Motivically functorial coniveau spectral sequences; direct summands of cohomology of function fields

Motivically functoria l coniv eau sp ectral sequences for cohomolog y; direct sum m a nds of cohomolo g y of f unction ﬁelds M.V. Bondark o ∗ Marc h 1, 2022 Abstract The goal of this pap er is to pro ve that coniveau s pectral sequences are motivically functorial for all cohomolog y theories that can b e fac- torized through motiv es. T o this end the motif of a smo oth v ariet y o ve r a coun table ﬁeld k is decomp osed (in the sense of P ostnik o v to w ers) in to tw isted ( co)motiv es o f its poin ts; this is generalized to a rbitrary V o ev o dsky’s motives. In order to study the functor ialit y of this con- struction, w e use the formalism of w eig ht structur es (in tro duced in the previous paper). W e also dev elop this formalism (for general triangu- lated categories) further, and relate it with a new notion of nic e duality of triangula ted categories (that’s a sort of a pairi ng f or tw o distinct categories); this p iece of ho mological algebra c ould b e in teresting for itself. W e construct a certain Gersten w eigh t structure for a triangulated category of c omotives that con tains D M ef f g m as well as (co)motiv es of function ﬁelds o v er k . I t turns out that the corresp onding weight sp e c- tr al se quenc es genera lize the classical coniv eau ones (to cohomology of arbitrary motives). When a cohomological functor is represent ed b y a Y ∈ Ob j D M ef f − , the corresp onding conivea u sp ectral sequences can ∗ The author gratefully ackno wledges the supp ort from Deligne 2004 Balza n prize in mathematics. The work is also supp orted by RFBR (grants no. 08-01- 00777a and 10 -01- 00287 ) a nd INT AS (grant no. 05-10 00008-81 18). 1 b e expressed in terms of the (homotop y) t -trun cations of Y ; thi s ex- tends to motiv es the seminal coniv eau s pectral sequence computations of Blo c h and Ogus. W e also obtain that the comotif of a smo oth connected semi-lo cal sc heme is a direct summand of the comotif of its generic p oin t; co- motiv es of function ﬁelds con tain t wisted comotiv es of their residue ﬁelds (for all geometric v aluations). Hence similar results hold f or an y cohomology of (semi-lo cal) sc hemes men tioned. Con ten ts 1 Some prelimina ries on triangulated categories a nd motiv es 14 1.1 t -structures, Pos tnik ov tow ers, idemp oten t completions, and an embedding theorem of Mitc hell . . . . . . . . . . . . . . . . 14 1.2 Extending cohomological functors from a triangulated sub cat- egory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 1.3 Some deﬁnitions of V o ev o dsky: reminder . . . . . . . . . . . . 19 1.4 Some prop erties o f T ate tw ists . . . . . . . . . . . . . . . . . . 20 1.5 Pro-motiv es vs. comotiv es; the description of o ur strategy . . 22 2 W eigh t structures: reminder, truncations , w eight sp ectral sequences, and dualit y with t -structures 25 2.1 W eigh t structures: basic deﬁnitions . . . . . . . . . . . . . . . 27 2.2 Basic pro p erties of w eigh t structures . . . . . . . . . . . . . . 29 2.3 Virtual t -tr uncations of (cohomolog ical) functors . . . . . . . . 37 2.4 W eigh t sp ec tral sequences and ﬁltrations; relation with virtual t -truncations . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 2.5 Dualities of triang ulated categor ies; orthogo nal w eigh t and t - structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 2.6 Comparison of w eigh t sp ectral sequences with those coming from ( o rthogonal) t - trunc ations . . . . . . . . . . . . . . . . . 51 2.7 ’Change of w eigh t structures’; comparing w eigh t sp ec tral se- quences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 3 Categories of comotiv es (main prop erties) 57 3.1 Comotiv es: an ’axiomatic description’ . . . . . . . . . . . . . . 58 3.2 Pro-sc hemes and their comotiv es . . . . . . . . . . . . . . . . . 61 3.3 Primitiv e sc hemes: reminder . . . . . . . . . . . . . . . . . . . 63 2 3.4 Basic motivic prop erties of primitiv e sche mes . . . . . . . . . . 64 3.5 On morphisms b et ween the comotiv es of primitiv e sc hemes . . 65 3.6 The G ys in distinguished triang le for pro-sc hemes; ’Gersten’ P ostnik ov to w ers for the comotiv es of pro-sc hemes . . . . . . . 66 4 Main motivic results 68 4.1 The G ers ten w eigh t structure for D s ⊃ D M ef f g m . . . . . . . . . 70 4.2 Direct summand results for comotiv es . . . . . . . . . . . . . . 72 4.3 On cohomology of pro-sc hemes, and its direct summands . . . 73 4.4 Coniv eau sp ectral sequence s for cohomology of (co)moti v es . . 