorem 2.3.8, which m ake s precise the statement s concer ning H ure wicz homo morphisms mentioned abo ve. W ith the exceptio n of Section s § 2 and th e end of § 4, the reader so incline d can enti rely av oid the stable A 1 -homoto py categor y . Second, via Spanier -Whitehea d duality or Poincar ´ e duality , homologic al compu tations can be turned into cohomolo gical comput ations. For smooth project iv e v arieties, there is an analog of 4 1 Introduction Atiyah’ s classical explicit desc ription of the Spanier -Whitehead dual in terms of Thom spaces of the negati ve tangen t b undle; the precise results we need are described in § 3.5. T heorem 3.5.4 giv es an explici t descriptio n of the section s of the 0 -th stable A 1 -homolo gy sheaf of a smooth project iv e scheme in terms of the Thom space of a stable normal b undle. Third, the cohomol ogy of the Thom spaces in question ﬁts into a general theory of twisted Thom is omorphisms that we de ve lop here—in the unorient ed setting the Thom isomorp hisms w e constr uct in v olve cohomology with coef ﬁcients twisted by a local system just like Poincar ´ e duality on unorie nted ma nifolds. Howe ver , rather than deﬁning the notion of a loc al system in A 1 -homoto py theory , which wou ld requ ire some additional effor t, w e tak e a shortcut that in v olv es delving into the formalism of Chow-W it t group s twisted by a line bu ndle. W ithout the twist (or , rather , with the twist by the canonical line b undle ) this theory was deﬁned by Bar ge-Morel [BM00]. A more genera l theory that keep s track of the line bund le twist was de vel oped by Fasel [Fas08, Fas07], and in v olv es the Balmer -W itt g roups of a triangul ated catego ry w ith duality . Beginni ng with a rev iew of the analog of our computa tion in classical algebraic topolog y , the theory and results just mentioned are de velop ed in S ection 4. The functorialit y st atements regarding the zeroth stable A 1 -homoto py sheaf are summarized in the follo wing result. Theor em 3 (See T heorem 4.3.2) . The isomorphisms of Theorem 2 sa tisfy th e fo llowing comp ati- bilitie s. A) The “Hur ewicz” homomorphism π s 0 (Σ ∞ P 1 X + ) → H S 0 ( X ) induces the for get ful map g C H 0 ( X L ) → C H 0 ( X L ) upon e valuation on section s over a sepa rable ﬁnitel y gener ated exte nsion L/k . This homomorphis m is surje ctive for any suc h ﬁeld extens ion L /k . B) The p ushforwar d map π s 0 (Σ ∞ P 1 X + ) → π s 0 (Σ ∞ P 1 Sp ec k + ) coincide s with the pu shforwar d map g deg : g C H 0 ( X L ) → g C H 0 (Sp ec L ) vi a More l’ s identiﬁ cation of π s 0 (Σ ∞ P 1 Sp ec k + )( L ) with the GW ( L ) . C) The two identiﬁca tions just mentioned ar e compatible , i.e., th e diagra m g C H 0 ( X L ) / / g deg   C H 0 ( X L ) deg   GW ( L ) rk / / Z commutes. The proof T heorem 3 follo ws from Theorem 2 together with good choice s of deﬁnitions of the objects under consideratio n. Indeed, Theorem 3A follo ws fr om the m ain computati on and the deﬁnitio n of th e Hurewicz ho momorphism together with the disc ussion of abelian ization in § 2.3. Theorem 3B is a consequenc e of the du ality formal ism an d 3C follo ws by combinin g parts B and A. W e de velo p the necessary preliminaries in § 2.1. F inally Theorem 1 follo ws from T heorem 2 by carefu l analysis of the Gersten resolut ion of the zeroth stable A 1 -homoto py sheaf. 5 2 Preliminaries and r eductions Relationship with other work In [AH11], we sho wed that existenc e of a k -rational point was detected by the stable A 1 -homoto py cate gory of S 1 -spect ra or e ven the rational ized v ariant of this categ ory . Combining Theorem 1 with the main result from [A H11], we see that the dif ference between ration al poin ts and 0 -cycl es of degree 1 arises fro m the passa ge from S 1 -spect ra to P 1 -spect ra. Since the latter ca tegory “h as transfe rs” in a sense we descri be here, this result corr oborates a principle suggest ed in [Le v10]. There are a number of tools a v ailable to analyze this trans ition and one can hope to construct an obstru ction theory for lifting 0 -cy cles of degree 1 to k -ratio nal points. Acknowledgements This paper originally began as a joint project between the authors and Fabien Morel; a preliminary ver sion of the wo rk herein was pr esented as such by th e ﬁrst named aut hor in a tal k at the Oberwol - fach works hop “Moti ves and the homotopy th eory of schemes” in M ay 2010 ( c f. [AHM10]). W e thank Morel for his collabor ation in the early stages of the project. T he ﬁrst author would also like to thank the Uni versit y of Esse n, and especially Marc Levi ne, for providin g an exce llent worki ng en vi ronment while portions of this work were completed. W e would also like to thank Richard El- man an d Sasha Merkurje v for answeri ng our question s regardin g the theory of qu adratic forms, and Jean Fasel for an swering our questions regard ing Chow-W i tt theory . 2 Pr eliminaries a nd r eductions In this section , we recal l (/dev elop) the formalism necessa ry in the rest of the paper . In Section 2.1, we recall the deﬁnitio ns and necessa ry propertie s of all the dif ferent categori es of motivi c orig in that are required in our study . This section will also serv e to ﬁx the notati on for the rest of the paper . In Sectio n 2.2, we recall the pr oof of the analog of ou r m ain th eorem in the conte xt of Suslin homolog y; the main resul ts are L emmas 2.2.1 and 2.2.2. The proofs of these results will be used as template s for the proofs of our more general results in stable A 1 -homoto py theory . Finally Section 2.3 studies the H ure wicz homomorphism (introduced in S ection 2.1) from stable A 1 -homoto py shea ves of gr oups to stable A 1 -homolo gy shea ves. The main result, i.e., Theorem 2.3.8 al lows us to focus our attentio ns on the stabl e A 1 -deri ved cate gory . 2.1 Motivic categories In this paper , we use a number of diff erent catego ries of motiv ic or A 1 -homoto py theoretic origin. For the reader’ s beneﬁt, Diagr am 2.1 displays the categorie s under consider ation and their mutual relatio nships. 6 2.1 Motivic categories (2.1) H Nis s ( k )   / / SH S 1 ( k ) / /   D ( Ab ( k )) / /   D ( Cor ( k ))   Sm k / / Spc k / / : : v v v v v v v v v H ( k ) / / SH s ( k ) / /   D eﬀ A 1 ( k ) o o / /   DM eﬀ ( k ) o o   SH ( k ) / / D A 1 ( k ) / / DM ( k ) In the diagram, the nine categor ies belo w and to the right of SH S 1 ( k ) are homotop y cate gories of stable monoidal model categor ies and the functors in dicated by arr ows pointin g do wn o r to th e right are m onoida l in an appropriate sense. The rightward pointing arro ws in this group (and their composi tes) are the Hure wicz fu nctors referr ed to in the introdu ction. The cate gories in the third column from the left are homotopy categorie s of model catego ries that are not stable, and the cate- gories in the ﬁrst and second column are the geometri c ca tegorie s fr om which all of the homotopy cate gories are b uilt. T he passa ge from the homotop y cate gories in the ﬁ rst ro w to the homotop y cate gories in the second and third rows is a chie ved by means of a Bousﬁeld localiza tion. The unstable categ ories 2.1.1 (Smooth schemes and spaces) . W e write Sm k for the catego ry of schemes separated, smooth and of ﬁnite type over k . The categ ory Spc k is the categ ory of simplicial Nisne vich sheav es of sets o n Sm k ; we refer to objects of Spc k as k -spac es or simply spaces if k is cle ar from context. The fu nctor Sm k → Spc k of D iagram 2.1 is deﬁned as follo ws: send a smooth sch eme X to the corres ponding representab le functor Hom Sm k ( · , X ) and th en tak e the a ssociated con stant simplici al object (all face and degenera cy maps are the identity). T he Y oneda lemma s hows that the functor so deﬁned is fully faithfu l. W e systemat ically ab use notation by identifying the catego ry of smooth schemes with its essenti al image in Spc k . W e al so introd uce the cate gory of po inted k -space s , denoted Spc k , • ; obj ects of thi s cate gory are pairs ( X , x ) co nsisting of a k -space X togethe r with a morphism of spaces x : Sp ec k → X . The for getful functor Sp c k , • → Spc k has a left adjoin t sending a space X to X + , w hich is X with a disjoi nt base-p oint attached. If X is a space and F • is a c omplex of Nisne vich she av es of abelian gr oups on Sm k , th en we de- ﬁne hyper cohomology groups H p ( X , F • ) as (hyp er-)e xtensio n groups in the category of Nisne vich shea ves of abelian groups on Sm k : H p ( X , F • ) = Ext p Nis ( Z ( X ) , F • ) . If X is a scheme, this hyperco homology is canonically isomorp hic to that obtained by the usua l deﬁnitio n. 2.1.2 . The ca tegory H Nis s ( k ) is the (unpointed ) homotopy c ategory of Nisnev ich simplicial s hea ves as construc ted by Joyal-Ja rdine (see, e.g., [MV99, § 2 Theorem 1.4]). One equips Spc k with the injecti v e lo cal model structure (coﬁbrations are monomorph isms, weak equi vale nces are stalkwise 7 2.1 Motivic categories weak equiv alen ces of simplic ial sets, and ﬁbratio ns are determined by the right liftin g prope rty) and realizes H Nis s ( k ) as the associated ho motopy catego ry . Giv en two spaces X and Y , we w rite [ X , Y ] s for the set of m orphis ms between the resulting objects in H Nis s ( k ) . W e write H Nis s, • ( k ) for the poin ted vari ant of H Nis s ( k ) , and if it is not alrea dy clear from cont ext, to dist inguish m orphis ms in this categ ory we will explic itly specify the base-point. 2.1.3 (Unstab le A 1 -homoto py cate gory) . The catego ry H ( k ) is the (unpointed) More l-V oe v odsky A 1 -homoto py category , as constructed in [MV99, § 2 Theorem 3.2 and § 3 Deﬁnition 2.1]. V ery brieﬂy , one equips Spc k with the A 1 -local injec tiv e m odel structu re (coﬁbration s are monomor- phisms, weak equi val ences are the A 1 -local weak equi v alences , and ﬁbrations are determin ed by the right lifting propert y) and realize s H ( k ) as the associ ated h omotopy catego ry . More precisely , H ( k ) is equ iv alent to the full-su bcatego ry of H Nis s ( k ) consisti ng of A 1 -local objec ts; this inclu- sion admits a left adjoint L A 1 called the A 1 -locali zation functor , which giv es rise to the functo r H Nis s ( k ) → H ( k ) in Diagra m 2.1. The fu nctor Spc k → H ( k ) also se nds a space X to its A 1 - localiz ation L A 1 ( X ) . Giv en two spaces X and Y , w e w rite [ X , Y ] A 1 for the set of m orphis ms between the resulti ng object s in H ( k ) . W e write H • ( k ) fo r the pointed v ariant of H ( k ) and as abo ve, if it is not clea r from conte xt, to distingui sh morphisms in this categ ory we will explic itly specify the base-point. The passage from unstable to stable categori es is by no w standar d: we replace spaces by sp ectra. T o do this in a fashion that giv es a tensor structure that is commutati ve and associati v e on the nose, we use the theor y of symmetric spectra . Symmetric sequences Through out this work C is assumed to be a symmetric monoidal categ ory tha t is complete and cocompl ete. In our example s, C will be one of the follo wing four examples . i) the cate gory of simpli cial sets ∆ ◦ Set equipped with the cartesi an product, or the categ ory ∆ ◦ Set • of poin ted simplici al sets equipped with the smash product. ii) the cate gory of space s Spc k equipp ed with the cartesian produ ct, or the category Spc k , • of pointe d spaces equipp ed with the smash produ ct. iii) for a “nice” abelian cate gory A , the cate gory Ch k ( A ) of (bounded belo w) cha in compl exes (i.e., dif ferential of degree − 1 ) of object s in A equi pped with the shuf ﬂe produc t of com- ple xes. The importance of t he ab ove choice of tensor p roduct wil l be e vide nt when we discuss the Dold -Kan corres pondence below . Every thing we say below is a recap itulation of resu lts from [HS S00] or [A yo07 ]. Deﬁnition 2 .1.4 ([HSS00, Deﬁnition 2.1.1]) . For ea ch integ er n ≥ 0 , deﬁne n to be ∅ if n = 0 , and the set of inte gers 1 ≤ i ≤ n fo r i > 0 . Let S be the gr oupoid whose objects are the sets n and whose morphisms are the bijec tions n → n . A symmetri c sequence in C is a funct or S → C . Notation 2.1.5. Write Fun ( S , C ) for the catego ry of symmetric sequences in C . 