75 4.5 An extension of results of Blo c h and Og us . . . . . . . . . . . 76 4.6 Base ﬁeld c hange for coniv eau sp ectral sequences; functorialit y for an uncoun table k . . . . . . . . . . . . . . . . . . . . . . . 78 4.7 The Chow w eigh t structure fo r D . . . . . . . . . . . . . . . . 80 4.8 Comparing Chow -we ight a nd coniv eau sp ectral sequences . . . 83 4.9 Birational motiv es; constructing the G ers ten we ight structure b y gluing; ot her p ossible w eigh t structures . . . . . . . . . . . 84 5 The c onstruction of D and D ′ ; base cha nge and T ate twists 88 5.1 DG-categories and mo dules o v er them . . . . . . . . . . . . . 88 5.2 The deriv ed category of a diﬀeren tia l graded categor y . . . . . 90 5.3 The construction of D ′ and D ; the pro of of Prop osition 3.1.1 . 91 5.4 Base c hange and T ate tw ists fo r comotiv es . . . . . . . . . . . 93 5.4.1 Induction and restriction for diﬀeren tial g r a ded mo d- ules: reminder . . . . . . . . . . . . . . . . . . . . . . . 93 5.4.2 Extension and restriction of scalars f or comotiv es . . . 94 5.4.3 T ensor pro ducts a nd ’co-internal Hom’ fo r comotiv es; T ate twis ts . . . . . . . . . . . . . . . . . . . . . . . . 95 6 Supplemen ts 95 6.1 The wei ght complex functor; relation with generic motiv es . . 96 6.2 The relation of the heart of w with H I (’Brow n r epresen tabilit y’) 97 6.3 Motiv es and comotiv es with rational and torsion co eﬃcien ts . 99 6.4 Another p ossibilit y for D ; motiv es with compact support of pro-sc hemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 6.5 What happ ens if k is uncoun table . . . . . . . . . . . . . . . . 101 3 In tro duction Let k b e our p erfect base ﬁeld. W e recall tw o v ery imp o rtan t stateme nts concerning coniv eau spectral sequenc es. The ﬁrst one is the calculation o f E 2 of the coniv eau sp ec tral se- quence for cohomological theories that satisfy certain conditions; see [BOg94] and [CHK97]. It w as pro v ed by V o ev o dsky that these conditions are fulﬁlled b y an y theory H represen ted by a motivic complex C (i.e. an ob ject of D M ef f − ; see [V o e00a]); then the E 2 -terms o f the sp ectral sequence could b e calculated in terms of the (homotopy t -structure) cohomology of C . This result implies the second one: H -cohomology of a smoot h connected semi- lo cal sc heme (in the sense of §4.4 of [V o e00b], i.e. actually an aﬃne essen- tially smo oth o ne ) injects in to the cohomolog y of its generic p oin t; the la tt er statemen t w as extended to all (smo oth connected) primitiv e sc hemes b y M. W alk er. The main g oal of the presen t pap er is to construct (motivically) functorial coniv eau sp ectral sequence s con v erging to cohomology of arbitrary motives ; there should exist a description of these sp ec tral sequences (starting from E 2 ) that is similar to the description for the case of cohomology of smo oth v arieties (men tioned ab o v e). A related ob jectiv e is to clarify the nature of the injectivit y result men- tioned; it t urned our that (in the case of a coun table k ) the cohomology of a smo oth connected semi-lo cal (mor e g enerally , primitiv e) sc heme is actu- ally a direct summand of the cohomolog y of its g ene ric p oin t. Moreo v er, the (twiste d) cohomology of a residue ﬁeld of a function ﬁeld K /k (for any geometric v aluation of K ) is a direct summand of the cohomology of K . W e actually prov e more in §4.3. Our main homological algebra to ol is the theory of weig ht structur es ( in triangulated categories; w e usually denote a w eigh t structure b y w ) in tro- duced in the previous pap er [Bon10]. In this article w e dev elop it further; this part of the pap er could b e interes ting also to readers not acquain ted with motiv es (and could b e r ead indep enden tly from the rest of the pap er). In particular, w e study nic e dualities (certain pairings) o f (tw o distinct) trian- gulated categories; it seems that this sub ject was not previously considered in the literature at all. This allo ws us to generalize the concept of adjac ent w eigh t and t -structures ( t ) in a t ria ngulated category (dev elop ed in §4.4 of [Bon10]): w e in tro duce the notion of ortho g o nal structur es in (t w o p ossibly distinct) triangulated categories. If Φ is a nice dualit y of triangulated C , D , 4 X ∈ O bj C , Y ∈ O bj D , t is o rthogonal to w , then the sp ectral sequence S conv erging to Φ( X, Y ) that comes fro m the t -truncations of Y is natu- rally isomorphic (starting from E 2 ) to the weight sp e ctr al se quenc e T for the functor Φ( − , Y ) . T comes from weight trunc ations of X (note that those gen- eralize stupid truncations for complexes). Our approac h yields an abstract alternativ e t o the metho d of comparing similar sp ectral sequences using ﬁl- tered complexe s (dev elop ed by Deligne and P aranjap e, and used in [Par96] , [Deg09], and [Bon10]). Note also that w e relate t -truncations in D with vir- tual t -trunc ations of cohomological functors on C . Virtual t -truncations for cohomological functors a r e deﬁned for any ( C , w ) (w e do not need any tri- angulated ’categories of functors’ or t -structures for them here); this notion w as introduced in §2.5 of [Bon10] and is studied further in the current pap er. No w, we explain wh y we really need a certain new category o f c om o tives (con taining V o ev o dsky’s D M ef f g m ), and so the theory of adjacen t structures (i.e. ort hog onal structures in the case C = D , Φ = C ( − , − ) ) is not suﬃcien t for our purposes. It was already prov ed in [Bon10 ] that w eigh t structures pro vide a p o w erful to ol for constructing sp ectral sequenc es; they a ls o relate the cohomology of ob jects of triangulated categories with t -structures adja- cen t to them. Unfor tunately , a w eigh t structure corresp onding to coniv eau sp ec tral sequences cannot exist on D M ef f − ⊃ D M ef f g m since these categories do not con tain (a n y) motiv es for function ﬁelds ov er k (as w ell as motiv es of other sc hemes not of ﬁnite t yp e o v er k ; still cf. Remark 4.5.4(5)). Y et these motives should (co)generate the he art of this weigh t structure (since the ob jects of this heart should corepresen t co v ariant exact functors f rom the category of homotop y in v arian t shea v es with tr a ns fers to Ab ). So, we need a category that w ould contain certain homotop y limits of ob jects of D M ef f g m . W e succee d in constructing a triangulated categor y D (of c omotives ) that allow s us to reac h the ob jectiv es listed. Unfortunately , in o r der to con tro l morphisms betw een the homotop y limits men tioned w e ha v e to assume k to b e coun table. In this case there exists a large enough triangulated category D s ( D M ef f g m ⊂ D s ⊂ D ) endo w ed with a certain Ger- sten weight structur e w ; its heart is ’cogenerated’ b y comotiv es of function ﬁelds. w is (left) orthogona l to the homotop y t -structure on D M ef f − and (so) is closely connected with coniv eau sp ectral sequence s and Gersten resolutions for shea v es. Not e still: w e need k to b e coun table only in order to construct the Gersten w eight structure. So those readers who w o uld just w ant to hav e a category that contains reasonable homotopy limits of g eometric motiv es (including comotiv es of function ﬁelds and of smo oth semi-lo cal sc hemes), 5 and consider cohomology theories for this category , ma y freely ignore this restriction. Moreov er, for an arbitrary k one can still pass to a countable homotop y limit in the G ys in distinguished triangle (a s in Prop osition 3.6.1). Y et for a n uncoun table k coun table homotopy limits don’t seem to b e in- teresting; in particular, they deﬁnitely do not allow to construct a G ers ten w eigh t structure (in this case). So, w e consider a certain triangulated category D ⊃ D M ef f g m that (roughly!) ’consists of ’ (co v arian t) homological functors D M ef f g m → Ab . In particular, ob jects of D deﬁne cov ariant functors S mV ar → Ab (whereas another ’big’ motivic category D M ef f − deﬁned by V o ev o dsky is constructed from certain shea ves i.e. con trav ariant functors S mV ar → Ab ; this is also tr ue for all mo- tivic homoto p y catego ries of V o ev o dsky and Morel). Besides, D M ef f g m yields a family of (w eak) co compact cogenerators f o r D . This is why w e call ob- jects of D comotiv es. Y et note that the em b edding D M ef f g m → D is co v ariant (actually , we in v ert the arrow s in the correspo ndin g ’category of functors’ in o r der to mak e the Y oneda em b edding functor co v arian t), as w ell as the functor that sends a smo oth sc heme U (not necessarily of ﬁnite type ov er k ) to its comotif (whic h coincides with its motif if U is a smo oth v ariety ). W e also recall the Cho w we ight structure w ′ C how in tro duced in [Bon10]; the corresp onding Chow-w eight sp ectral sequences a re isomorphic to the clas- sical (i.e. Deligne’s) w eigh t sp ectral sequenc es when the latter are deﬁned. w ′ C how could b e naturally extended to a w eigh t structure w C how for D . W e alw a ys hav e a natural comparison morphism from the Cho w-w eigh t sp ectral sequenc e for ( H , X ) to the corresp onding coniv eau one; it is an isomorphism for any birational cohomology theory . W e consider the