8 2.1 Motivic categories Lemma 2.1.6. If C is compl ete, cocomplete , or monoidal, so is Fun ( S , C ) , with all corr espo nding notion s deﬁned “levelwise . ” The monoidal structure on symmetric sequences is explain ed in detail in, e.g., [HSS00, Deﬁni- tion 2.1.3 and Cor ollary 2.2.4] or [CD09, § 7.3]. In particul ar , it mak es sense to talk about a mono id R in Fun ( S , C ) , and of (left or right) m odule s over R [ML98, § VII.3-4]. As a consequenc e, if R is a commutati ve monoi d object i n Fun ( S , C ) , giv en two R -modu les, we ha ve an inte rnal hom and tensor produc t satisfying the usual hom-tenso r adjunct ion [H SS00, Lemma 2.2.8]. If C ′ is anoth er monoida l categ ory , and Φ : C → C ′ is a monoidal functor (see [ML98, § XI. 2]), then Φ induces a monoidal functor Fun ( S , C ) → Fun ( S , C ′ ) . Symmetric spectra W e writ e S n s for the simpli cial n -sphe r e , i.e., t he cons tant shea f in Spc k , • arising fro m the simplicial set ∆ n /∂ ∆ n . The space S n s has a natural action of the symmetric group Σ n . One deﬁnes a functor S → Spc k , • by sending n → S n s equipp ed with this action of Σ n . W e write Σ ∞ s S 0 s for this symmetric sequence; it admit s a nat ural structu re of commutati ve ring object in Fun ( S , Spc k , • ) . The subcateg ory o f Fun ( S , Spc k , • ) consisting of modules ov er Σ ∞ s S 0 s will be called the category of symmetric spectr a in k -spaces ; we write Sp Σ ( Spc k ) for the resulting categ ory . Example 2 .1.7 . Suppose X is a po inted k -space. Deﬁne a functor S → Spc k , • by n 7→ S n s ∧ X and where the action of Σ n is induced by th e permut ation on S n s as expla ined abov e. W e write Σ ∞ s X for this symmetric sequence . As in [HSS 00, Example 1.2.4], one checks that Σ ∞ s X is a symmetric spectr um in k -spaces that we call the s uspension symmetric spectr um of X . In the sequel, w e will refer to Σ ∞ s X as the simpl icial suspension spectrum of X . Example 2 .1.8 . If X is a k -space, write Z [ X ] for th e free she af of a belian grou ps on X and ˜ Z [ X ] for the kern el of the morphism of shea ves Z [ X ] → Z induced by the structure morphism X → Sp ec k . Deﬁne a functor S → Spc k , • by n 7→ ˜ Z [ S n s ] ; w rite H Z for this functor . As in [HS S00, Example 1.2.5], one checks that H Z is a symmetric spectrum in k -spaces. More generally , gi ven a sheaf of abelia n groups M , we can deﬁne the Eilenber g-Mac Lane spectrum H M . Stable homotopy catego ries W e no w disc uss the v arious st able ho motopy categori es that arise in our discussion. W e want our theory to ha ve well-deﬁned internal hom and tensor pro ducts, and so we follo w the now standard constr uctions of stable A 1 -homoto py theory . The cate gory of symmetric spectra in k -spaces can also be vie wed as the catego ry of Nisne vich sheav es of symmetric spectra. The cat egory of ordinary symmetric sp ectra has the structure of a mon oidal model ca tegory (for the deﬁniti on of monoidal model cate gory see [SS00, Deﬁ nition 3.1]). There are sev eral model structu res one natura lly consi ders. For the stable m odel structu re, the weak equi vale nces, coﬁbra- tions and ﬁbrations are giv en in [HSS00, Deﬁnitions 3.1.3, 3.4.1, and 3.4.3] and [HS S00, Theorem 3.4.4] es tablishes that the se actuall y deter mine a model structure and ib id. Corollary 5.3.8 es tab- lishes that this model structu re is actua lly m onoida l. 9 2.1 Motivic categories 2.1.9 ( S 1 -stable simplicial homotop y cat egory) . A youb expl ains how the catego ry of Nisnev ich shea ves with v alue s in a mon oidal mode l cate gory can naturally be equipped with a monoidal mode l structu re; see [A yo07, Deﬁnition 4.4.4 0, Corollar y 4.4.42 and Proposition 4.4.62] . A pplyin g this constr uction to th e monoid al model ca tegory of symmetri c spec tra sho ws that Sp Σ ( Spc k ) has the structu re of a mono idal mo del ca tegory ; we write SH S 1 ( k ) for th e ho motopy cat egory of th is mode l cate gory and refer to it as the stable homotopy catego ry of S 1 -spect ra. By constr uction the functor sendin g a space X to the symmetric s pectrum Σ ∞ s X + fact ors through H Nis s ( k ) inducing the fu nctor H Nis s ( k ) → SH S 1 ( k ) of Diagram 2.1. 2.1.10 ( S 1 -stable A 1 -homoto py catego ry) . The catego ry SH s ( k ) of Diagram 2.1 , called the stable A 1 -homoto py catego ry of S 1 -spect ra, is obtained from SH S 1 ( k ) by the procedu re of Bousﬁeld localiz ation. The ﬁrst constructi on is due to Jardine (see [Jar00, T heorems 4.3.2 and 4.3.8] for two dif ferent model structu res). A youb giv es an equi v alent presentati on. The catego ry Sp Σ ( Spc k ) can be equi pped with an A 1 -local model stru cture; see [A yo07, Deﬁnition 4.5.12] . By ibid. Prop osition 4.2.76, it follo ws that the resulting model structure is again a symmetric monoidal model structure . The homotopy catego ry of this model structure is SH s ( k ) . The homotopy category so construct ed is equi v alent to the one described in [Mor05b, Deﬁnition 4.1.1] by [Jar00, Theorem 4.40]. One can constr uct an A 1 -resolu tion functo r that commutes with ﬁnite products using the Gode- ment resolutio n func tor and the si ngular co nstruction S ing A 1 ∗ of [MV99, § 2 Theorem 1.66 and p. 88]. Using this A 1 -resolu tion functo r we can speak of s ymmetric sequences of A 1 -local objec ts. One deﬁnes th e A 1 -local symmetric sp here spectrum by taki ng the functor n 7→ L A 1 ( S n s ) equipped with the in duced ac tion of the symmetric gr oups. W e then co nsider the cate gory of modules over the A 1 -local symmetri c sphere spectrum; by construc tion this is a model of Sp Σ ( Spc k ) . In an anal- ogous fashion, one deﬁnes the A 1 -local symmetric suspens ion spect rum of a pointed space ( X , x ) as the symmetric sequen ce n 7→ L A 1 ( S n s ∧ X ) equipped with the induced actions of the symmetric group s. The functor Spc k → SH s ( k ) is induc ed by the functor sendin g a space X to the A 1 -local symmetric suspens ion spectrum of X + . Deﬁnition 2.1.11. Suppose E is an A 1 -local symmetric spectru m in k -spaces. T he i -th S 1 -stable A 1 -homoto py sheaf of E , denoted π s i ( E ) , is the Nisne vich shea f on Sm k associ ated w ith the preshe af U 7→ Hom SH s ( k ) ( S i s ∧ Σ ∞ s U + , E ) . The main structu ral property of these shea ves π s i ( X ) is summarized in the follo wing result, which will be used without mention in the sequel. Pro position 2.1.12 ([Mor05b, T heorem 6.1.8 and Corollary 6.2.9]) . If E is an A 1 -local symmetric S 1 -spect rum in k -spaces , the sheave s π s i ( X ) are stri ctly A 1 -in va riant. 2.1.13 ( P 1 -stable A 1 -homoto py cat egory) . The categor y SH ( k ) is th e s table A 1 -homoto py cat egory of P 1 -spect ra (see, e.g., [Mor04a, § 5]). This cate gory was also con structed in [Jar00, Theorem 4.2] , b ut again we f ollo w A youb for co nsistenc y . Now , in stead of conside ring symmetric S 1 -spect ra, one consid ers symmetric P 1 or T -spe ctra. In o ther word s, we conside r the cate gory Fun ( S , Spc k , • ) of symmetric se quences. Consider P 1 is a pointed spa ce with base point ∞ . Equip P 1 ∧ n with an action of Σ n by permuta tion of the factor s. The assignment n 7→ P 1 ∧ n determin es a symmetric sequen ce. Moreov er , this object has the structu re of a commutativ e ring object in the categ ory 10 2.1 Motivic categories of symmetric sequenc es (this object is the “free” symmetric spectrum generated by ( P 1 , ∞ ) ); this spectr um, denoted S 0 k , is the sp her e symmetric P 1 -spect rum or just spher e spectrum if no co nfusion can arise . A symmetric P 1 -spect rum is a symmetric sequ ence hav ing the structu re of a module ov er S 0 k . Write Sp Σ P 1 ( Spc k ) for the full subcate gory of Fun ( S , Spc k , • ) consist ing of symmetric P 1 -spect ra. One equips the cat egory Sp Σ P 1 ( Spc k ) with a model structure as in [A yo07, Deﬁnition 4 .5.21]. The category SH ( k ) is the homotop y ca tegory of this model structure. By [Jar00, Theorem 4.31], the category so constructed is equi vale nt to the categ ory of P 1 -spect ra considered by V oe vo dsky in [V oe98] and Morel in [Mor04a, § 5]. Gi ve n a pointed space ( X , x ) , the suspe nsion symmetric P 1 -spect rum, denote d Σ ∞ P 1 X is giv en by the func tor n 7→ P 1 ∧ n ∧ X with the symmetric group acting by permuting the ﬁ rst n -fact ors. W e write S i k for the suspen sion symmetric P 1 -spect rum of the simplicia l i -sphere S i s . Deﬁnition 2.1.14. Suppose E is a symmetric P 1 -spect rum. The i -th stable A 1 -homoto py sheaf of E , denoted π s i ( E ) is the Nisnev ich sheaf on Sm k associ ated w ith the preshea f U 7→ Hom SH ( k ) ( S i k ∧ Σ ∞ P 1 U + , E ) . Strict ring and module structures at the lev el of spaces induce ring and module structures at the le vel of homoto py catego ries and thus w e deduc e the followin g result. Pro position 2.1.15. If E is a symmetric P 1 -spect rum, the sheaf π s 0 ( E ) is a sheaf of modules ove r the sheaf of rings π s 0 ( S 0 k ) ; a morphism f : E → E ′ of symmetric P 1 -spect ra induces a morphis m of π s 0 ( S 0 k ) -module s. The cate gory SH ( k ) can be viewed as SH s ( k ) with the ( A 1 -locali zed) suspen sion spectrum of G m formally in v erted. More preci sely , we hav e the follo wing result. Pro position 2.1.16. F or any smooth sc heme U , and any pointed space ( X , x ) the ca nonical m or - phism colim n Hom SH s ( k ) (Σ ∞ s G m ∧ n ∧ Σ ∞ s ( U + ) , Σ ∞ s G m ∧ n ∧ Σ ∞ s X ) − → Hom SH ( k ) (Σ ∞ P 1 U + , Σ ∞ P 1 X ) is an isomorph ism. Pr oof. A youb shows that the cycl ic permutation (123) acts as the identity on T ∧ 3 in [A yo07, 4.5.65]. The result then follo ws from [A yo07, Theorems 4.3.61 and 4.3.79]. Remark 2.1.17 . One usefu l Corollary to this Proposit ion is that the stable A 1 -homoto py shea ves π s i (Σ ∞ P 1 X + ) of a k -space X are always strictly A 1 -in v arian t since th ey are c olimits of strictly A 1 - in v ariant shea ves. Stable A 1 -homotopy shea ves of sphere s Deﬁnition 2.1.18. For e very inte ger n ∈ Z we set K M W n := π s 0 (Σ ∞ P 1 G m ∧ n ) 11 2.1 Motivic categories Remark 2.1.19 . Since stable A 1 -homoto py groups are strictly A 1 -in v arian t, the sheav es K M W n are strictl y A 1 -in v arian t. What we refer to as “Morel’ s computation of the zeroth stable A 1 -homoto py shea ves ” is a description of the shea ves K M W n in concrete terms. More preci sely , the sectio ns of K M W n ov er ﬁeld s are precisely the Milnor-W itt K-theory g roups d eﬁned by Hopkins an d Morel (see [Mor04a, § 6 .3]). In [Mor11, § 2.1-2], Morel exp lains ho w to deﬁne these s heav es in an “elementary” fash ion in terms of re sidue maps for Mil nor- W itt K-theory . Deﬁnition 2.1.1 8 will be used i n a more concre te fash ion by means of Morel’ s computa tions in T heorem 4.1.8. The sheaf-theo retic D old-Kan corr espondence Let ∆ ◦ Ab k denote the category of simplicia l Nisne vich shea ves of abelian group s. Let Ch ≥ 0 Ab k denote the cate gory of chain complex es (differ ential of degree − 1 ) of Nisnev ich shea ves of abelian group s situated in homological degree ≥ 0 . The functo r of normalized chain complex determin es a functo r N : ∆ ◦ Ab k − → Ch ≥ 0 Ab k . One can also const ruct a fun ctor K : Ch ≥ 0 Ab k → ∆ ◦ Ab k . The sheaf-t heoretic Dold-Kan cor - respon dence shows that the functors N and K gi ve mutually in v erse equi v alences of categori es. Because N and K are mutually in vers e, they are als o adjoint to each oth er on bo th sides; this ob- serv ation is impo rtant if we consider monoidal str uctures on the ca tegorie s in q uestion. W e v iew N as the right adjoint. Then, N is a (lax) sy mmetric mon oidal func tor with respect to th e shuf ﬂe produ ct, though not with respect to the “usual” tensor product on chain complex es, and it sends the unit object in th e categ ory of simpli cial abelian gro ups to the unit ob ject in the catego ry of chain comple xes of abelian groups. A 1 -deriv ed categories W e no w descr ibe the cate gories in the second to la st column of Diagra m 2.1. W e fo llow the p resen- tation of Cisinski-D ´ eglise [CD09, § 7] or [CD10, § 5.3]. W rite Ch Ab k for the categ ory of bounded belo w chain complex es (i.e., dif ferentia l of degree − 1 ) of Nisnev ich sheav es of abelian groups. Begin by co nsidering the space of symmetric sequences Fun ( S , Ch Ab k ) . There is an ob vious inclusio n func tor Ch ≥ 0 Ab k ֒ → Ch Ab k , and we exten d N from the pre - vious se ction to a fu nctor N : ∆ ◦ Ab k − → Ch Ab k . The functor N is sti ll (lax) monoi dal. As a conseq uence, the induce d composite functor ˜ N : Fun ( S , ∆ ◦ Ab k ) → Fun ( S , Ch Ab k ) is also (lax) monoida l. 2.1.20 (Der iv ed cate gory of Nisne vich sheav es of abel ian groups) . The Eilen ber g-MacLane sym- metric spectrum H Z ∈ Fun ( S , Sp c k , • ) is naturally an object in Fun ( S , ∆ ◦ Ab k ) and by compo- sition with ˜ N we get a symmetric sequence ˜ N (H Z ) . Since H Z is a ring object, and the functor ˜ N is (lax) monoidal , it follo ws that ˜ N (H Z ) is a ring object in Fun ( S , Ch Ab k ) . W e let Sp Σ ( Ch Ab k ) denote th e full subc ategory of Fun ( S , Ch Ab k ) cons isting of module s over ˜ N (H Z ) . One equi ps Sp Σ ( Ch Ab k ) with a le vel -wise model str ucture [CD09, § 7.6 and P ropos ition 7.9]; this model struc - ture is furth ermore monoidal, and w e write D − ( Ab k ) for the resultin g homotop y category . 12 2.1 Motivic categories 2.1.21 . The functor N in the Dold-Kan correspond ence is part of a Quillen equi va lence of model cate gories. O ne also has the (monoidal) free abelian group func tor Z ( · ) : Spc k → ∆ ◦ Ab k . The composi te of these two func tors gi ves a functo r Fun ( S , S pc k , • ) → Fun ( S , Ch Ab k ) . By con- structi on, the image of th e sphere symmetri c sequence und er this composi te functor is ˜ N (H Z ) , an d one obtain s a functor between corresp onding cate gories of symmetric spectra: (2.2) Sp Σ ( Spc k ) → Sp Σ ( Ch Ab k ) . By [Hov0 1, Theorem 9.3], this functor induces a Quillen functor SH S 1 ( k ) → D( Ab k ) that w e refer to alternati vel y as a Hurewicz functor or the deri ved functor of abelia nization. In any case, we write Z [ n ] for the image of S n s under this functo r . 2.1.22 (Ef fecti ve A 1 -deri ved cat egory) . The cate gory D eﬀ A 1 ( k ) of Diagr am 2.1 is ca lled the ef fecti ve A 1 -deri ved cate gory . This category is constructe d from Sp Σ ( Ch Ab k ) by A 1 -locali zation. Thus, one gets an A 1 -local model structure on Sp Σ ( Ch Ab k ) ; again this model structure is monoidal, and we write D eﬀ A 1 ( k ) for the resulting homotop y categor y . Cisinski and D ´ eglise prov e (see [CD10, Proposit ion 5.3.19 and 5.3.20]) that the homotop y cate gory of this model structur e is equi v alent to the A 1 -deri ved category as deﬁned in [Mor11, § 3.2]. Again, one construct s an A 1 -locali zation functo r , which can be assumed to commute with ﬁnite product s. 2.1.23 . It is not true that Functor 2.2 takes A 1 -local ob jects to A 1 -local objec ts. Ne ve rtheless, if ( X , x ) is a point ed space, we can consider the suspensio n symmetric spectrum Σ ∞ s X . App lying Functor 2.2 we get an object in Sp Σ ( Ch Ab k ) that we can A 1 -locali ze. W e w rite ˜ C A 1 ∗ ( X ) for the resulti ng object. On the other hand, w e could A 1 -locali ze Σ ∞ s X and then apply Functor 2.2. T here is a na tural morphism from the latter to t he former , and since all f unctors in questio n preserve ﬁnite produ cts, this functor is monoidal. As a consequen ce, there is an induced functor SH S 1 ( k ) → D eﬀ A 1 ( k ) . If X is not pointe d, we write C A 1 ∗ ( X ) for ˜ C A 1 ∗ ( X + ) . If ( X , x ) is a pointed space, we write ˜ C A 1 ∗ ( X )[ n ] for the tensor product ˜ C A 1 ∗ ( X ) ⊗ Z [ n ] . Deﬁnition 2.1.24. The i -t h A 1 -homolo gy sheaf of a spa ce X , denoted H A 1 i ( X ) , is the Nisnev ich sheaf on Sm k associ ated w ith the preshea f U 7→ Hom D eﬀ A 1 ( k ) ( C A 1 ∗ ( U )[ n ] , C A 1 ∗ ( X )) . In th e c ategory D eﬀ A 1 ( k ) , th e s hift func tor coin cides with tensoring with Z [1] . F ix a r ational p oint on P 1 and ca ll it ∞ . Point G m by 1 ; the open co ver of P 1 by two cop ies of A 1 with intersection G m gi ves rise to an isomorphism S 1 s ∧ G m ∼ → P 1 in H ( k ) . This giv es an identiﬁcatio n ˜ C A 1 ∗ ( S 1 s ∧ G m ) ∼ → ˜ C A 1 ∗ ( P 1 ) . Since the functor from the S 1 -stable A 1 -homoto py cate gory to the effect iv e A 1 - deri ved cate gory is monoidal, we may identify this object w ith ˜ C A 1 ∗ ( G m )[ − 1] . Deﬁnition 2.1.25. The enhanc ed T ate comple x, denoted Z h 1 i , is the object ˜ C A 1 ∗ ( P 1 )[ − 2] . 2.1.26 (S table A 1 -deri ved category ) . Next, w e deﬁne the catego ry D A 1 ( k ) —the stable A 1 -deri ved cate gory—from Diagram 2.1. The ca tegory D A 1 ( k ) is ob tained from D eﬀ A 1 ( k ) by formall y in ver ting 13 2.1 Motivic categories the enhanced T ate complex. Again, this task is acco mplished by m eans of the theory of symmet- ric spect ra. The simples t thing to do, ensu ring compatib ility with the const ruction of the stabl e A 1 -homoto py catego ry gi ven above is to proc eed as follo ws. Consider the A 1 -locali zation of th e normaliz ed chain comple x of the free ab elian group on the sphere symmetric P 1 -spect rum. This determin es a monoid in the cate gory of symmetric sequences , and we can consid er the full sub- cate gory , denot ed Sp Σ P 1 ( Ch Ab k ) , of Fun ( S , Ch Ab k ) co nsisting of modules ov er th is object. As abo ve, we equip th is ca tegory with a monoidal model str ucture an d let D A 1 ( k ) den ote th e a ssociated homotop y category . The symmet ric P 1 -suspe nsion spectrum of a pointed space ( X , x ) deﬁnes an object ˜ C s A 1 ∗ ( X ) of Sp Σ P 1 ( Ch Ab k ) : take the A 1 -locali zation of the normalized chain complex of the free abelian group on the susp ension symmetric P 1 -spect rum of X . W e call this ca tegory th e cate gory of symmetric P 1 -chain complex es. Write 1 k for the unit obj ect in D A 1 ( k ) for the induced ten sor struc ture, i.e., the comple x ˜ C s A 1 ∗ ( S 0 k ) , and write 1 k [ n ] for the space ˜ C s A 1 ∗ ( S n s ) , and ˜ C s A 1 ∗ ( X )[ n ] for the space ˜ C s A 1 ∗ ( X ) ⊗ 1 k [ n ] . If X is not pointed , write C s A 1 ∗ ( X ) for ˜ C s A 1 ∗ ( X + ) . Deﬁnition 2 .1.27. For any ob ject X of Sp Σ P 1 ( Ch Ab k ) , th e i -th stable A 1 -homolo gy sheaf H s A 1 i ( X ) is the sheaf on Sm k associ ated with the preshe af U 7→ Hom D A 1 ( k ) ( C s A 1 ∗ ( U )[ i ] , C s A 1 ∗ ( X )) . Pro position 2.1.28. If X is an object of Sp Σ P 1 ( Ch Ab k ) , the she af H s A 1 0 ( X ) is a shea f of (left or righ t) modules over the sheaf of ring s H s A 1 0 ( 1 k ) ; a morphism f : X → Y of objec ts in Sp Σ P 1 ( Ch Ab k ) induces a morphism of H s A 1 0 ( 1 k ) -module s. W e woul d lik e to kno w that m orphis ms in the categor y so deﬁned can actually be obtained by in v erting Z h 1 i in D eﬀ A 1 ( k ) . Pro position 2.1.29. F or any smooth scheme U , any i and any pointed space ( X , x ) the canonical morphism colim n Hom D eﬀ A 1 ( k ) ( C A 1 ∗ ( U ) h n i [ i ] , ˜ C A 1 ∗ ( X ) h n i [ i ]) − → Hom D A 1 ( k ) ( C s A 1 ∗ ( U ) , ˜ C s A 1 ∗ ( X )) is an isomorph ism. Pr oof. The pro of is identical to th e corre sponding statement in the sta ble A 1 -homoto py catego ry , i.e., Propositio n 2.1.16. Hurewic z fu nctors Composing the free abelian grou p functor and the normalized chain comple x functo r with A 1 - localiz ation, there are functo rs Sp Σ ( Spc k ) − → Sp Σ ( Ch Ab k ) , and Sp Σ P 1 ( Spc k ) − → Sp Σ P 1 ( Ch Ab k ) . 14 2.1 Motivic categories By cons truction, both func tors are m onoida l and ind uce functors (which we refe r to as Hur ewicz functo rs ) on homotopy cate gories SH s ( k ) − → D eﬀ A 1 ( k ) , and SH ( k ) − → D A 1 ( k ) . Consequ ently , if X is a space, there are indu ced m orphis ms π s i (Σ ∞ s X + ) − → H A 1 i ( X ) , and π s i (Σ ∞ P 1 X + ) − → H s A 1 i ( X ) (2.3) that we refer to as Hur ewicz mor phisms . Pro position 2.1.30. If X is any k -space, the following diagr am commutes: π s 0 ( S 0 k ) ⊗ π s 0 (Σ ∞ P 1 X + ) / /   π s 0 (Σ ∞ P 1 X + )   H s A 1 0 ( 1 k ) ⊗ H s A 1 0 ( X ) / / H s A 1 0 ( X ) . Pr oof. The Hurewicz functor is induced by a mo noidal functor . The result then follo ws immediately by combinin g P roposi tion 2.1.15 and 2.1.28 . Motiv ic categories 2.1.31 (Deriv ed cate gory of correspo ndences) . Let Cor k denote the categ ory whose objects are smooth k -schemes an d whe re th e s et o f mo rphisms fr om X to Y is the f ree ab elian gro up o n i ntegral closed subsch emes of X × k Y ﬁnite and surjec tiv e o ver a componen t of X . Sending a mor phism f : X → Y of smooth k -schemes to its graph determines a functo r Sm k → Cor k . Precompo sing with this functor , an y (say abel ian group v alued) presh eaf on Cor k can be viewed as a preshea f on Sm k . Write Ab Nis ( Cor k ) for the full subcate gory of abelian group v alued preshea ve s on Cor k that are Nisne vich shea v es. Wri te Z tr ( X ) for the re presentab le presheaf U 7→ Cor k ( U, X ) ; this preshe af can be shown to be a Nisnevi ch sheaf on Sm k . Abusi ng notat ion, we write D − ( Cor k ) for the de riv ed cate gory of th e abelian ca tegory Ab Nis ( Cor k ) . This category can also be constructe d using symmetric spect ra. 2.1.32 (Ef fecti ve moti vic co mplexe s) . W e let DM eﬀ ( k , R ) be V oe vo dsky’ s triang ulated category of ef fecti v e motivi c complex es; when R = Z we su ppress it from th e notatio n and write DM eﬀ ( k ) . The catego ry DM eﬀ ( k , R ) is th e A 1 -locali zation of the der iv ed cate gory of N isne vich sheav es of R -modules with transfers. W e write M ( X ) for the complex C ∗ R tr ( X ) vie wed as an object of DM eﬀ ( k , R ) . The categor y DM eﬀ ( k , R ) can b e vie wed a s the subcatego ry of the deriv ed ca tegory of Nisnev ich shea ves of R -modul es with transfers co nsisting of A 1 -local objects. The functor for get transfe rs preser ves A 1 -local objects and therefo re induc es a functo r DM eﬀ ( k , R ) − → D eﬀ A 1 ( k , R ) . The nex t result sho ws that this functor is part of an adjunc tion of model cate gories. 15 2.2 Suslin homology and 0 -cycles of degree 1 Pro position 2.1.33. Ther e is an adjunction D eﬀ A 1 ( k , R ) / / DM eﬀ ( k , R ) o o The adjoin t functor pr eserves tensor pr oduct s. Pr oof. See [CD10, 10.4.1] Deﬁnition 2.1.34. Assume k is a ﬁeld and X is a smooth k -variet y . T he 0 -th S uslin homology sh eaf H S 0 ( X ) is the 0 -th homology sheaf of the complex L A 1 C ∗ Z tr ( X ) . Pro position 2.1.35. Supp ose X is a smooth k -scheme and k is a perfec t ﬁeld. The canonica l map H A 1 0 ( X ) → H S 0 ( X ) factors thr ou gh a m orphis m H s A 1 0 ( X ) → H S 0 ( X ) . Pr oof. Use the functor of Proposition 2.1.33. By constr uction, giv en a smooth scheme X , thi s functo r se nds C A 1 ∗ ( X ) to M ( X ) and is compatible with shifts. In particula r , Z h 1 i is sent to Z (1) . W e ca n identify H A 1 0 ( X ) with H om D eﬀ A 1 ( k ) ( Z , C A 1 ∗ ( X )) . For eve ry inte ger i ≥ 0 , we th erefore hav e functo rial m aps: (2.4) Hom D eﬀ A 1 ( k ) ( Z h i i , C A 1 ∗ ( X ) h i i ) − → Hom DM eﬀ ( k ) ( Z ( i ) , M ( X )( i )) . By V oe v odsky ’ s cancelati on theorem, we kno w that the canonical map Hom DM eﬀ ( k ) ( Z ( j ) , M ( X )( j )) − → Hom DM eﬀ ( k ) ( Z ( j + 1) , M ( X )( j + 1)) is an isomorph ism for any j ≥ 0 . In particula r , we get an isomorphism H S 0 ( X ) ∼ → Hom DM eﬀ ( k ) ( Z ( i ) , M ( X )( i )) for a ny i ≥ 0 . In lig ht of Propo sition 2.1.29, the c olimit of th e maps of F ormula 2.4 followed b y the in v erse to the isomorphis m of the pre vious line giv es the morphism of the statement. 2.2 Suslin homology and 0 -cycles of degr ee 1 As a model for our study of the 0 -th A 1 -homolo gy sheaf, in this sectio n we revie w the relationship between the 0 -th Suslin homology sheaf and 0 -cy cles of degree 1 . Lemma 2.2.1. Assume k is a perfect ﬁeld, and X is a smooth pr op er k -variet y . F or any separab le ﬁnitel y gener ated extens ion L/k , we have a canonical identiﬁcati on H S 0 ( X )( L ) = C H 0 ( X L ) . Pr oof. W e hav e the follo wing seq uence of ide ntiﬁcations. Let n = dim X . For simplicit y of 16 2.2 Suslin homology and 0 -cycles of degree 1 notati on, we w rite H S 0 ( X L ) for H S 0 ( X )( L ) . H S 0 ( X L ) ( a ) = Hom DM eﬀ k ( Z , M ( X L )) ( b ) = Hom DM eﬀ L ( Z , M ( X L )) ( c ) ∼ = Hom DM L ( Z , M ( X L )) ( d ) ∼ = Hom DM L ( M ( X L ) ∗ , Z ) ( e ) ∼ = Hom DM L ( M ( X L )( − n )[ − 2 n ] , Z ) ( f ) ∼ = Hom DM eﬀ L ( M ( X L ) , Z ( n )[2 n ]) ( g ) = H 2 n Nis ( X L , Z ( n )) ( h ) ∼ = H n Nis ( X L , K M n ) ( i ) = C H 0 ( X L ) . Identi ﬁcation (a) follo ws from the deﬁnition of M ( X ) . Identiﬁcation (b) follo ws from, e.g., [MVW06, Exerci se 1.12]. That (c) is an isomorp hism is the statement of V oev odsk y’ s cancel ation theorem [V oe10, Corollary 4.10], which implies th at homomorph isms in DM eﬀ L and DM L coinci de. Iso- morphism (d) follo ws from the fa ct that the catego ry DM L admits duals. Isomorphism (e) is a conseq uence of the explicit description of duals in DM k ; this can be vi ewed as a form of Atiyah dualit y (we w ill explain this in detail later). For smooth varie ties, Friedlande r-V oe vods ky pro ved a dualit y theorem under the assumption that k admits resolution of singulari ties (see, e.g., [MVW 06, Theorem 16.27 ] for some recolle ctions). Howe ve r , the assumpt ion that X is, in additio n, proper allo ws us to weaken the hypoth esis on k to merely assuming it is perfect. Isomorph ism (f) follo ws again from V oe vo dsky’ s cancelati on theorem. Identiﬁcati on (g) fol- lo ws from the fact that Z ( n )[2 n ] is A 1 -local; see [MVW06, Corollary 14.9]. Identiﬁcation (h) requir es m ore argumen t. One kno ws [MVW06, T heorem 5.1 ] that th ere is a canonical isomor phism H n ( Z ( n ))( L ) = K M n ( L ) (t his is essentia lly the Nestere nko-Susl in-T otaro theore m but the afo re- mentione d proof is self-con tained). Moreove r , this arg ument shows that H i ( Z ( n ))( L ) van ishes for i > n . The hyperco homology spectra l sequenc e togethe r with dimension al van ishing gi v e th e requir ed identi ﬁcation. Ident iﬁcation (i) is a result of Kato [Kat86] and is a consequenc e of the exi stence of the Gersten resolut ion for the sheaf K M n . Lemma 2.2.2. Assume k is a perfect ﬁeld, and X is a smooth pr oper k -variet y . The pushforwa rd map H S 0 ( X ) → H S 0 (Sp ec k ) = Z coincides w ith the de gr ee map C H 0 ( X L ) → Z upon evaluati on on section s over ﬁnitely gener ated exte nsions L/k . Pr oof. This follo ws from the explicit constru ction of the duality map; see [V oe03, Theor em 2.11 ] togeth er with the identi ﬁcations of Lemm a 2.2.1. Cor ollary 2.2.3. Assume k is a perfect ﬁeld, and X is a smooth pr oper k -variety . Ther e is a 17 2.3 Abelianization canon ical bijection between splittings s : Z → H S 0 ( X ) of the de gr ee homomorphism and 0 -cycles of de g r ee 1 given by s 7→ s (1) . Pr oof. Write Ab A 1 k ,tr for the categor y of strictly A 1 -in v arian t Nisne vich shea ve s with transfers . By the Y oneda lemma, for any ﬁeld k , there is a canonic al bijection H S 0 ( X )( k ) = Hom Spc k (Sp ec k , H S 0 ( X )) ∼ − → Hom Ab A 1 k,tr ( Z , H S 0 ( X )); the map sends an element x ∈ H S 0 ( X )( k ) to the free abelian group g enerated by multi ples of x . The result is then a consequ ence of Lemma 2.2.2. 2.3 Abelianization The goal of this section is to identify the 0 -th stable A 1 -homoto py sheaf of a smooth scheme with an an alogous object of “ homological ” nature, b ut in a man ner compatibl e w ith th e all the ad ditional structu re. The S 1 -stable story The follo wing result was stated as [Mor0 4b , Theorem 4.3.2], but a proof w as not gi ven there. W e write Ab A 1 k for the cate gory of strictly A 1 -in v arian t shea ves of abelian groups. Lemma 2.3.1. Assume k is a ﬁeld, and sup pose X i s a space. The Hur e wicz homomorphism π s 0 (Σ ∞ s X + ) → H A 1 0 ( X ) of Formul a 2.3 is an isomorphism. Pr oof. Suppose M is a strictly A 1 -in v arian t sheaf of abeli an group s, and let H M denot e the as soci- ated Eilen berg -MacLane S 1 -spect rum. The S 1 -suspe nsion spectrum Σ ∞ s X + is ( − 1) -connected and hence its A 1 -locali zation is again ( − 1) -connect ed by the stable A 1 -conne ctiv ity theorem [Mor05b, Theorem 6.1.8]. Using the fact that H M is also ( − 1) -connec ted, existence and functoriali ty of the Postnik ov tower gi ves rise to a ca nonical isomorphism Hom Ab A 1 k ( π s 0 (Σ ∞ s X ) , π s 0 (H M )) ∼ − → H 0 Nis ( X , M ) . The analog ous constructio n in D eﬀ A 1 ( k ) gi ves rise to an isomorp hism Hom Ab A 1 k ( H A 1 0 ( X ) , M ) ∼ − → H 0 Nis ( X , M ) . The result follo ws immediately from the Y oned a lemma. Morel’ s stable A 1 -conne ctiv ity the orem used in the proof of Lemma 2.3.1 also equips the cate- gory Ab A 1 k with a symmetric monoida l structure for which we will write ⊗ A 1 . Deﬁnition 2.3.2. If M and M ′ are strictly A 1 -in v arian t shea ves, then M ⊗ A 1 M ′ := H A 1 0 ( M ⊗ M ′ ) . 