category of birational comotiv es D bir i.e. the lo calization of D b y D (1) (that con tains the cat- egory of birational geometric motive s in tro duced in [KaS02 ]; though some of the results of this unpublished preprin t are erroneous, this make s no dif- ference for the curren t pap er). It turns our that w and w C how induce the same we ight structure w ′ bir on D bir . Con v ersely , starting f rom w ′ bir one can ’glue’ (fro m slic es ) t he we ight structures induced by w and w C how on D / D ( n ) for a ll n > 0 . Moreo v er, these structures b elong to an in teresting family o f w eigh t structures indexed by a single integral parameter! It could b e in- teresting to consider other mem b ers of this f a mily . W e relate brieﬂy these observ ations with those of A. Beilinson (in [Bei98] he prop osed a ’g eometric’ c haracterization of the conjectural motivic t -structure). No w we des crib e the connection of our results with r elated results of F. De glise (see [Deg08a], [Deg08b], and [Deg09 ]). He considers a certain 6 category of pro-motiv es whose ob jects are naiv e in v erse limits of o b jects of D M ef f g m (this category is not triangulated, though it is pr o-triangulate d in a ce rtain sense). This approach allows to obtain (in a univ ersal w ay ) classical coniv eau sp ectral sequences for cohomology of motiv es of smo oth v arieties; Deglise also prov es their relation with the homotopy t -truncations for cohomology represen ted b y an ob ject of D M ef f − . Y et for cohomology theories not coming from motivic complexes, this metho d do es not seem to extend to (sp ectral sequenc es for cohomology of ) arbitrary mot ives ; motivic functorialit y is no t o b vious a ls o. Moreo ver, Deglise didn’t pro v e that the pro-motif of a (smo oth connected ) semi-lo cal sc heme is a direct summand of the pro-motif of its g ene ric p oint (though this is true, at least in the case of a coun ta ble k ). W e will tell m uc h more ab out our strategy and on the relation of our results with those of Deglise in §1.5 b elow . Note also that our metho ds are m uc h more con v enien t f or studying functorialit y (of coniv eau sp ectral sequences ) than the metho ds applied b y M. Rost in the related conte xt of cycle mo dules (see [Ros96] and §4 of [Deg08b]). The author would lik e to indicate the interd ep endencies of the parts o f this text (in o rder to simplify reading for those who are not in terested in all of it). Those readers who are not (ve ry m uc h) in terested in (coniv eau) sp ec tral seq uences, ma y a v oid most of section 2 a nd read only §§2.1 – 2.2 (Remark 2.2.2 could also b e ignored). Moreov er, in order to prov e our direct summands results (i.e. Theorem 4.2.1, Corollary 4.2.2, and Prop osition 4.3.1) one needs only a small p ortion of the theory of w eigh t structures; so a reader v ery reluctan t to study this theory may try to deriv e them from the results of §3 ’b y hand’ without reading §2 at all. Still, for motivic functoriality of coniv eau sp ectral sequenc es and ﬁltrations (see Prop osition 4.4.1 and Remark 4.4.2) one needs more of w eigh t structures. On the other hand, those readers who are more intere sted in the (general) theory of triangulated categories ma y restrict their atten tion to §§1.1– 1.2 and §2; y et note that the rest of the pap er describ es in detail an imp ortan t (and quite non-trivial) example of a w eigh t structure whic h is orthogonal to a t -structure with resp ect to a nice duality (o f triangulated categories). Moreov er, m uc h of section §4 could also b e extended to a general setting of a triangulated category satisfying prop erties similar t o those listed in Prop osition 3.1.1; y et the author c hose not to do this in o rder to mak e the pap er somewhat less abstract. No w we list the con ten ts of the pap er. More details could b e found at the b eginnings of sections. W e start §1 with the recollection of t -structures, idemp oten t completions, 7 and Postni k ov to w ers for triangulated categories. W e describ e a metho d for extending cohomological functors from a full triangulated sub category to the whole C (after H. Krause). Next we recall some results and deﬁnitions f or V o ev o dsky’s motiv es (this includes certain prop erties o f T ate twis ts for mo- tiv es and cohomological functors). Lastly , w e deﬁne pro -motiv es (follow ing Deglise) and compare them with our triangulated category D of comotiv es. This allow s to explain our strategy step b y step. §2 is dedicated to w eigh t structures . First w e remind the basics of this theory (dev elop ed in §[Bon10]). Next w e recall that a