18 2.3 Abelianization Contractio ns and G m -loop spaces W e begin by recallin g the deﬁnition of contrac tions of a pointed sheaf, and then stating some prop- erties that follo w immediately from the deﬁnitions. Deﬁnition 2.3.3. Suppose F is presheaf of pointe d se ts on Sm k . The presheaf F − 1 is the intern al pointe d function presheaf Hom • ( G m , F ) , i.e., for any smooth scheme U we ha ve F − 1 ( U ) = k er( F ( G m × U ) ev 1 − → F ( U )) , where the map ev 1 is the pullback along the map U id × 1 → U × G m . Lemma 2.3.4. Suppo se F is a pr es heaf of pointed sets . If F is a sh eaf (r esp. sheaf of gr oups, sheaf of abeli an gr ou ps), then so is F − 1 . Suppose E is an S 1 -spect rum and Σ ∞ s G m is the symmetric spectru m associ ated with the space G m pointe d by 1 . W e can consider the internal function spectrum Hom (Σ ∞ s G m , E ) . The diagonal U → U × U ind uces a map U + → U + ∧ U + and consequent ly a morphism of spectra Σ ∞ s U + → Σ ∞ s U + ∧ Σ ∞ s U + . Any morphism U → G m induce s a morphism of pointed spaces U + → G m and conseq uently a morphism of sp ectra Σ ∞ s U + → Σ ∞ s G m . Combining these two observ ations, we get a map [Σ ∞ s U + ∧ Σ ∞ s G m , E ] s × Ho m Sm k ( U, G m ) − → [Σ ∞ s U + ∧ Σ ∞ s U + , E ] s − → [Σ ∞ s U + , E ] s . Sheaﬁfying for the Nisne vich topology and keeping track of basepo ints, this corresponds to a map π s 0 (Hom (Σ ∞ s G m , E )) ∧ G m → π s 0 ( E ) . By adjunc tion, such a morphism is equi v alent to a m orphis m of sheav es π s 0 (Hom (Σ ∞ s G m , E )) → π s 0 ( E ) − 1 . Similarly , for any inte ger n ∈ Z there is an induced morphism (2.5) π s i (Hom (Σ ∞ s G m , E )) → π s i ( E ) − 1 . Assuming E is A 1 -local prod uces a correspon ding map of stable A 1 -homoto py shea ves of E . Pro position 2.3.5 ([Mor04a, Lemma 4 .3.11]) . If E is an A 1 -local S 1 -spect rum, then for any i ∈ Z the morphism π s i (Hom (Σ ∞ s G m , E )) → π s i ( E ) − 1 of Equation 2.5 is an isomorph ism. In part icular , if E is a ( − 1) -connecte d A 1 -local s pectrum, then for any inte g er n ≥ 0 , Hom (Σ ∞ s G m ∧ n , E ) is also a ( − 1) -conne cted A 1 -local spec trum. Cor ollary 2.3.6. If M is a strictly A 1 -in va riant sheaf of gr oups, then so is M − 1 . Pr oof. The sheaf M is a stric tly A 1 -in v arian t sheaf if and only if the Eilenber g-MacLane spec- trum H M [ i ] is A 1 -local for all i . Since H M [ i ] is A 1 -local, so is Hom (Σ ∞ s G m , H M [ i ]) . T he shea ves π s i ( X ) are strictly A 1 -in v arian t if X is A 1 -local. The res ult then follo ws immediately from Proposit ion 2.3.5. 19 2.3 Abelianization Contractio ns in the A 1 -deriv ed setting W e now rew ork the res ults of the prev ious sec tion in the setting of the A 1 -deri ved categ ory . Suppose C is an object of Sp Σ ( Ch Ab k ) . If Z h 1 i is the enhanced T ate comple x introd uced before, we study the inter nal hom object Hom ( Z h 1 i [1] , C ) . Mirroring the cons truction of the prev ious section in D eﬀ A 1 ( k ) , there is an induced morphism (2.6) H A 1 i (Hom ( Z h 1 i [1] , C )) → H A 1 i ( C ) − 1 . W e no w show tha t this morphism is an isomorphism. Pro position 2.3.7. If C is an A 1 -local object in Sp Σ ( Ch Ab k ) , then the morphism of Formula 2.6 is an isomorph ism. Pr oof. The pr oof is formally ide ntical to [Mor04a, Lemma 4.3.11]. One ﬁrst prov es th e resu lt in the case w here M is a strictly A 1 -in v arian t sh eaf of groups. In that case, it sufﬁces to check the result on section s over ﬁelds. W e then just hav e to sho w that Hom D( Ab k ) ( Z , Hom ( Z h 1 i [1] , M )[ i ]) − → Hom D( Ab k ) ( Z , M − 1 [ i ]) is an isomorph ism. By adjuncti on the map abov e can be rewritten as: Hom D( Ab k ) ( Z h 1 i [1] , M [ i ]) − → Hom D( Ab k ) ( Z , M − 1 [ i ]) . Ho wev er , the sheaf Hom ( Z h 1 i [1]) is pr ecisely th e A 1 -chain complex of G m , and sinc e M is A 1 - local the group on the left is an ordinary cohomolog y group. T o ﬁnish , we observe that ˜ C A 1 ∗ ( P 1 ) = ˜ C A 1 ∗ ( G m )[ − 1] , and th at bot h G m and P 1 ha ve Nisnev ich cohomo logical dimens ion ≤ 1 . T o treat the genera l case, we reduce to the one abov e by means of a Postniko v tower ar gument. The P 1 -stable story W e no w prove the P 1 -stable versio n of the Hurewicz theo rem. Theor em 2.3.8. Suppose X is a k -space . The Hur ewicz morph ism of Formul a 2.3 π s 0 (Σ ∞ P 1 X + ) − → H s A 1 0 ( X ) is an isomorphism covaria ntly functor ial in X . When X = Sp ec k , the afor ementione d isomorphism r eads K M W 0 ∼ − → H s A 1 0 ( 1 k ) . Mor eover , via this id entiﬁcati on, the Hur e wicz iso morphism is com- patibl e with the (left) action of K M W 0 on both the sour ce and tar get in the sense that the following dia gram commutes: K M W 0 ⊗ A 1 π s 0 (Σ ∞ P 1 X + ) / /   π s 0 (Σ ∞ P 1 X + )   K M W 0 ⊗ A 1 H s A 1 0 ( X ) / / H s A 1 0 ( X ) . 20 3 Duality and A 1 -homology Pr oof. The last statemen t regardin g module structures follo ws from the Hurewic z isomorphis m via Proposit ion 2.1.30 and Deﬁnition 2.1.18. T hus, it sufﬁces to pro ve that the Hure wicz homomor - phism is an isomorph ism. By Propositi on 2.1.16, π s 0 (Σ ∞ P 1 X + ) is the sheaf associated with the presheaf U 7→ colim n Hom SH S 1 ( k ) (Σ ∞ s G m ∧ n ∧ Σ ∞ s U + , Σ ∞ s G m ∧ n ∧ Σ ∞ s X + ) . Equi va lently , by adjun ction, we can ident ify π s 0 (Σ ∞ P 1 X + ) with the sheaf colim n π s 0 ( R Hom • (Σ ∞ s G m ∧ n , Σ ∞ s G m ∧ n ∧ Σ ∞ s X + )) . Proposit ion 2.3.5 gi ves an isomor phism π s 0 ( R Hom • (Σ ∞ s G m ∧ n , Σ ∞ s G m ∧ n ∧ Σ ∞ s X + )) ∼ − → π s 0 (Σ ∞ s G m ∧ n ∧ Σ ∞ s X + )) − n . Lemma 2.3.1 sho ws that the map π s 0 (Σ ∞ s G m ∧ n ∧ Σ ∞ s X + ) → ˜ H A 1 0 ( G m ∧ n ∧ X + ) is an isomorph ism and consequen tly that the induced map π s 0 (Σ ∞ s G m ∧ n ∧ Σ ∞ s X + ) − n → ˜ H A 1 0 ( G m ∧ n ∧ X + ) − n is an isomorph ism. On the oth er hand, Propo sition 2.1.29 together with an adju nction ar gument sh ow that H s A 1 0 ( X ) can be written as colim n H A 1 0 (Hom ( Z h n i [ n ] , C A 1 ∗ ( X ) h n i [ n ])) . Proposit ion 2.3.7 then identi ﬁes this group w ith colim n H A 1 0 ( C A 1 ∗ ( X ) h n i [ n ]) − n . Ho wev er , the groups in the colimit are precisely ˜ H A 1 0 ( X + ∧ G m ∧ n ) − n . Since the simplicial Hure wicz homomorph ism is compati ble with the tenso r structures the result follo ws. 3 Duality and A 1 -homology Suppose k is a ﬁeld. Section 3.1 studies T hom spaces and the ir cohomology and connecti vity proper ties. In Section 3.2, we intr oduce complex es Z h n i in the A 1 -deri ved cate gory (see Deﬁnition 3.2.1) that are formally very similar to the usual motivic comple xes (see, e.g., [MVW06, § 3]). W e then in v estigate a cohomolog y theory associat ed with these comple xes, anal ogous to moti vic cohomol ogy , paying special attentio n to the case of Thom spaces of vecto r b undl es. W e introduce two v ersions of this theory in the unstable and stable settings (see Deﬁnition 3.2.3). By co nstraining the degree and dimensio ns of the spaces in question, we sho w in Section 3.3 that sometimes the unstabl e group s coin cide with the stable group s (see Corollary 3.3.6). In order to do thi s, we need to recall facts ab out homotopy modu les (see Deﬁnition 3.3.1) , which are o bjects of the heart of the so-calle d homoto py t -structur e on D A 1 ( k ) . In addition to their use in p roving 21 3.1 Properties of Thom spaces the afor ementioned stable represen tability result, homoto py modules serv e a second purp ose: using formal properties of homotopy m odules and the homotopy t -struct ure, w e sho w in Section 3.4 that the z eroth A 1 -homolo gy sheaf is a birational in v ariant of smoo th proper v arieti es; thus, it suf ﬁces to descri be this sheaf for smooth proj ectiv e varie ties by Chow’ s lemma. Finally , having reduced to the study of A 1 -homolo gy of smooth projecti ve vari eties, in S ection 3.5 we study the analog of Atiyah duality in the stable A 1 -deri ved cate gory (see Proposition 3.5.2). Combining all of these result s, we provide the analogs of steps (a) - (h) of the outline gi ven in the proof of Lemma 2.2.1 in the conte xt of stable A 1 -homolo gy sheav es. The key issue w e ha ve to consid er is that duality is only deﬁned in the stable A 1 -deri ved cate gory , while essenti ally all of the other constructio ns and computation s we mak e are “unstable. ” The main result of the section is Theorem 3.5.4, which pro vides the ﬁrst part of our descrip tion of the stable A 1 -homoto py shea ves of grou ps. 3.1 Pr operties of Thom spaces If ξ : E → X is a vector bund le w ith zero section i : X → E , recall that T h ( ξ ) = E /E − i ( X ) . If ξ ′ : E ′ → X is an other vector bund le, and f : E → E ′ is a morphism of vecto r bu ndles that is injecti ve on ﬁber s, then there is an ind uced morph ism T h ( ξ ) → T h ( ξ ′ ) . W e use the foll owing results about Thom spaces repeate dly in the sequel. Pro position 3.1.1 ([MV99, § 3 Propositio n 2.17]) . Suppose X 1 and X 2 ar e smooth k -schemes, and ξ 1 : E 1 → X 1 and ξ 2 : E 2 → X 2 ar e vecto r bund les. i) Ther e is a canoni cal isomorphis m of poin ted space s T h ( ξ 1 × ξ 2 ) ∼ → T h ( ξ 1 ) ∧ T h ( ξ 2 ) . ii) If ξ 1 is a trivial rank n vecto r bundl e, ther e is an A 1 -weak equ ivalence T h ( ξ 1 ) ∼ − → P 1 ∧ n ∧ X 1 + . Pro position 3.1.2. Suppose X and X ′ ar e smooth sc hemes, Z ⊂ X is a smoo th closed subsc heme, f : X ′ → X is a morp hism, and Z ′ = f − 1 ( Z ) . Let ν Z ′ /X ′ : N Z ′ /X ′ → Z ′ and ν Z/X : N Z/X → Z be the associa ted normal bund les. If the induced m ap N Z ′ /X ′ → f ∗ N Z/X is an isomorphis m, then the diag ram X ′ / ( X ′ − Z ′ ) / /   X/X − Z   T h ( ν Z ′ /X ′ ) / / T h ( ν Z/X ) , wher e the vertical maps ar e the puri ty isomorph isms and the horizo ntal maps are indu ced by f , commutes. Thom spaces in short exact sequences Pro position 3.1.3. Suppos e ξ : E → X , ξ ′ : E ′ → X and ξ ′′ : E ′′ → X ar e thr ee vector bundl es ove r X and assume ther e is a short ex act sequence of vector bu ndles 0 − → E ′ − → E − → E ′′ − → 0 . 22 3.1 Properties of Thom spaces Ther e is an isomorp hism in the A 1 -homotop y cate gory T h ( ξ ) ∼ − → T h ( ξ ′ ) ∧ T h ( ξ ′′ ) that is unique up to unique isomorph ism. Pr oof. Since X is a smooth v ariety we can ch oose a smooth a fﬁne vecto r b und le torsor π : ˜ X → X . Pulling back the exac t sequence from X to ˜ X , we get a short exact sequ ence 0 − → ˜ E ′ − → ˜ E − → ˜ E ′′ − → 0 , where ˜ ξ : ˜ E → ˜ X (resp. ˜ ξ ′ , ˜ E ′ , ˜ ξ ′′ , ˜ E ′′ ) are the pullbacks of E (resp. E ′ , E ′′ ) to ˜ X . Now , since ˜ X is af ﬁne, this sho rt ex act seque nce of ve ctor b undles spli ts. Choice of a sp litting gi ves an isomorphis m ˜ E ∼ − → ˜ E ′ ⊕ ˜ E ′′ , and by Proposit ion 3.1.1 (i), we get an isomorp hism T h ( ˜ ξ ) ∼ → T h ( ˜ ξ ′ ) ∧ T h ( ˜ ξ ′′ ) . Since th e morphis m π is an A 1 -weak e quiv alen ce, and pusho uts of A 1 -weak e quiv alen ces along coﬁbratio ns are ag ain A 1 -weak equiv alen ces (the A 1 -local m odel str ucture is proper), it follows th at the induced morphisms T h ( ˜ ξ ) → T h ( ξ ) (resp. T h ( ˜ ξ ′ ) → T h ( ξ ′ ) , T h ( ˜ ξ ′′ ) → T h ( ξ ′′ ) ) are again A 1 -weak equi v alences. Thus there is an isomorphi sm in H • ( k ) T h ( ξ ) − → T h ( ξ ′ ) ∧ T h ( ξ ′′ ) . One can check that the spac e of splittings is an inducti v e limit of linear spaces and thus A 1 - contra ctible. It follo ws that the in duced map in the A 1 -homoto py cate gory is unique up to unique isomorph ism. Finally , we sho w that the constr uction abov e does not depend on the choice of afﬁne vector b undle torsor π . If π ′ : ˜ X ′ → X is another afﬁne vector bundle torsor , then the ﬁber prod uct ˜ X ′′ := ˜ X ′ × X ˜ X → X is also an af ﬁne vector bund le torsor that maps to both ˜ X and ˜ X ′ by projec tions. Moreov er , these pro jections mak e ˜ X ′′ into a vector bundl e ov er ˜ X or ˜ X ′ since af ﬁne vec tor bund le torsors over afﬁne v arieties are simply vector b undles. Running the constructio n abo ve on each of these spaces (i.e., pulling back in sequence to ˜ X , ˜ X ′ and ˜ X ′′ ) gi ves the necess ary indepe ndence. Thom spaces of virtual bundles If ξ : E → X is a vector bun dle, we can consider the suspension spectrum Σ ∞ P 1 ( T h ( ξ )) , w hich we refer to as the Thom spectrum of ξ . Give n a virtua l vecto r bun dle on a smooth variet y X , that is, a formal dif ference ξ = ξ 1 − ξ 2 of ve ctor bundle s, we can construc t an associate d Thom spectrum as follo ws. First, we cho ose an af ﬁne v ector b undle torso r ˜ X → X ; let ˜ ξ 1 and ˜ ξ 2 be the pullbac k v ector b undles. S ince ˜ X is af ﬁne, we can ﬁnd a vector bund le ξ 3 ov er ˜ X such that ˜ ξ 2 ⊕ ξ 3 ∼ − → O ⊕ n X for some n . Now we deﬁne th e Thom spectrum Σ ∞ P 1 ( T h ( ξ )) (n ote this is an abuse of notatio n as this spectr um is not in fact a sus pension spectrum) by Σ ∞ P 1 ( T h ( ξ )) = Σ − n P 1 (Σ ∞ P 1 ( T h ( ξ 1 )) ∧ Σ ∞ P 1 ( T h ( ξ 3 ))) . By Proposi tion 3.1.3, the Thom sp ectrum is well-deﬁned up to unique isomorphis m in the st able homotop y category SH ( k ) , and its isomorphis m class only depends on the class of ξ in K 0 ( X ) . In 23 3.1 Properties of Thom spaces an an alogous f ashion, we can con struct Thom objects ˜ C A 1 ∗ ( T h ( ξ )) in D A 1 ( k ) for any virtual b undle ξ , well deﬁned up to unique isomorph ism. The follo wing example indicat es the way in which this discussio n will be used. Example 3.1.4 . Suppose X is a smooth scheme, and ξ : E → X and ξ ′ : E ′ → X are vector bund les on X . Suppose we hav e an equation [ E ⊕ O ⊕ n X ] = [ E ′ ] in K 0 ( X ) . A pplyin g P roposit ion 3.1.1(i) and (ii), we get an identiﬁcatio n Σ ∞ P 1 T h ( ξ ) ∧ P 1 ∧ n ∼ → Σ ∞ P 1 T h ( ξ ′ ) , or equi va lently an identiﬁca tion Σ ∞ P 1 ( T h ( ξ )) ∼ → Σ − n P 1 Σ ∞ P 1 T h ( ξ ′ ) . Connectivi ty and cohomologica l dimension of Th om space s Pro position 3.1.5. If X is a smooth k -sche me, and ξ : E → X is a rank n vector b undle over X , then T h ( ξ ) is ( n − 1) - A 1 -conne cted. Pr oof. If ξ : E → X is a trivia l b und le, the n we ha ve the identiﬁcatio n T h ( ξ ) ∼ − → X + ∧ ( P 1 ) ∧ n ; the result th en follo ws immediately from the unsta ble A 1 -conne ctiv ity theorem [MV99, § 2 Coro llary 3.22]. If U → V is an open imm ersion , then there is an obviou s m orphis m of vector bun dles ξ | U → ξ . This induces a map T h ( ξ | U ) → T h ( ξ ) . Functo riality of pushout s thus gi v es rise to a morphism fro m the pusho ut of T h ( ξ | U ) and T h ( ξ | V ) along T h ( ξ | U ∩ V ) to T h ( ξ ) ; this mor phism is an isomorphi sm. Thus, suppose we hav e an open cov er U, V of X and assume we kno w that T h ( ξ | V ) , T h ( ξ | U ) and T h ( ξ | U ∩ V ) are all n -connected. W e then ha ve a push out diagram T h ( ξ | U ∩ V ) / /   T h ( ξ | V )   T h ( ξ | U ) / / T h ( ξ ) . If n = 1 , the result is an immediate conseque nce of the unstable A 1 -conne ctiv ity the orem [Mor11, Theorem 5.37]. If n = 2 , we cho ose ba se-points and the result follows by an ar gument usi ng the A 1 -v an Kampen theo rem [Mor11, Theorem 6.12]. If n > 2 , we use th e (unstable) A 1 -Hure wicz theore m [Mor1 1 , Theorem 5.36] to deduce the res ult by induction . Indee d, if X is an i -connecte d space ( i ≥ 1 ), then the induced map from π A 1 i +1 ( X , x ) → H A 1 i +1 ( X , x ) is an isomorph ism for i > 1 . In that case, the result follo ws from a Mayer- V ietoris argu ment. Remark 3.1.6 . Assume hypoth eses are as in the statement of Proposi tion 3.1.5. If M is a strictl y A 1 -in v arian t sheaf, one can sho w that H i Nis ( T h ( ξ ) , M ) v anishe s for i ≤ n − 1 in a “purely stable” manner . By the stable A 1 -conne ctiv ity theorem, one kno ws that the Thom spect rum of a trivi al b undle is stably ( n − 1) -conne cted. Mayer -V ie toris can be u sed to conclude the argumen t in general. In the se quel, we will only eve r u se th is weak er vanish ing statemen t. In a differ ent direc tion, anot her conseq uence of the unstabl e Hure wicz theorem is that the canonical morphism π A 1 n ( T h ( ξ )) → H A 1 n ( T h ( ξ )) is an isomorphism. Recall that fo r an y X an ob ject of Spc k and any complex of N isne vich sh eav es o f abe lian grou ps F • on Sm k we can deﬁne the hyper cohomology groups H p Nis ( X , F • ) ; we say that a space X has cohomol ogical dimension ≤ d provided that for ev ery sheaf F (viewed as comple x concentrated in deg ree 0 ), the cohomology groups H p Nis ( X , F ) = 0 for all p > d . 24 3.2 Enhan ced mo tivic complexes and cohomology Lemma 3.1.7. If ξ : E → X is a rank r vector bun dle over a smooth k -scheme X , then cd Nis ( T h ( ξ )) < = dim X + r . Pr oof. If i : X → E denotes the zero section of ξ , then by deﬁnition of T h ( ξ ) , we hav e a coﬁbration sequen ce E − i ( X ) − → E − → T h ( ξ ) − → Σ 1 s ( E − i ( X )) + − → · · · . This coﬁbration sequence induces a long exact sequence in cohomology with coefﬁcie nts in an arbitra ry abelian sheaf. The resu lt then follo ws from the usual Nisne vich cohomolo gical dimensio n theore m, i.e., that the Nisnev ich cohomologica l dimen sion of a sche me is bounded abov e by its Krull dimension . 3.2 Enh anced motivic complexes and cohomology If k is a perfect ﬁeld, motivic cohomology of a smooth k -scheme X can be deﬁned in terms of the moti vic comple xes Z ( n ) and th e motiv e of X by the formula (se e, e.g., [MVW06, Deﬁnition 1 4.7]) H i,n ( X, Z ) := Hom DM eﬀ ( k ) ( M ( X ) , Z ( n )[ i ]) . There is a canon ical m ap Hom DM eﬀ ( k ) ( M ( X ) , Z ( n )[ i ]) → Hom DM ( k ) ( M ( X ) , Z ( n )[ i ]) comparin g th e “unsta ble” moti vic cohomolog y grou ps with their “stable” count erparts. In the mo- ti vic conte xt, this map is known to be an isomorphism for arbitrary pairs ( i, n ) by V oe vod sky’ s cancel ation theore m [V oe10, Corollary 4.10]. Our goal in this section is to d eﬁne an ana log of motivic cohomolo gy where DM eﬀ ( k ) (resp. DM ( k ) ) is replaced by D eﬀ A 1 ( k ) (re sp. D A 1 ( k ) ). T he analog of M ( X ) in this conte xt is th e A 1 - chain complex C A 1 ∗ ( X ) . Producing a theory that has a hope of being stably represen table will in v olv e some cont ortions. Enhanced motivi c complexes W e now deﬁne enhanced motivic complex es; th ese complex es play the role of the m oti vic comple xes just discussed in the A 1 -deri ved cate gory . The resulting comp lexe s are mapped to the m oti vic comple xes by the deriv ed functor of “adding transfers” from the A 1 -deri ved category to V oev odsk y’ s deri ved cate gory of effe ctiv e motivi c complex es. Deﬁnition 3.2.1. The enhance d motivic comple x of weight n is deﬁned by the formula Z h n i := ˜ C A 1 ∗ ( P 1 ∧ n )[ − 2 n ] . Lemma 3.2.2. The map Z h 1 i ⊗ n → Z h n i induce d b y the morphism of spaces P 1 × n → P 1 ∧ n is an isomorph ism. 25 3.2 Enhan ced mo tivic complexes and cohomology Deﬁnition 3.2.3. If X is an object of Spc k , the unstable ( i, n ) enhance d moti vic cohomolo gy of X is th e group Hom D eﬀ A 1 ( k ) ( C A 1 ∗ ( X ) , Z h n i [ i ]) . T he sta ble ( i, n ) enhan ced motivic coho mology of X is the group Hom D A 1 ( k ) ( C s A 1 ∗ ( X ) , Z h n i [ i ]) . Remark 3.2.4 . O bserv e that acco rding to Deﬁnition 3.2.1, the con ven tion that P 1 ∧ 0 = S 0 s im- plies that Z h 0 i := ˜ C A 1 ∗ ( S 0 s ) = Z . One conseque nce of this ob serv ation is that the unstabl e group Hom D eﬀ A 1 ( k ) ( C A 1 ∗ (Sp ec k ) , Z h 0 i ) = Z does not coincide with its stable counte rpart Hom D A 1 ( k ) ( C s A 1 ∗ (Sp ec k ) , Z h 0 i ) . Indeed , the latter sheaf has been compute d by Morel to be K M W 0 . Basic pro perties of enhanced m otiv ic cohomology The follo wing result summarizes some properties of the enhanced motivic cohomol ogy groups and the comple xes Z h n i . Pro position 3.2.5. Suppose X is a space . i) F or any inte gers i, n , ther e is a canonic al identiﬁca tion H i Nis ( X , Z h n i ) ∼ − → Hom D eﬀ A 1 ( k ) ( C A 1 ∗ ( X ) , Z h n i [ i ]) . ii) F or i > n , the cohomolog y sheaves H i ( Z h n i ) vanish. iii) F or any inte ger n > 0 , ther e is a canonica l isomorph ism H n ( Z h n i ) = K M W n . Pr oof. Statement (i) follo ws immediately from the observ ation that the comple x Z h n i is A 1 -local, which is implicit in its deﬁnitio n. Statements (ii) and (iii) are equi vale nt to the assertions that (ii)’: ˜ H A 1 i ( G m ∧ n ) = 0 for i < 0 and (iii’): ˜ H A 1 0 ( G m ∧ n ) = K M W n after trading cohomologic al indexing for its homologica l counterpart . Statemen t (ii) ’ is a conse quence of the stable A 1 -conne ctiv ity theore m [Mor05b, Theorem 6.1.8] s ince the compl ex Z ( G m ∧ n ) is ( − 1) -connected . Statement (ii i)’ is a cons equence of the stable Hure wicz theorem 2.3.8 and Deﬁnition 2.1.18. Statements (ii) and (i ii) of Proposition 3.2.5 imply that the hyp ercohomolo gy spectral sequence gi ves rise to a well-deﬁned morphism (3.1) H p Nis ( X , Z h n i ) − → H p − n Nis ( X , K M W n ) . Since the construct ion of the hypercohomol ogy sp ectral sequence is functorial in X , it follo ws that the morphi sm abo ve is actually natural in X . The next result gi ves a condit ion under which this morphism is an isomorp hism. Cor ollary 3.2.6. Suppose X is a space having Nisnevic h cohomolo gical dimension ≤ d . The m ap of Equation 3.1 is an isomorph ism for p ≥ n + d . Pr oof. The kernel and coker nel of the map in Equation 3.1 are buil t out of groups of the form H p − i Nis ( X , H i ( Z h n i )) and H p − i +1 Nis ( X , H i ( Z h n i )) with i < n . By assump tion p − n ≥ d , so if i < n , then p − i > p − n ≥ d . Since the Nisne vich coho mological dimen sion of X is ≤ d , it follo ws both of the aforemention ed groups are triv ial. 26 3.3 Stable repre sentabili ty in top codimension Comparison with motiv ic cohomology Assume k is a perfect ﬁ eld. For e very inte ger n ≥ 0 , the functor D eﬀ A 1 ( k ) → DM eﬀ ( k ) sends the comple xes Z h n i to Z ( n ) by constructio n. V ie w the complex Z ( n ) as an object in D eﬀ A 1 ( k ) via the functo r of “for getting transfers. ” Lemma 3.2.7. Assume k is a perfec t ﬁeld . Sup pose X is a sp ace of co homolog ical dimensio n d > 0 . F or every pair of inte gers ( p, n ) the following dia gram commutes H p Nis ( X , Z h n i ) / /   H p − n Nis ( X , K M W n )   H p Nis ( X , Z ( n )) / / H p − n Nis ( X , K M n ) . wher e the left vertical map is induced by the morphism of complex es Z h n i → Z ( n ) , the horizonta l maps ar e the induced maps in the hyper cohomo logy spec tral sequences and the right vertical m ap is the morphism of sheaves K M W n → K M n induce d by applying the functor n -th cohomolog y sheaf to the morphis m of comple xes Z h n i → Z ( n ) . Pr oof. This follo ws immediately from functoria lity of the hyper cohomology spectral sequen ces: observ e that the ind uced m aps o n cohomolog y sheav es of degr ee > n are isomorp hisms since these shea ves v anish for both complex es. If X has N isne vich cohomol ogical dimensi on ≤ d , and p ≥ n + d , then both horiz ontal maps are isomorphisms. A s a consequenc e, in this range of deg rees th e left vertical m ap coincides with the map on cohomolo gy induc ed by the morphi sm of sheav es K M W n → K M n . Enhanced motivi c cohomology of Thom spaces Cor ollary 3.2.8. If ξ : E → X is a rank r vector bu ndle over a smooth k -sc heme X of dimension d , then the canoni cal m orphis m Hom D eﬀ A 1 ( k ) ( C A 1 ∗ ( T h ( ξ )) , Z h d + r i [2( d + r )]) − → H d + r Nis ( T h ( ξ ) , K M W d + r ) of Equation 3.1 is an isomorph ism. Pr oof. Combine Corollary 3.2.6 and Lemma 3.1.7. 3.3 Stable r epr esentability in top codimension A nai ve anal og of V oe v odsky’ s cancelatio n theorem in the settin g of the A 1 -deri ved cate gory is fals e, as observ ed in Remark 3.2.4. Thus, on e must be some what careful in attemp ting to formulate any kin d of stable representab ility re sult. Nev erthele ss, the go al of this section is to sho w that ce rtain unstab le enha nced motiv ic coho- mology groups can be identiﬁed w ith their stable counterpart s. Our result amounts to a ver y weak form of V oe vo dsky’ s cancelatio n theore m. T o prov e the result , we need to rec all va rious facts abou t the interac tion between G m -loopi ng and Nisne vich cohomolog y . 27 3.3 Stable repre sentabili ty in top codimension Homotopy modules The cate gory D A 1 ( k ) admits a t -struc ture w hose heart we no w describe . Deﬁnition 3.3.1 . A homotopy modu le ove r k is a pair ( M ∗ , ϕ ∗ ) co nsisting of a Z -grade d strictly A 1 -in v arian t sheaf M ∗ and, for each n ∈ Z , an isomorph ism of abelia n shea ves ϕ n : M n ∼ − → ( M n +1 ) − 1 . W e write Ab s A 1 k for the categ ory of homotop y m odules . Lemma 3.3.2. If C is an object of D A 1 ( k ) , then for any inte ger i , the sheaves H A 1 i ( C h n i [ n ]) form a homotop y module with structur e maps given by the isomorphi sms of Proposition 2.3.7 . Proposit ion 2.3.7 sh ows th at if C is a ( − 1) - connected A 1 -local co mplex then all o f the com- ple xes C h n i [ n ] are also ( − 1) -con nected. One can use th is obse rvat ion to deﬁne a t -structure on the catego ry D A 1 ( k ) : the “positi ve” objects are those for w hich H A 1 i ( C h n i [ n ]) vanis hes for i < 0 and for all n ∈ Z . S imilarly , the “negat iv e” objects are those for which H A 1 i ( C h n i [ n ]) van ishes for i > 0 and for all n ∈ Z . The fo llowin g result is an analog of [Mor04a, Theorem 5.2.6] in the stable A 1 -deri ved cate gory; the proof follo ws from this result via Theorem 2.3.8. Theor em 3.3.3. The functor C 7→ L n ∈ N H A 1 0 ( C h n i [ n ]) identiﬁe s the heart of the homotopy t - struct ur e on D A 1 ( k ) with the cate gory of homotopy modules. Contractio ns of Milnor -W itt K -theory shea ves Pro position 3.3.4. W e have a canoni cal identiﬁcatio n ( K M W n ) − 1 ∼ − → K M W n − 1 . Pr oof. This is a conseque nce of Proposition 2.3.7. Stable r epres entability Suppose ( X , x ) is a point ed k -space. W e ha ve a natural map Hom D eﬀ A 1 ( k ) ( C A 1 ∗ ( X ) , Z h n i [2 n ]) − → Hom D eﬀ A 1 ( k ) ( C A 1 ∗ ( X ) ⊗ Z h 1 i [ 2] , Z h n i [2 n ] ⊗ Z h 1 i [2] ) . W e hav e id entiﬁcation s Z h n i [2 n ] ⊗ Z h 1 i [2] ∼ → Z h n + 1 i [2( n + 1)] and C A 1 ∗ ( X + ∧ P 1 ) ∼ → C A 1 ∗ ( X ) ⊗ Z h 1 i [2] . W ith these identiﬁcat ions, we ha ve a map H 2 n Nis ( X , Z h n i ) − → H 2( n +1) Nis ( X ∧ P 1 , Z h n + 1 i ) . The group H 2( n +1) Nis ( X ∧ P 1 , Z h n + 1 i ) is a summand of H 2( n +1) Nis ( X × P 1 , Z h n + 1 i ) . Pro position 3.3.5. F or an y pointed space ( X , x ) ther e is a canonic al isomorph ism H n Nis ( X , K M W n ) − → H n +1 Nis ( X ∧ P 1 , K M W n +1 ) . 