cohomological func- tor H from an (arbitrary t r ia ng ulated category) C endo w ed with a w eight structure w could b e ’truncated’ a s if it b elonged to some triangulated cat- egory of functors (from C ) that is endo wed with a t -structure; w e call the correspo ndin g pieces of H its virtual t -trunc ations . W e recall the notion of w eigh t sp ec tral seq uence (introduces in ibid.). W e prov e that the deriv ed exact couple for a weigh t sp ectral sequence could b e describ ed in terms of virtual t -t r uncations. Next w e in tro duce the deﬁnition of a (nic e) duality Φ : C op × D → A (here D is triangulated, A is ab elian), and of ortho g- onal w eigh t and t - struc tures (with resp ect to Φ ). If w is orthogonal t o t , then the virtual t - t r uncations (corresponding to w ) of functors of the type Φ( − , Y ) , Y ∈ O bj D , are exactly the functors ’represen ted via Φ ’ by the a c- tual t - trunc ations of Y (corresp onding to t ). Hence if w and t are orthogonal with resp ect to a nice dualit y , the wei ght sp ectral sequen ce con v erging to Φ( X, Y ) (for X ∈ O bj C , Y ∈ O bj D ) is naturally isomorphic (starting from E 2 ) to the one coming from t - truncations of Y . W e also men tion some a l- ternativ es and predece ssors of our results. Lastly we compare w eigh t decom- p ositions, virtual t -truncations, and w eigh t sp ectral sequences corresponding to distinct w eigh t structures (in p ossibly distinct triangulated categories). In §3 we describ e the main prop erties of D ⊃ D M ef f g m . The exact ch oice of D is not imp ortan t for most of this pap er; so w e just list the main pro perties of D (and its certain enhanc ement D ′ ) in §3.1. W e construct D using the formalism of diﬀeren tial graded mo dules in §5 later. Next w e deﬁne como- tiv es for (certain) sc hemes and ind-sc hemes of inﬁnite t yp e ov er k (we call them pro-sc hemes). W e recall the notion of primitiv e sc heme. All (smo oth) semi-lo cal pro-sc hemes are primitiv e; primitiv e sc hemes hav e all nice ’mo- tivic’ prop erties of semi-lo cal pro- sc hemes. W e prov e that there are no D - morphisms of p ositiv e degrees b et w een the comotiv es of primitiv e sche mes (and also b et w een certain T ate twis ts of those). In §3.6 we pro v e that the Gysin distinguished triangle for motiv es of smo oth v arieties (in D M ef f g m ) could 8 b e naturally extended to comotiv es of pro-sche mes. This allows to construct certain P ostnik o v tow ers for the comotiv es of pro - sc hemes; these to w ers ar e closely related with classical coniv eau sp ec tral sequences for cohomology . §4 is cen tral in this pap er. W e in tro duce a certain Gersten weight struc- tur e for a certain triangulated category D s ( D M ef f g m ⊂ D s ⊂ D ). W e pro v e that P ostnik o v tow ers constructed in §3.6 are actually weight Postnikov tow- ers with respect to w . W e deduce our (in teresting) results on direct sum- mands of the comotiv es of function ﬁelds . W e tr a ns late these results to cohomology in the obv ious w ay . Next w e pro v e that w eight sp ectral sequenc es for the cohomology of X (corresp o ndin g to the Gersten w eight structure) are naturally isomorphic (starting from E 2 ) to the classical coniv eau sp ectral sequen ces if X is the motif of a smo oth v ariety ; so w e call these sp ectral sequence coniv eau ones in the general case also. W e also prov e that the Gersten w eigh t structure w (on D s ) is orthogonal t o the homotopy t -structure t on D M ef f − (with respect to a certain Φ ). It follo ws that for an arbitra r y X ∈ O bj D s , for a cohomolog y theory represen ted b y Y ∈ O bj D M ef f − (an y choice of ) the coniv eau sp ectral sequenc e that conv erges to Φ( X , Y ) could be describ ed in terms of the t - truncations of Y (starting from E 2 ). W e a lso deﬁne coniv eau sp ectral sequences for cohomology of motives ov er uncoun ta ble base ﬁelds as the limits of the corresp onding conive au sp ectral sequenc es o v er countable p erfect subﬁelds of deﬁnition. This deﬁnition is compatible with the class ical one; so we establish motivic functoriality o f coniv eau sp ectral sequences in this case also. W e also pro v e that the Ch o w weight structur e for D M ef f g m (in tro duced in §6 of [Bon10]) could b e extended to a w eigh t structure w C how on D . The cor- respo ndin g Cho w -weight spectral sequences are isomorphic to the classical (i.e. Deligne’s) ones when the latter are deﬁned (this was pro v ed in [Bon10] and [Bo n0 9 ]). W e compare Cho w-w eigh t sp ectral sequences with coniv eau ones: w e alw ay s ha v e a comparison morphism; it is a n isomorphism