28 3.4 Projectivity and Base change Pr oof. This is a conseque nce of the fact that the shea ves K M W n ﬁt togeth er to form a homotop y module (which is, in turn, a conse quence of Proposition 3.3.4). In fact, the result holds for any such homotopy module. N e verthel ess, here are the details . For any pointed space X w e ha ve identi ﬁcations H n Nis ( X , K M W n ) = [ X , K ( K M W n , n )] s ∼ − → [ X , K ( K M W n , n )] A 1 . Indeed , the ﬁrst eq uality is [ MV99, § 2 Propo sition 1.26], an d the seco nd identiﬁcat ion follo ws from the fact that the sheaf K M W n is strictly A 1 -in v arian t, w hich implies that K ( K M W n , n ) is A 1 -local. No w , we can re place X ∧ P 1 by S 1 s ∧ G m ∧ X up to A 1 -weak equi val ence. In the S 1 -stable homotop y cate gory the suspension isomorphis m has been in ve rted and by adjunctio n we get identiﬁcat ions [ S 1 s ∧ G m ∧ X , K ( K M W n +1 , n + 1)] A 1 ∼ − → [ G m ∧ X , K ( K M W n +1 , n )] A 1 ∼ − → [ X , Ω G m K ( K M W n +1 , n )] . No w , th ere is an ob vious map G m ∧ K M W n → K M W n +1 coming fro m the de ﬁnition of the sh eav es K M W n as free st rictly A 1 -in v arian t shea ves gene rated by the shea ves of se ts G m ∧ n . For an a rbitrary smooth scheme U , we ha ve H n Nis ( U ∧ G m , K M W n +1 ) ∼ − → H n Nis ( U, ( K M W n +1 ) − 1 ) = H n Nis ( U, K M W n ) by means of P roposi tion 3.3 .4. Under these identiﬁcation s, the map of the statement is induc ed by the morphism of spaces K M W n → Ω G m G m ∧ K M W n . Thus, it sufﬁces to compute [ X , Ω G m K ( K M W n +1 , n )] . The result then follo ws immediately from [Mor04a, Lemm a 4.3.11 (p. 427)]. Cor ollary 3.3.6. F or a space X of cohomolo gical dimension n > 0 , w e have a canonica l isomor - phism Hom D eﬀ A 1 ( k ) ( C A 1 ∗ ( X ) , Z h n i [2 n ]) ∼ − → Hom D A 1 ( k ) ( C A 1 ∗ ( X ) , Z h n i [2 n ]) . Pr oof. The seemingly odd hypoth esis o n cohomologic al dimensio n is necessi tated by Remark 3.2.4. The isomorphism Hom D eﬀ A 1 ( k ) ( C A 1 ∗ ( X ) , Z h n i [2 n ]) ∼ − → H n Nis ( X , K M W n ) obtained by combining Propo- sition 3.2.5(i) and C orollar y 3.2.6 is contra v ariantly func torial in X . V ia this isomorphis m, the result follo ws immediately from Proposition 3.3.5. 3.4 Pr ojectivity and Base change In this section, we prove that the sheav es H s A 1 0 ( X ) are birational in varia nts of smooth p roper v a- rieties if k is an inﬁnite ﬁeld. The sheaf H s A 1 0 ( X ) is by construction an object in the hea rt of the homotop y t -structure on D A 1 ( k ) (see Deﬁnitio n 3.3.1 and Theorem 3.3 .3 ). Moreov er , almost by its constr uction, it is initial for such objects . Precisely , we hav e the followin g result. Pro position 3.4.1. If M ∗ is a homotopy module , then ther e is a canonical bijection H 0 Nis ( X , M 0 ) ∼ − → Hom Ab s A 1 k ( H s A 1 0 ( X ) ∗ , M ∗ ) contr avarian tly functorial in X . 29 3.5 Atiyah duality in the stable A 1 -derive d catego ry Pr oof. By deﬁnition M ∗ is an objec t lying in the heart of D A 1 ( k ) . Applying the functor H s A 1 0 induce s a map H 0 Nis ( X , M 0 ) ∼ − → Hom D A 1 ( k ) ( C s A 1 ∗ ( X ) , M ∗ ) − → Hom Ab s A 1 k ( H s A 1 0 ( X ) ∗ , M ∗ ) . A standard ar gument in volvi ng the axioms o f a t -str ucture sh ows the second map is a bijection as well. Pro position 3.4.2. If X and X ′ ar e biratio nally eq uivalent smooth pr oper varieties, then H s A 1 0 ( X ) ∼ = H s A 1 0 ( X ′ ) . Pr oof. Suppose M ∗ is a ho motopy module . By Propos ition 3.4.1 and the Y oneda lemma, i t suf ﬁces to sh ow that M 0 ( X ) = M 0 ( X ′ ) . Howe ver , the sheaf M 0 is str ictly A 1 -in v arian t. S ince k is inﬁnite , the asserti on follo ws immediately from [CTH K97, Theor em 8.5.1]. Base Change The analog of the follo wing result in the stable A 1 -homoto py cate gory of S 1 -spect ra is a conse - quenc e of [Mor05b, Lemma 5.2.7]. Pro position 3.4.3. Suppo se X is a k -space , and L/k is a ﬁnitely gen erated separ able exte nsion. Let X L denote X × Spec k Sp ec L . W e then have an identi ﬁcation H A 1 0 ( X )( L ) ∼ − → Hom D eﬀ A 1 ( L ) ( Z , C A 1 ∗ ( X L )) . Pr oof. Suppose X → Sp ec k is a smooth scheme such that L = k ( X ) . T aking a directed system of neighborho ods of the generic point of X , we can write L as an essentia lly smooth scheme, i.e., a ﬁltering limit o f smoo th k -schemes with afﬁne ´ etal e trans ition morphisms. For an y smoo th morphism f : S → S ′ , there is a pullback functo r f ∗ : D A 1 ( S ′ ) − → D A 1 ( S ) Use the analog of [Mor05 b , Lemma 5.1.1] to conclude. 3.5 Atiyah duality in the stable A 1 -deriv ed category The work of Cisinski-D ´ eg lise sho ws that one can apply the duality formalism to the cat egory D A 1 ( k ) . In partic ular , the categ ories D A 1 ( k ) conta in mapping objects . Give n a smooth va riety X , we deﬁne C s A 1 ∗ ( X ) ∨ to be the interna l functio n object Hom ( C s A 1 ∗ ( X ) , 1 ) . Atiyah duality Our next goal is to provid e a “concrete ” de scription of the dual. What is now called Atiyah duality was intro duced in [Ati61], and has been stud ied in the contex t of stable A 1 -homoto py theory by Hu [Hu05, Appendix A] and Riou [Rio05]. W e begin by recalling the follo wing result due to V oev odsk y . 30 3.5 Atiyah duality in the stable A 1 -derive d catego ry Theor em 3.5.1 ([V oe03, Theorem 2.11]) . Let X be a smooth pr ojec tive va riety of pur e dimension d over a ﬁeld k . Ther e e xists an inte g er n and a vecto r bundl e ν : V → X of r ank n such that i) if τ X : T X → X is the tangen t bundl e to X , then [ V ⊕ T X ] = [ O ⊕ n + d X ] in K 0 ( X ) , and ii) ther e e xists a morphis m T n + d → T h ( ν ) in H • ( k ) such tha t the induce d m ap H 2 d,d ( X ) → Z coinci des w ith the de gr ee map. As a conse quence of Theore m 3.5.1(i) and Example 3.1.4, we see that the Thom spectr um of the ne gati ve tangen t b undle − τ X coinci des with the ( n + d ) -fold P 1 -desus pension of the Thom spectr um of ν . As a con sequence of Theorem 3.5.1(ii), we deduce the exis tence of a morphism 1 k → Σ ∞ P 1 T h ( − τ X ) . Pro position 3 .5.2. If X is a smooth pr ojective va riety , then C s A 1 ∗ ( X ) is a str ongly dual izable obje ct of D A 1 ( k ) and its str on g dual is ˜ C s A 1 ∗ ( T h ( − τ X )) . Pr oof. By the deﬁnition of strong duality , we need to construc t maps η : 1 k → ˜ C s A 1 ∗ ( T h ( − τ X )) ⊗ C s A 1 ∗ ( X ) and ǫ : ˜ C s A 1 ∗ ( T h ( − τ X )) ⊗ C s A 1 ∗ ( X ) → 1 k . By deﬁnition of the tensor product, this is equiv alen t to specifying a map 1 k → ˜ C s A 1 ∗ ( T h ( − τ X ) ∧ X + ) . W e identify X + with the Thom space of th e trivi al rank 0 b undle on X . This map is then induce d by V oev ods ky’ s duality map. Similarly , the map ǫ is induce d by the Thom collapse map. One then needs to check that a number of diagrams commute; this is accomplish ed in the same manner as [Hu05, Appendix A]. For later con ve nience, we record the follo wing consequence of Proposition 3.5.2 and the hom- tensor adjunc tion. Lemma 3.5.3. If X is a smooth p r ojective variety we have a ca nonical an d fun ctorial iso morphism: Hom D A 1 ( k ) ( 1 k , C s A 1 ∗ ( X )) ∼ − → Hom D A 1 ( k ) ( C s A 1 ∗ ( X ) ∨ , 1 k ) . Consequences of duality The next result is a vari ant of [Ati61, Lemma 2.1] in the contex t of A 1 -homolo gy . W e use the notati on of Theorem 3.5.1. Theor em 3.5.4. F or any smooth pr op er k -variety X of dimensi on d , and for any se parabl e, ﬁnitely gen erated exte nsion L/k ther e is a canonic al isomorp hism H s A 1 0 ( X )( L ) ∼ − → H n + d Nis ( T h ( ν L ) , K M W n + d ) , functo rial w ith r espect to ﬁeld e xtension s. 31 4 T wisted Chow-W itt groups and Thom isomor phi sms Pr oof. First, combining Chow’ s lemma w ith Propositio n 3.4.2, we can (and will) assume that X is a smooth projecti v e k -v ariety . W e then proceed in a fashio n mirroring the proof of Lemm a 2.2.1: H s A 1 0 ( X )( L ) ( a ) = Hom D A 1 ( k ) ( Z , C s A 1 ∗ ( X L )) ( b ) = Hom D A 1 ( L ) ( Z , C s A 1 ∗ ( X L )) ( c ) = Hom D A 1 ( L ) ( C s A 1 ∗ ( X L ) ∨ , Z ) ( d ) = Hom D A 1 ( L ) ( ˜ C s A 1 ∗ ( T h ( − τ X L )) , Z ) ( e ) = Hom D A 1 ( L ) ( ˜ C s A 1 ∗ ( T h ( ν L )) , Z h n + d i [2 ( n + d )]) ( f ) = H 2( n + d ) Nis ( T h ( ν L ) , Z h n + d i ) ( g ) = H n + d Nis ( T h ( ν L ) , K M W n + d ) Identi ﬁcation (a) is the de ﬁnition of A 1 -homolo gy . Identiﬁcati on (b) is a c onsequen ce of Proposition 3.4.3. Identiﬁcatio n (c) is Lemma 3.5.3. Identiﬁcation (d) is a co nsequenc e of Atiyah duality in the A 1 -deri ved catego ry , i.e., Proposition 3.5.2. Ident iﬁcation (e) follo ws from the discussion of Example 3.1.4. Identiﬁcati on (f) is a conseque nce of Corollary 3.3.6. Finally , identi ﬁcation (g) is a consequenc e of Corollary 3.2.8. F unctor iality with respect to ﬁeld extensi ons is eviden t from the constr uctions of all the isomor phisms. Pro position 3.5.5. The morphism T n + d → T h ( ν ) deﬁned by V oevodsk y induces a map Hom D A 1 ( k ) ( C s A 1 ∗ ( T h ( ν )) , Z h n + d i [2( n + d )]) − → Hom D A 1 ( k ) ( C s A 1 ∗ ( P 1 ∧ n + d ) , Z h n + d i [2( n + d )]) that coinc ides w ith the push forwar d map H s A 1 0 ( X ) → H s A 1 0 ( 1 k ) . Pr oof. This is a conseque nce of the fact that this map induc es the duality map. 4 T wisted Cho w-Witt gr oups and Thom isomorphisms Suppose X is a smooth k -scheme, and ξ : E → X is a v ector bun dle of rank n . At the end of Section 3, we consid ered grou ps of the form H i Nis ( T h ( ξ ) , K M W j ) . The ﬁrst goa l of this section, achie ve d at the end of Section 4.2 is to describe these groups more concret ely in terms of shea f cohomol ogy on X . T o ac hiev e this, we begin by re vie wing the theory of twist ed Chow-W itt grou ps in S ection 4.1; here the twist refers to a choice of line bun dle on X , which, as we explain belo w , can be thought of as an “ A 1 -local system on X . ” W e then prov e a twisted Thom isomorphism (Theorem 4.2.7) tha t identiﬁes cohomolo gy of Milnor- W itt K-theory shea ves on the Thom s pace of a vector bund le in terms o f t wisted Cho w-W itt groups. Muc h o f the work he re con sists of unpacking deﬁnitio ns, but to connect wit h the work o f pre vious sec tions, we n eed to identify the largely for mal deﬁnitio n of Milnor -W itt K-theory shea ves giv en in Deﬁnition 2.1.18 with someth ing concrete ly computa ble; as we expl ain, this identiﬁcation uses Morel’ s comput ation of the stabl e A 1 -homoto py group s of spheres, and the theorem of Orlov-V ishik-V oev odsk y proving Milnor’ s conje cture on 32 4.1 T wisted Chow-W itt groups quadra tic forms. Thus, taken together , these identiﬁcatio ns can be vie wed as the “reduction to Morel’ s computations ” alluded to in the introduct ion. The twisted Thom isomorphism we use constitut es a translatio n of Atiyah’ s classical theory of Thom isomorphisms for not necessaril y oriented manifolds [Ati61]. W e brieﬂy recall this; suppose M is a smooth manifold . If ξ : E → M is a rank n vecto r bundl e over M , then ξ is classiﬁed by a map M → B O ( n ) . O ne knows that π 1 ( B O ( n )) = π 0 ( O ( n )) = Z / 2 induced by the determinant. The cla ssifying map of ξ thus indu ces a homomorph ism π 1 ( M ) → Z / 2 , i.e., an ori entation charac- ter of M , and only depends on the determinant of ξ . The orientat ion chara cter therefore induces a rank 1 local system Z [det ξ ] on M . One can deﬁne a Thom class τ ( ξ ) in H n ( T h ( M ) , Z [det ξ ∨ ]) . The Thom isomorph ism theorem can be phrase d as saying that the the cup produc t τ ( ξ ) ∪ · : H i ( M , Z [det ξ ]) ∼ − → H n + i ( T h ( M ) , Z ) . induce s a canonical isomorphis m between source and tar get. After these preliminaries, Sections 4.3 and 4.4 are dev oted to the proofs of all the results stated in the intro duction. The reader should be warne d that throu ghout this section, unless otherwise mentione d, the base ﬁ eld k will be assumed to ha ve char acteristi c unequa l to 2 ; this require ment is imposed by the theory of C ho w-W itt groups via its dependen ce on Balmer’ s W itt theory where the requir ement that 2 be in ve rtible is imposed from the begin ning. 4.1 T wisted Cho w-Witt gr oups The Gersten r esolution for a strict ly A 1 -in v ariant sheaf Pro position 4.1.1. Suppo se M is a strictly A 1 -in va riant sheaf of gr ou ps and k is an inﬁnite ﬁeld. If X is a smooth k -variet y , then the re striction of M to th e small Zar iski site o f X , which we denote M | X Z ar admits the followin g r esolutio n: M | X Z ar − → M x ∈ X (0) M ( κ ( x )) − → M x ∈ X (1) M − 1 ( κ x ) − → · · · − → M x ∈ X ( i ) M − i ( κ x ) − → · · · . Pr oof. Since M is st rictly A 1 -in v arian t, w e k now that M | X Z ar admits a Cousin resolu tion, further - more we kno w that all cohomology preshea ves in the Zariski and Nisne vich topolog y agree (these statemen ts are a c onsequen ce of [CTHK97, T heorem 5.1.10 and Propositio n 5.3.2a] and [CTHK97, Theorem 8.3 .1] respecti v ely). W e need to sho w that H q x ( X, M ) = M − q ( κ x ) . By deﬁniti on, the group on the left hand side is colim H q ¯ x ∩ U ( U, M | U ) . For any smooth closed immersion Z ֒ → X , we hav e H q Z ( X, M ) = H q Nis ( T h ( N Z/X ) , M ) by the purity isomorp hism. By shrinking X and Z and usi ng Nisne vich e xcision , we can assu me N Z/X is trivial and th e Lemma 4.1.2 sho ws that H q ( T h ( N Z/X ) , M ) = H q − codimZ/X ( X, M ) − codimZ/X . Lemma 4.1.2 ([MVW06, Exerc ise 23.4]) . If M is stri ctly A 1 -in va riant in the Zariski topolo gy , for any smooth sc heme X we have a canonical isomorphism H m X ×{ 0 } ( X × A m , M ) ∼ − → H m − n ( X, M ) − n 33 4.1 T wisted Chow-W itt groups Theor em 4.1.3. F or an y smooth scheme X , the sheaves K M W n admit a r esolut ion of the form K M W n | X − → M x ∈ X (0) K M W n ( κ x ) − → · · · − → M x ∈ X ( i ) K M W n − i ( κ x ) − → · · · ; we write C ∗ ( X, K M W n ) for this comple x. Pr oof. This is immediate from Proposition 4.1.1 and Propositio n 3.3.4. Deﬁnition 4.1.4. W e recall from [Fas08, Section 9] that for each reg ular scheme X , line bu ndle L on X and inte ger j we ha ve a comple x C ∗ ( X, I j , L ) which in degre e k is C k ( X, I j , L ) = M x ∈ X ( k ) I j ( O X,x , L x ) , where I j ( O X,x , L x ) is the j -th power of th e fundamenta l ideal of the W itt ring of modules of ﬁnite length with dual ity gi ven by the line bundl e L at the point x . In this way w e can cl early deﬁne a comple x of shea ves on X N is , which will be denot ed by C ∗ ( − , I j , L ) . Remark 4.1.5 . The reader s hould be wa rned that Fas el uses sl ightly dif ferent notatio n, and dif ferent inde xing, for these complex es. Deﬁnition 4.1.6. T he Gersten resolu tion of the sheaf K M n of unramiﬁed Milnor K -theory is of the form K M n | X − → M x ∈ X (0) K M n ( κ x ) − → · · · − → M x ∈ X ( i ) K M n − i ( κ x ) − → · · · , and similarly for the sh eav es K M n / 2 . In li ght of Propositio n 4.1.1 the exist ence of this reso lution follo ws fr om the fact (see [Mor04a]) th at K M ∗ is a homotopy module. W e write C ∗ ( − , K M n ) and C ∗ ( − , K M n / 2) for these Gersten complex es of shea ves. As describ ed in [Fas 08, S ection 10], there is a map of comple xes of shea ves from C ∗ ( − , I j , L ) to C ∗ ( − , I j /I j +1 , L ) and also a map from C ∗ ( X, K M j / 2) → C ∗ ( X, I j /I j +1 , L ) . The aforemen- tioned map is an isomorphism by the Orlo v-V is hik-V oev odsk y theorem proving Miln or’ s conjectur e on quadrati c forms; se e [O VV07, Theor em 4.1] , or [Mor05a, Theorem 1.1 and § 2.3] for a discus- sion in the spirit of this paper . As a conse quence, we can compute the cohomology shea ves of the comple xes C ∗ ( − , I j , L ) . Pro position 4.1.7. The comple x of sheaves C ∗ ( − , I j , L ) has the following cohomolo gy: 1. H i C ∗ ( − , I j , L ) = 0 for i < 0 , 2. H 0 C ∗ ( − , I j , L ) = I j ( L ) nr (the unr amiﬁed sheaf associat ed to I j ) for i = 0 , and 3. H i C ∗ ( − , I j , L ) = 0 for i > 0 . 