fo r a bir ational cohomology theory . W e consider the category of birational como- tiv es D bir i.e. the lo calization o f D by D (1) . w and w C how induce the same w eigh t structure w ′ bir on D bir ; one almost can glue w and w C how from copies of w ′ bir (one may sa y that these we ight structures could almost b e glued from the same slices with distinct shifts). §5 is dedicated to the construction of D and the pro of of its prop erties. W e apply the f ormalism of diﬀeren tial graded categories, mo dules ov er them, and of the corr esp onding deriv ed categories. A reader not in terested in these 9 details may skip (most of ) this section. In fa ct, the a uthor is not sure that there exists only one D suitable for our purp oses; y et the c hoice of D do es not aﬀ ect cohomology of (t he comotiv es of ) pro- sc hemes and of V o ev o dsky’s motiv es. W e a ls o explain how the diﬀeren tial gra ded mo dules formalism can b e used to deﬁne base c hange (extension and restriction of scalars) for comotiv es. This allo ws to extend our results on direct summands o f the comotive s (and cohomology) of function ﬁelds to pro-sc hemes obtained from them via base c hange. W e also deﬁne tensoring of comotiv es by motiv es (in particular, this yields T a t e t wist fo r D ), as w ell as a certain coin ternal Hom (i.e. the correspo ndin g left adjoin t functor). §6 is dedicated to prop erties of comotiv es that are not (directly) related with the main results of the pap er; we a ls o mak e sev eral commen ts. W e recall the deﬁnition of the additiv e category D g en of generic motives (studied in [Deg08a]). W e prov e that the exact conserv ative weight c omplex f unc tor correspo ndin g to w (that exists by the g eneral theory of w eigh t structures) could b e mo diﬁed to an exact conserv ativ e W C : D s → K b ( D g en ) . Next we pro v e that a cofunctor H w → Ab is represen table b y a homotopy inv ariant sheaf with transfers whenev er is con v erts all pro ducts in to direct sums. W e also note that our theory could be easily extended to (co)motiv es with co eﬃcien ts in an arbitrary ring. Next we note (after B. Kahn) that reasonable mot ives of pro- sc hemes with compact supp ort do exist in D M ef f − ; this observ ation could b e used for the construction of an alternative mo del for D . Lastly w e describe whic h parts of o ur argument do not work (and whic h do w ork) in the case of an uncountable k . A caution: the notion of we ight structure is quite a general formalism fo r triangulated categories. In particular, one t r ia ng ulated category can suppo rt sev eral distinct we ight structures (note that there is a similar situation with t -structures). In fact, w e construct an example fo r suc h a situation in this pap er (certainly , muc h simpler examples exist): w e deﬁne the Gersten w eigh t structure w for D s and a Cho w w eigh t structure w C how for D . Moreov er, w e sho w in §4.9 that these w eigh t structures a r e compatible with certain we ight structures deﬁned on the lo calizations D / D ( n ) (for a ll n > 0 ). These tw o series of w eigh t structures are deﬁnitely distinct: note that w yields coniv eau sp ec tral sequen ces, whereas w C how yields Cho w-w eigh t sp ectral sequen ces, that generalize Deligne’s w eigh t sp ectral sequenc es for étale and mixed Ho dge cohomology (see [Bon10 ] and [Bon09]). Also, the w eigh t complex functor constructed in [Bon09] and [Bon10] is quite distinct fr o m the o ne considered 10 in §6.1 b elo w (ev en the targets of the functors men tioned are completely distinct). The author is deeply gra teful to prof. F. Deglise, prof. D. Héb ert, prof . B. Kahn, prof. M. Ro vinsky , pro f . A. Suslin, prof. V. V o ev o dsky , and to the referee fo r their in teresting remarks. The author gratefully ac kno wledges the suppo rt f r o m Deligne 200 4 Balzan prize in mathematics. The w ork is also suppo rted b y RFBR (grants no. 08-01- 00777a and 10- 01-00287a). Notation. F or a category C , A, B ∈ O bj C , w e denote b y C ( A, B ) the set of C -morphisms from A to B . F or categories C , D w e write D ⊂ C if D is a full sub category of C . F or additiv e C , D w e denote by AddF un( C , D ) the category of additive functors from C to D (w e will ignore set-theoretic diﬃculties here since t hey do not aﬀect our argumen ts seriously). Ab is the category o f ab elian groups. F or an additiv e B w e will denote b y B ∗ the category AddF un( B , Ab ) and b y B ∗ the category AddF un( B op , Ab ) . Note that both of these are ab elian. Besides , Y oneda’s lemma giv es full em b eddin gs of B in to B ∗ and of B op in to B ∗ (these