34 4.1 T wisted Chow-W itt groups Pr oof. The ﬁ rst two assertio ns are immediate from the deﬁnition. Since we are dealing w ith a local problem, we may ass ume that L is triv ial. First consider the ca se where j ≤ 0 . In this case the comple x in questio n is, by deﬁnition, simply the Gersten-W itt complex of Balmer -W alter (see [BW02, Theorem 7.2]), which is a resolut ion. W e proceed by induc tion o n j , employin g the short exa ct sequence of complex es (see [Fas08, D ´ eﬁnitio n 9.2.10]) 0 − → C ∗ ( − , I j , L ) − → C ∗ ( − , I j − 1 , L ) − → C ∗ ( − , I j − 1 /I j , L ) → 0 . It follo ws fr om th e Orlov -V ishik -V oe vo dsky theore m (refere nces just prior to the statement of the propo sition) that the third complex is isomor phic to the Gersten comple x of Milnor K -theory K M j − 1 / 2 , and hence a resolutio n; and th e middle complex is a resolution by induction . Therefore the ﬁrst comple x is a resolutio n as well. The ne xt r esult ties together all of the deﬁnitions made so f ar . For context , recall that w e deﬁned Milnor -W itt K-theory sh eav es in ter ms of zero th stable A 1 -homoto py shea ves of spectra (Deﬁnition 2.1.18). This deﬁnition has t he be neﬁt of exp laining why the sh eav es K M W ∗ are strictly A 1 -in v arian t and giv e rise to a homotopy module (see Deﬁnition 3.3.1). W e obse rved in Theorem 4.1.3 that this structu re is suf ﬁcient to giv e a Gersten resolution for K M W n . T o connect this a priori abstract de - scripti on to the discussi on of Fasel that w e hav e revie wed above requires Morel’ s explicit descrip tion of the shea ve s K M W n . Theor em 4.1.8 (Mor el) . A ssume k is an inﬁnite perf ect ﬁeld of char acteristic unequa l to 2 . F or any smooth sc heme X , ther e is a (functorial in X ) quas i-isomorphi sm of comple xes C ∗ ( − , K M W j ) = C ∗ ( − , K M j ) × C ∗ ( − ,I j /I j +1 , O X ) C ∗ ( − , I j , O X ) . Pr oof. Consider the zeroth co homology she af H 0 ( C ∗ ( − , K M j ) × C ∗ ( − ,I j /I j +1 , O X ) C ∗ ( − , I j , O X )) . In [Mo r04c, Theore m 5.3], Mor el gi ve s an explici t descrip tion of th e sect ions of the af orementione d cohomol ogy sheaf ov er ﬁnitely ge nerated e xtensio ns L/k in terms of w hat are called Milnor -W itt K-theory of ﬁ elds; this implicitly uses the O rlov -V ishik -V oe vo dsky theorem provin g Milnor’ s con- jecture on quadratic forms. This terminolog y is justiﬁed by [Mor11, Theorem 5.39 and R emark 5.41], which implicitly uses the assumptio n that k is perfect, and asserts that these Milnor- W itt K-theory of ﬁelds are precisely the sections of wha t we called Miln or-W itt K-theory shea ves ove r ﬁelds (recall Deﬁnition 2.1.18). Moreo ver , there is a morphism of sheav es fr om K M W j to the zeroth cohomology sheaf abov e; see [Mor05a, § 2.2] for an expl anation of this construct ion. Since the zeroth cohomology sheaf consid ered abov e is strictl y A 1 -in v arian t by its very constructio n, the discussi on of the pre vious paragr aph sh ows that the aforementi oned morph ism is an isomorp hism. Thus, both complex es are (funct orially in X ) resolut ions of the sheaf K M W j and are ther efore quasi-isomor phic. Again follo wing Fasel, we deﬁne (with sligh tly diff erent notation and indexing ): Deﬁnition 4.1.9. The j -th Chow-W itt comple x of X with twist L is the pull-back complex of shea ves C ∗ ( − , K M W j , L ) := C ∗ ( − , K M j ) × C ∗ ( − ,I j /I j +1 , L ) C ∗ ( − , I j , L ) . 35 4.2 The Thom isomorphism theorem Theor em 4.1.10. The only non-vanish ing coho mology sheaf of the comple x of sheaves C ∗ ( − , K M W j , L ) occur s in de gr ee zer o. Pr oof. This is a straightfo rward consequenc e of P roposi tion 4.1.7. Deﬁnition 4.1.11. Given a re gular sc heme X and a line b undle L on X , we deﬁne the L -twisted Milnor -W itt K -theory shea ves on X by means of the formula: K M W j ( L ) := H 0 C ∗ ( − , K M W j , L ) . By Theorem 4.1.10, C ∗ ( − , K M W j , L ) is a ﬂasque resolution of the sheaf K M W j ( L ) , so that we deduc e the follo wing result. Cor ollary 4.1.12. F or any clos ed subsche me Y ⊆ X with ope n complement U = X − Y , we hav e an isomorph ism H ∗ Y ( X, K M W j ( L )) = H ∗ (k er { C ∗ ( X, K M W j , L ) − → C ∗ ( U, K M W j , L ) } ) . Deﬁnition 4.1.13. For a smooth p roper k -vari ety X of di mension d , we deﬁne th e Cho w-W itt group of quad ratic zero cycles as g C H 0 ( X ) := H d Nis ( X, K M W d ( ω X )) . Note that, by the Gersten resolu tion, g C H 0 ( X ) is a quotient of L x ∈ X ( d ) K M W 0 ( κ x /k ) . Remark 4.1.14 . Theorem 4.1.10 immedia tely implies that there is a canonical isomorphism H p Nis ( X, K M W p ( L )) = g C H p ( X, L ) , where the right-han d side is the twisted C ho w-W itt group deﬁned by Fas el in [Fas08, D ´ eﬁnition 10.2.16]. In particular , for a smooth proper varie ty X , we hav e a canonical identiﬁcatio n g C H 0 ( X ) = g C H d ( X, ω X ) , and this exp lains our choice of notation. 4.2 The Thom isomorphism th eorem Recollec tion: functoriality . Fasel deﬁnes (in [Fas0 8, Section 10.4] and [Fas0 7, Sections 5 throug h 7]) pullbacks, products and proper pushfo rwards for the complex es C ∗ ( − , K M W j , L ) as follo ws: Theor em 4.2.1. L et X , Y and Z be re gular sch emes and suppose f : X → Y is a morphism and g : X → Z is pr oper of r elativ e dimension n . Then: 1. Assume f is ﬂat. F or any line b undle L on Y , we h ave pullbac k homomorphisms f ∗ : C ∗ ( Y , K M W j , L ) − → C ∗ ( X, K M W j , f ∗ L ) which ar e fu nctorial for composition of ﬂat morphisms. 36 4.2 The Thom isomorphism theorem 2. F or f arbit rary , for all i , and f or any lin e bun dle L on Y , we have fu nctorial Gysin pullb acks f ! : H i ( C ∗ ( Y , K M W j , L )) − → H i ( C ∗ ( X, K M W j , f ∗ L )) . If f is ﬂat, then f ! coinci des w ith the map indu ced on cohomolo gy by the ﬂat pullbac k f ∗ . 3. F or any line bun dle L on Z we ha ve pushforwar d homomorphisms g ∗ : C ∗ ( X, K M W j , g ∗ L ⊗ ω X ) − → C ∗− n ( Z, K M W j − n , L ⊗ ω Y ) which ar e fu nctorial for composition of pr oper morphisms. 4. Let L an d L ′ be li ne b undles on X . Ther e is a fu nctorial (with r esp ect to t he Gysin pu llbac ks) gra ded associat ive pr oduc t H ∗ ( C ∗ ( X, K M W j , L )) ⊗ H ∗ ( C ∗ ( X, K M W k , L ′ )) − → H ∗ ( C ∗ ( X, K M W j + k , L ⊗ L ′ )) deﬁne d fr om an e xterior pr odu ct via pullbac k along the diagon al. Interlude: the p r ojection formula W e will need a pr ojection formula for the cohomology of th e Cho w-W itt compl exes . In some o f the results below we will, for notational con v enience , suppress the line bu ndle twists that are implicit; to recov er the twists relev ant to a giv en diagram, we refer the reader to T heorem 4.2.1. T he crucial ingred ient in the pro of is a proper base ch ange theorem; we thank Jean Fas el for sho wing us the proof of the follo wing result. Pro position 4.2.2. If X ′ v / / g   X f   Y ′ u / / Y is a cartesia n squar e of smooth sc hemes with f pr o per , then u ! f ∗ = g ∗ v ! . Pr oof. The proof follows the same pattern as [Ros96, Proposition 12.5]. Any morphism of smooth schemes c an be fa ctored as a cl osed immersion fo llowed by a smooth (in pa rticular , ﬂat) morphis m. Then, we contemp late the diagram X ′ Γ v / / X ′ × X p X / / g × id   X X ′ / / g   Y ′ × X / / id × f   X f   Y ′ Γ u / / Y ′ × Y p Y / / Y . 37 4.2 The Thom isomorphism theorem Using the fact that the Gysin pullb ack coincid es w ith the usual ﬂat pullback for ﬂat morphis ms [Fas07, Proposition 7.4] , the functorialit y of pull backs [Fas07, Theorem 5 .11] reduc es us to prov ing the result in the case where u is ﬂat or a closed immersion. If u is ﬂat, this is [Fas08, Corollair e 12.3.7]. For a closed immersion, it follo ws, in light of the naturality of the defo rmation to the n ormal con e cons truction, from the way the Gysin map is deﬁned (see [Fas0 7 , Deﬁnition 5.5]) and anoth er applic ation of [Fas08, Corollai re 12.3.7]. Cor ollary 4 .2.3. Suppose f : X → Y is a pr oper morphism of smooth sche mes. F or any classes η , η ′ (with arbitr ary twists) we have a canonica l identiﬁ cation f ∗ ( f ! η ∪ η ′ ) = η ∪ f ∗ η ′ . Pr oof. Giv en base change, the proof is standar d. Applying Proposition 4.2.2 to the square X Γ t f / / f   Y × X 1 × f   Y ∆ Y / / Y × Y , we conclu de that ∆ ! Y (1 × f ) ∗ = f ∗ (Γ t f ) ! = f ∗ ∆ ! X ( f × 1) ∗ where the second equality comes from the equatio n Γ t f = ( f × 1)∆ X . Remark 4.2.4 . The produc t structur e on the cohomology of Cho w-W itt complexe s is not (graded ) commutati ve (cf. [Fas07, Remark 6.7]). Nev ertheless the abov e pr oof cle arly a pplies mutatis mu- tandis to sho w that f ∗ ( η ∪ f ! η ′ ) = f ∗ η ∪ η ′ . Thom isomor phisms Suppose ξ : E → X is a vector bun dle w ith zero section i : X → E . Let d et ξ be the determinan t b undle. Wi th th e deﬁnition of det ξ ∨ -twisted Cho w-W itt groups ab ove , we hav e the followin g pur ity result; see [Fas08, Remar que 10.4.8]. Pro position 4.2 .5. F or a re gular subsc heme Y ⊆ X of codimension c with normal bundl e ν and complement U = X − Y , th er e is a quasi-is omorphism C ∗− c ( Y , K M W j , det ν ) − → k er { C ∗ ( X, K M W j ) − → C ∗ ( U, K M W j ) } . Deﬁnition 4.2.6. Let X be a smooth scheme and ξ : E → X a vector b undle of rank r and with zero section i : X → E . By Propositio n 4.2.5, we hav e an isomorphism i ∗ : H 0 Nis ( X, K M W 0 ) → H r i ( X ) ( E , K M W r (det ξ ∨ )) . The image τ ( ξ ) = i ∗ (1 X ) of 1 X under this isomorphism i s called t he Thom class o f ξ . P ull back along ξ and cup produ ct wit h the T hom class deﬁnes homomorphisms τ p,q ( ξ ) : H p Nis ( X, K M W q (det ξ )) → H p + r i ( X ) ( E , K M W q + r ) , and the latter group is canonically isomorphic (since E − i ( X ) → E → T h ( ξ ) is a coﬁbrat ion sequence, cf. Lemma 3.1.7) to H p + r Nis ( T h ( ξ ) , K M W q + r ) . 38 4.3 The zero th stable A 1 -homotopy sheaf Theor em 4.2.7 (Thom isomorphism) . The bigrad ed H ∗ Nis ( X, K M W ∗ ) -module H ∗ i ( X ) ( E , K M W ∗ (det ξ ∨ )) is fr ee of r ank one on the Thom class τ ( ξ ) . Consequ ently we have a collection of Thom isomor - phisms τ p,q ( ξ ) : H p Nis ( X, K M W q (det ξ )) → H p + r Nis ( T h ( ξ ) , K M W q + r ) deﬁned by pullbac k followed by cup-pr oduct w ith the Thom class . Pr oof. Let ξ : E → X be a v ector bundl e on X . Consider the followin g diagram: H p Nis ( X, K M W q (det ξ )) i ∗ / / ξ ∗   H p + r i ( X ) ( E , K M W q + r ) H p Nis ( E , K M W q ( ξ ∗ det ξ )) . ∪ τ ( ξ ) 5 5 j j j j j j j j j j j j j j j The top horizo ntal morphism is an isomorphism by Proposition 4.2.5. By [Fas07, Propositio n 6.8], the fundamental class 1 X is both a left and right u nit. Moreov er , i ! ξ ∗ = id by [Fas07, L emma 5.10]. Therefo re, for any class η , we hav e η = i ! ξ ∗ η ∪ 1 . Combining the observ atio n just made with the projecti on formula 4.2.3, we then hav e i ∗ ( η ) = i ∗ ( i ! ξ ∗ η ∪ 1) = ξ ∗ η ∪ i ∗ (1) , that is, the diagra m commutes. 4.3 The zer oth stable A 1 -homotopy sheaf The goal of this section is to pro ve Theorems 2 and 3 from the introduc tion. Theor em 4.3.1. Suppose k is an inﬁnite perfect ﬁeld having char acterist ic une qual to 2 , and X is a smooth pr oper k -variety . F or any separ able, ﬁn itely gener ated exten sion L /k , ther e ar e isomor- phisms π s 0 (Σ ∞ P 1 X + )( L ) ∼ − → g C H 0 ( X L ) , functo rial w ith r espect to ﬁeld e xtension s. Pr oof. By Theorem 2.3.8, w e only need exhibit a natural isomorphi sm H s A 1 0 ( X )( L ) ∼ − → g C H 0 ( X L ) . W e can a ssume X has dimension d . By T heorem 3.5.4 we hav e a canonical i somorphism (fu nctorial with respe ct to ﬁeld ex tensions) H s A 1 0 ( X )( L ) ∼ − → H n + d Nis ( T h ( ν L ) , K M W n + d ) , where ν : V → X is a v ector b undle of rank n such th at [ T X ⊕ V ] = n + d in K 0 ( X ) . By th e Thom isomorph ism 4.2.7, the right h and side is can onically isomorphic to H d Nis ( X L , K M W d (det ν L )) , and by properties of the determinant , det ν L = ω X L (the canonica l b undle) . Finally , H d Nis ( X L , K M W d ( ω X L )) = g C H o ( X L ) by deﬁnition . Theor em 4.3.2. The isomorphisms of Theorem 4.3.1 satisfy the following compatibiliti es. A) The “Hur ewicz” homomorphism π s 0 (Σ ∞ P 1 X + ) → H S 0 ( X ) induces the for get ful map g C H 0 ( X L ) → C H 0 ( X L ) upon eval uation on sections over a separ able ﬁnitely gener ated extens ion L/k . 