send X ∈ O bj B to X ∗ = B ( − , X ) and to X ∗ = B ( X, − ) , respectiv ely). F or a category C , X , Y ∈ O bj C , w e sa y that X is a r etr act of Y if id X can b e factorized through Y . Note that when C is t ria ngulated or ab elian then X is a retract of Y if and only if X is its direct summand. F or an y D ⊂ C the sub catego r y D is called Kar oubi-c l o se d in C if it con tains all retracts of its ob jects in C . W e will call the smallest Karoubi-closed sub category o f C con taining D the Kar oubi-closur e o f D in C ; sometimes w e will use the same term f o r the class of ob jects of the Karo ubi-clos ure of a full sub category of C (corresp onding to some subclass of O bj C ). F or a category C w e denote by C op its opp osite category . F or an additiv e C an ob ject X ∈ O bj C is called co compact if C ( Q i ∈ I Y i , X ) = L i ∈ I C ( Y i , X ) for an y set I a nd a n y Y i ∈ O bj C suc h that the pro duct exists (here w e don’t need to demand all pro ducts to exist, though they actually will exist b elo w). F or X , Y ∈ O bj C w e will write X ⊥ Y if C ( X , Y ) = { 0 } . F or D , E ⊂ O bj C w e will write D ⊥ E if X ⊥ Y f o r all X ∈ D , Y ∈ E . F or D ⊂ C w e will denote b y D ⊥ the class { Y ∈ O bj C : X ⊥ Y ∀ X ∈ D } . Sometimes we will denote by D ⊥ the corresp onding f ull sub category of C . 11 Dually , ⊥ D is the class { Y ∈ O bj C : Y ⊥ X ∀ X ∈ D } . This con v en tion is opp osite to the one o f §9.1 of [Nee01 ]. In this pap er all complexes will b e cohomological i.e. the degree of all diﬀeren tials is +1 ; resp ec tiv ely , we will use cohomological notation for their terms. F or an additiv e category B w e denote by C ( B ) the category of (un- b ounded) complexes o v er it. K ( B ) will denote the homotopy category of complexes . If B is also ab elian, we will denote b y D ( B ) the deriv ed catego r y of B . W e will also need certain b ounded analogues of these catego ries (i.e. C b ( B ) , K b ( B ) , D − ( B ) ). C and D will usually denote some tria ngulated categories. W e will use the term ’exact functor’ for a functor of triangulated categories (i.e. for a for a functor that preserv es the structures of triangulated categories). A will usually denote some ab elian category . W e will call a co v ar ia nt ad- ditiv e functor C → A for an ab elian A homolo gic al if it conv erts distinguished triangles into long exact sequen ces; homological functors C op → A will b e called c o homolo gic al when considered as con tra v ariant functors C → A . H : C op → A will alw a ys b e additiv e; it will usually b e cohomolog ical. F or f ∈ C ( X , Y ) , X , Y ∈ O bj C , w e will call the third v ertex of (an y) distinguished triangle X f → Y → Z a cone of f . Note that diﬀeren t c hoices of cones are connected by non-unique isomorphisms, cf. IV.1.7 of [GeM03]. Besides, in C ( B ) w e ha v e canonical cones of morphisms (see section §I I I.3 of ibid.). W e will often sp ecify a distinguished triangle b y tw o of its morphisms. When dealing with triangulated categories w e (mostly) use conv en tions and auxiliary statemen ts o f [GeM03]. F or a set of ob jects C i ∈ O bj C , i ∈ I , w e will denote b y h C i i the smallest strictly full triangulated sub category con taining a ll C i ; fo r D ⊂ C w e will write h D i instead of h C : C ∈ O bj D i . W e will sa y that C i generate C if C equals h C i i . W e will sa y that C i we akly c o gener ate C if for X ∈ O bj C w e ha v e C ( X, C i [ j ]) = { 0 } ∀ i ∈ I , j ∈ Z = ⇒ X = 0 (i.e. if ⊥ { C i [ j ] } contains only zero ob jects). W e will call a partially ordered set L a (ﬁltered) pr oje ctive system if for an y x, y ∈ L there exists some maxim um i.e. a z ∈ L suc h tha t z ≥ x and z ≥ y . By abuse of notation, we will iden tify L with the fo llowing category D : O bj D = L ; D ( l ′ , l ) is empt y whenev er l ′ < l , and consists of a single morphism otherwise; the composition of morphisms is the only one p ossible. If L is a pro jectiv e system, C is some category , X : L → C is a cov ariant 12 functor, w e will denote X ( l ) for l ∈ L by X l . W e will write X = lim ← − l ∈ L X l for the limit o f this functor (if it exists); w e will call it the inv erse limit of X l . W e will denote the colimit of a con trav ariant functor Y : L → C by lim − → l ∈ L Y l and call it the direct limit. Besides, w e will sometime s call the categorical image of L with resp ect to suc h an Y an inductive system . Belo w I , L will often b e pro j ectiv e systems; w e will usually require I t o b e coun table. A subsystem L ′ of L is a partia lly o rdere d subset in whic h maxim ums exist (we will also