39 4.3 The zero th stable A 1 -homotopy sheaf B) The pus hforwar d map π s 0 (Σ ∞ P 1 X + )( L ) → π s 0 (Σ ∞ P 1 Sp ec k + )( L ) induces by m eans of the Thom isomorphism Theorem 4.2.7 and M or el’ s identiﬁ cation of π s 0 (Σ ∞ P 1 Sp ec k + )( L ) w ith GW ( L ) a morphism g deg : g C H 0 ( X L ) → g C H 0 (Sp ec L ) that coincides with the corr espond- ing pushfo rwar d deﬁne d by F asel. C) The two identiﬁca tions just mentioned ar e compatible , i.e., th e diagra m g C H 0 ( X L ) / / g deg   C H 0 ( X L ) deg   GW ( L ) rk / / Z commutes. Pr oof. A) The homomorphism in the statement is the map from P roposi tion 2.1.35 composed with the stab le H ure wicz homo morphism π s 0 (Σ ∞ P 1 X + ) → H s A 1 0 ( X ) of 2.3. Now , combin e L emma 2 .2.1 and Theorem 4.3.1. B) Combining Theorem 3.5.4 and Theorem 4.2.7, ev alua tion of the pushfo ward map H s A 1 0 ( X ) → H s A 1 0 (Sp ec k ) on se ctions ov er a ﬁnit ely genera ted, separable exte nsion L/k giv es rise to a mor - phism H d Nis ( X L , K M W d ( ω X )) → H 0 Nis (Sp ec L, K M W 0 ) . By Theorem 4.3.1 and [Mor11, Lemma 2.10], this morphism can be vie wed as a morphism g C H 0 ( X L ) → GW ( L ) ; we wa nt to know that this coinc ides with the pushforwa rd deﬁned by Fasel. Unwinding the deﬁni tions, to iden tify the two pushforward s, we hav e to prov e commuta tivit y of the diagr am (4.1) H d ( X L , K M W d ( ω X L ))   g deg / / H 0 (Sp ec L, K M W 0 )   H n + d ( T h ( ν L ) , K M W n + d ) / / H n + d ( T n + d L , K M W n + d ) , where the left vertical ar row is th e Thom isomorp hism, the right v ertical arro w is the su spension isomorph ism, and the lower horiz ontal arro w is the m orphis m induced by V oev ods ky’ s du ality the- orem 3.5.1. T o simplify of notation, w e may use base change 3.4.3 and assume k = L in w hat follo ws. Step 1. W e claim it suf ﬁces to prov e the result fo r t he pu shforward induced b y a ﬁnite ﬁeld exte nsion Sp ec F → Sp ec k . T o see this, observ e that the morphism g deg has a local deﬁnition. Indeed , there are transfe r homomorphisms tr x : GW ( κ x , ω κ x /k ) → GW ( k ) and the map g deg is induced by the homomorphism L x ∈ X ( d ) tr x ; this follo ws by combin ing the local deﬁnition of deg for Chow groups with the study of the push-forwar d for W itt groups in 40 4.3 The zero th stable A 1 -homotopy sheaf [Fas08, § 6.4] togeth er with the identiﬁcatio n GW ( F ) ∼ = W ( F ) × Z / 2 Z . W e note here for later use that the map tr x is a Scharla u transfer with respect to the ﬁeld trace of κ x ov er k . On the other hand, collapsin g th e complement of x in X gi ves a morphism X → X/ ( X − x ) . Since the nor mal space to x is tri vial, the space X/ ( X − x ) is isomorph ic to T ∧ d ; to ﬁx such an isomorph ism we need a ch oice of tri vializatio n. Keeping tra ck of the twist by ω X , the co llapse map just mention ed gi ves rise to a homomor phism H d Nis ( X/X − x, K M W d ( ω X )) − → H d Nis ( X, K M W d ( ω X )) , where the gro up on the left hand side is deﬁned as the d -th co homology of the cone of the mor - phism C ∗ ( X − x, K M W ∗ , ω X − x ) → C ∗ ( X, K M W ∗ , ω X ) ; the resulting co mplex is supported at x by its very con struction. W e claim there is a suspensio n isomorphism H d ( X/X − x, K M W d ( ω X )) ∼ → GW ( κ x , ω κ x /k ) . In- deed, by inspect ion of the Gersten resolutio n (the only term that appears is K M W 0 ( κ x , ω κ x /k ) ), this is a cons equence of Morel’ s isomorphism. Using the fact that H d Nis ( X, K M W d ( ω X )) admits a surjection from a sum of groups of the form GW ( κ x , ω x/k ) , it sufﬁces to prov e that t r x coinci des with the homomorphism GW ( κ x , ω κ x /k ) → GW ( k ) giv en by composing the morphism of the previo us paragraph with the composite of the morphisms from the three unlabeled edges in Diagram 4.1 for any x ∈ X ( d ) . By functorial ity of the Thom isomorp hism, this co rresponds to pro ving the ini tial compatibil ity in the case where X = Sp ec F , as claimed. W e will write tr F /k for F asel’ s pushforw ard (the Scharl au transfer fo r the ﬁeld trace) . Step 2. W e no w assume that X = Sp ec F ; the dua lity cons truction simpliﬁes in this case. Pick an embe dding i : S p ec F ֒ → P 1 k . There is an in duced map P 1 k → T h ( ν i ) , sometimes called the “co-tra nsfer map, ” ( cf. [L e v10]) which is t he duality map. W e need to s how t hat the compos ite map t : H 0 Nis (Sp ec F , K M W 0 ( ν i )) ∼ − → H 1 Nis ( T h ( ν i ) , K M W 1 ) → H 1 Nis ( P 1 k , K M W 1 ) ∼ − → H 0 Nis (Sp ec k , K M W 0 ) , where the ﬁ rst map is the Thom iso morphism and the las t map is the in verse suspens ion isomor - phism, is precis ely the pushf orward map tr F /k of Fasel. Follo wing [Mor11, § 3], the composite map is precisely Morel’ s “cohomol ogical” transfe r . This transfe r can be descr ibed at the le ve l of the Gerste n resolutio n as follo ws. Cons ider A 1 k with co ordi- nate t . T here is a shor t exact sequen ce of the form 0 − → K M W 1 ( k ) − → K M W 1 ( k ( t )) P ν ∂ ν − → M ν K M W 0 ( κ ν , ω κ ν /k ) − → 0 . Here the maps ∂ ν are the residue maps in Milnor -W itt K -theory described by M orel in [Mor11, Theorem 2.15], and the short exact sequence is proven in [Mor11, T heorem 2.24]. Picking a prim- iti ve el ement θ of F with minimal polynomia l f ( t ) , we identify F with the closed point x of A 1 k corres ponding to f ( t ) . Give n an element α ∈ K M W 0 ( F ) = K M W 0 ( κ x ) the exact sequence above sho ws tha t we can lift α along the d iffer ential in the G ersten complex to some ˜ α ∈ K M W 1 ( k ( t )) . In other wor ds, w e hav e the equation ∂ can x ( ˜ α ) = α ; here ∂ can x is the x -comp onent of the dif- ferenti al in the Gersten comple x, which is obtained from ∂ x via twist with the canon ical bu ndle. No w t ( α ) = − ∂ can ∞ ( ˜ α ) . T o prov e our des ired compati bility it theref ore sufﬁce s to sho w tha t 41 4.4 Rational points up to stable A 1 -homotopy − ∂ can ∞ = tr F /k ◦ ∂ can x . Let β ∈ K M W 1 ( k ( t )) . It is cl ear fr om M orel’ s de scription of th e residues th at the ranks o f − ∂ can ∞ ( β ) an d tr F /k ◦ ∂ can x ( β ) coincide (th e sig n comes from the fact that t he pa rameter at ∞ is a multiple of 1 /t ). It is th erefore sufﬁcient to sho w that the equatio n − ∂ can ∞ = tr F /k ◦ ∂ can x holds on (cano nically twisted) W itt groups ; but that is an immedia te consequenc e of Schmid’ s reci- procit y theorem [Sch98, 2.4.5]. C) The Hure wicz maps are functorial by constructi on. 4.4 Rational points up to stable A 1 -homotopy Finally , in this section, w e pro ve Theorem 1. Lemma 4.4.1. If f : M → M ′ is a morphism of strictl y A 1 -in va riant sheaves of gr o ups, then f is an isomorphism (r esp. monomorphism, r es p. epimorphis m) if and only if it is bijective (re sp. injecti ve, re sp. surje ctive) on sections over any ﬁnitely gene rated, separab le e xtension L/k . Pr oof. This is immediate from the Gersten resolutio n 4.1.1. Pro position 4.4.2. Suppo se k is a perfect ﬁel d, and X is a smooth, pr oper k -scheme . The canonical morphism H s A 1 0 ( X ) → H S 0 ( X ) is an epimorphism. Pr oof. Since both shea ves in questio n are strictly A 1 -in v arian t, by Lemma 4.4.1 it suf ﬁces to prov e that the m ap in question is surjecti ve on sections ov er ﬁnitely generated extens ion ﬁelds L/k . Let d = dim X and ν : V → X the vect or bund le of T heorem 3.5.1. V ia duali ty , the homomorphism H s A 1 0 ( X )( L ) → H S 0 ( X )( L ) is identiﬁed with the homomor- phism H n + d Nis ( T h ( V ) , K M W n + d ) → H n + d Nis ( T h ( V ) , K M n + d ) induced by the epimorphis m of strictly A 1 -in v arian t she av es K M W n + d → K M n + d . Now , by Lemma 3.1.7 the cohomolo gical dimension of T h ( V ) is n + d so the functor H n + d Nis ( T h ( V ) , − ) is right exact. Deﬁnition 4.4.3. Suppose X is a smooth k -v ariety . W e say X has a r ational point up to stable A 1 -homotop y if the structure map Σ ∞ P 1 X + → S 0 k is a split epimorphism. If X has a rational point up to stable A 1 -homoto py , a choice of splitting S 0 k → Σ ∞ P 1 X + is called a ration al point up to stable A 1 -homoto py . Lemma 4.4.4. Suppose X is a smooth k -varie ty . The followin g conditi ons are equ ivalent. i) The variety X has a rati onal point up to stable A 1 -homotop y . ii) The structur e map H s A 1 0 ( X ) → H s A 1 0 (Sp ec k ) is a split epimorphism. iii) Ther e exists an elemen t x ∈ π s 0 ( X )( k ) lifting 1 ∈ π s 0 ( S 0 k ) . iv) Ther e exists an element x ∈ H s A 1 0 ( X )( k ) lifting 1 ∈ H s A 1 0 ( X )( k ) . Pr oof. The equi val ences (i) ⇔ (ii ) an d (iii) ⇔ (iv) are conse quences of the stable H ure wicz theo- rem; one only need s to note that the Hure wicz isomorphism is functorial in the input space . T o see that (i) ⇔ (iii), observ e that elements x ∈ π s 0 ( X )( k ) are precisely morp hisms of shea ves S 0 k → π s 0 (Σ ∞ P 1 X + ) . Under this identiﬁca tion, the element 1 ∈ π s 0 ( S 0 k ) corresp onds to the iden- tity morphism S 0 k → S 0 k . 42 4.4 Rational points up to stable A 1 -homotopy Theor em 4.4.5. Assume k is an in ﬁnite perfect ﬁ eld (having c harac teristic unequal to 2 ). A smoo th pr op er k -variety X has a rationa l point up to stable A 1 -homotop y if and only if X has a 0 -cycle of de gr e e 1 . Pr oof. Consider the diagram H s A 1 0 ( X ) / /   H S 0 ( X )   H s A 1 0 (Sp ec k ) / / H S 0 (Sp ec k ) . By P roposit ion 4.4.2 both horizontal ar rows are epimorphisms. By Lemma 4.4.1 it follo ws t hat upon taking sectio ns over k the ind uced horizontal maps are still epimorphisms . Moreo ver , by Theorems 4.3.1 and 4.3.2, the left vertical map coincides with the pushf orward map g deg : g C H 0 ( X ) − → GW ( k ) . If X has a rationa l point up to stable homot opy , by t he eq uiv alent c onditions of Lemma 4.4.4 w e kno w that there exists an element x ∈ H s A 1 0 ( X )( k ) lif ting 1 ∈ K M W 0 ( k ) . Since 1 ∈ H s A 1 0 (Sp ec k )( k ) is sent to 1 in H S 0 ( X )( k ) , it follo ws immediately that the image of x in H S 0 ( X )( k ) is mapped to a 0 -c ycle of degr ee 1 . Con v ersely , suppose X has a 0 -cycle of d egree 1 . By deﬁnitio n, the map H S 0 ( X ) → Z is a s plit epimorph ism. Let x ∈ H S 0 ( X )( k ) be the c orrespond ing lift of 1 . Since th e map H s A 1 0 ( X )( k ) → H S 0 ( X )( k ) is surj ecti ve, it follo ws that there exists ˜ x in H s A 1 0 ( X )( k ) lifting this element. By as- sumption , this element is not in the kernel of the induce d h omomorphism H s A 1 0 ( X )( k ) → K M W 0 ( k ) . Thus, let ¯ x be the image of ˜ x in K M W 0 ( k ) . W e kno w that K M W 0 ( k ) ∼ → GW ( k ) [Mor11, Lemma 2.10], and tha t GW ( k ) can be realiz ed as the ﬁber pro duct Z × Z / 2 W ( k ) , where the homomor phism Z → Z / 2 is just reduct ion mod 2 , while the map W ( k ) → Z / 2 is the mod 2 rank homomorphism. The kernel of W ( k ) → Z / 2 is precisely the f undamental ideal I ( k ) . By as sumption ¯ x can be written (1 , ¯ x ′ ) where ¯ x ′ is an element of W ( k ) whose image in Z / 2 is 1 . Case 1. Suppose k is not formally real, i.e., − 1 is a sum of squares in k . Under this hypothesis on k , the W itt ring W ( k ) is local with unique maximal ideal equal to the kernel of the m od 2 rank homomorph ism [EKM08, Proposit ion 31.4(2)]. In particu lar , ¯ x ′ is in v ertible. Since the oriented deg ree map is GW ( k ) -linear , after multiplicat ion by ¯ x ′− 1 , we can assume the image of the lift ˜ x of x is 1 . Case 2. S uppos e k is formally real. Cons ider the diagram H n Nis ( X, K M W n ( ω X )) g deg   / / H n Nis ( X, K M n ) deg   H 0 Nis (Sp ec k , K M W 0 ) / / H 0 Nis ( X, K M 0 ) . The Gersten resoluti on for K M W n ( ω X ) on X gi ves rise to a surjectio n M x ∈ X ( n ) GW ( κ x , Λ n m x / m 2 x ) − → H n Nis ( X, K M W n ( ω X )) . 43 REFERENCES The morphis m g deg can be lifted to a map M x ∈ X ( n ) GW ( κ x , Λ n m x / m 2 x ) − → GW ( k ) deﬁned as the sum of maps GW ( κ x , Λ n m x / m 2 x ) → GW ( k ) ; see [Fas08, C orollai re 6.4.3] for the constr uction of this homomorphism. Fixing a tri vializatio n of Λ n m x / m 2 x , the homomorph ism just mentione d can be identiﬁed with the transfer homomorphism GW ( κ x ) → GW ( k ) induce d by vie wing a symmetric bilin ear form ov er κ x as a linear map over k and compos ing with the ﬁeld trace. The map H n Nis ( X, K M W n ( ω X )) → H n Nis ( X, K M n ) is induced by the rank homomorph ism M x ∈ X ( n ) GW ( κ x , Λ n m x / m 2 x ) − → M x ∈ X ( n ) Z . If x ∈ X ( n ) , then the loca l contrib utio n at x to the degree map is the homomorphism Z → Z gi ven by multiplicat ion by [ κ x : k ] . A 0 -c ycle of deg ree 1 on X comes from an element of the form P x ∈ X ( n ) n x x where n x is zero for all b ut ﬁnitely many points in X ( n ) and the degrees of the extensio ns [ κ x : k ] are coprime. Fix such a representa tiv e of our 0 -cy cle of degree 1 and le t I be the ﬁnite set of points X ( n ) that appear in our represen tation. By deﬁnition, the re stricted map ⊕ x ∈ I Z → Z is still surjecti ve. Thus, to prov e our result, it sufﬁce s to prove that the correspondi ng map L x ∈ I GW ( κ x ) → GW ( k ) is surject iv e. Using the decomposit ion of GW ( k ) as a ﬁber product of Z and W ( k ) , since the κ x ha ve coprime deg rees, we can assume that some κ x has odd deg ree over k . No w , if k is formally real, ev ery ﬁnite ext ension L of k is simple. 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The 0-th stable A^1-homotopy sheaf and quadratic zero cycles

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