consider the corresp onding full sub category o f L ). W e will call L ′ un b ounded in L if for any l ∈ L there exists a n l ′ ∈ L ′ suc h that l ′ ≥ l . k will b e o ur p erfect base ﬁeld. Below w e will usually demand k t o b e coun table. Note: this yields that for any v ariety the set of its closed (or op en) subsc hemes is countable . W e also list cen tral deﬁnitions and main notation of this pap er. First w e list the main (general) homological algebra deﬁnitions. t -structures, t -truncations, and P ostnik ov tow ers in triangulated categor ies a re deﬁned in §1.1; w eight structures, w eigh t decomp ositions, w eigh t truncations, weigh t P ostnik ov to wers , and weigh t complex es are consid ered in §2.1; virtual t - truncations a nd nice exact complexes of functors are deﬁned in §2.3; w eigh t sp ec tral seque nces are studied in §2.4; (nice) dualities and orthogo nal we ight and t -structures a r e deﬁned in Deﬁnition 2.5.1 ; righ t and left we ight-exac t functors are deﬁned in Deﬁnition 2.7 .1 . No w we list notation (and some deﬁnitions) for motiv es. D M ef f g m ⊂ D M ef f − , H I and the homotop y t -structure for D M ef f g m are deﬁned in §1.3; T a te twi sts are considered in §1.4; D naive is deﬁned in §1.5; comotiv es ( D and D ′ ) are deﬁned in §3.1; in §3.2 we discuss pro-sc hemes and their como- tiv es; in §3.3 w e recall the deﬁnition of a primitiv e sc heme; in §4.1 we deﬁne the Gersten w eigh t structure w on a certain triangulated D s ; w e consider w C how in §4.7; D bir and w ′ bir are deﬁned in §4.9; sev eral diﬀeren tial graded constructions (including extension and restriction of scalars fo r comotiv es) are considered in §5; w e deﬁne D g en and W C : D s → K b ( D g en ) in §6.1. 13 1 Some preliminaries on triangulated categories and motiv es §1.1 w e recall the notion of t - structure (and in tro duce some notatio n for it), recall the notion o f the idemp oten t completion of an a dditiv e category; we also recall that any small ab elian category could b e f aithfully em b edded in to Ab (a w ell-kno wn result by Mitc hell). In §1.2 w e describ e (follo wing H. K ra us e) a natural metho d for extending cohomological functors f rom a full triangulated C ′ ⊂ C to C . In §1.3 w e recall some deﬁnitions and results o f V o ev o dsky . In §1.4 w e recall the not io n of T ate t wist; we study the prop erties of T ate t wists f o r motiv es and homotop y in v arian t shea v es. In §1.5 w e deﬁne pro-motives (f ollo wing [Deg08a] and [Deg08b]). These are not necessary f or our main result; y et they allo w to explain our metho ds step b y step. W e also describ e in detail the relation of our constructions and results with those of Deglise. 1.1 t -structures, P ostnik ov to w ers, i demp oten t comple- tions, and an em b edding theo rem of Mitc hell T o ﬁx the notation w e recall the deﬁnition of a t -structure. Deﬁnition 1.1.1. A pair o f sub classes C t ≥ 0 , C t ≤ 0 ⊂ O bj C for a triangulated category C will b e said to deﬁne a t -structure t if ( C t ≥ 0 , C t ≤ 0 ) satisfy the follo wing conditions: (i) C t ≥ 0 , C t ≤ 0 are strict i.e. con tain all ob jects o f C isomorphic to their elemen ts. (ii) C t ≥ 0 ⊂ C t ≥ 0 [1] , C t ≤ 0 [1] ⊂ C t ≤ 0 . (iii) Ort hogonality . C t ≤ 0 [1] ⊥ C t ≥ 0 . (iv) t -decomp osition . F o r an y X ∈ O bj C there exists a distinguished triangle A → X → B [ − 1] → A [1] (1) suc h t hat A ∈ C t ≤ 0 , B ∈ C t ≥ 0 . W e will need some more notation for t -structures. Deﬁnition 1.1.2. 1 . A category H t whose ob jects are C t =0 = C t ≥ 0 ∩ C t ≤ 0 , H t ( X , Y ) = C ( X , Y ) for X , Y ∈ C t =0 , will b e called the he art of t . Recall 14 (cf. Theorem 1.3.6 of [BBD82]) that H t is ab elian (short exact sequences in H t come from distinguishe d triangles in C ). 2. C t ≥ l (resp. C t ≤ l ) will denote C t ≥ 0 [ − l ] ( resp. C t ≤ 0 [ − l ] ). R emark 1.1.3 . 1. The axiomatics of t -structures is self-dual: if D = C op (so O bj C = O bj D ) then one can deﬁne the (opp osite) w eigh t structure t ′ on D b y taking D t ′ ≤ 0 = C t ≥ 0 and D t ′ ≥ 0 = C t ≤ 0 ; see part (iii) of Examples 1.3.2 in [BBD82]. 2. Recall (cf. Lemma IV.4.5 in [G eM03]) that (1) deﬁnes additiv e functors C → C t ≤ 0 : X → A and C → C t ≥ 0 : X → B . W e will denote A, B by X t ≤ 0 and X t ≥ 1 , resp ectiv ely . 3. (1) will b e called the t-de c omp osition of X . If X = Y [ i ] f o r some Y ∈ O bj C , i ∈ Z , then w e will denote A b y Y t ≤ i (it b elongs to C t ≤ 0 ) and B b y Y t ≥ i +1 (it b elongs to C t ≥ 0 ), resp ectiv ely . Sometimes w e will denote Y t ≤ i [ − i ] b y t ≤ i Y ; t ≥ i +1 Y = Y t ≥ i +1 [ − i − 1] . Ob jects o f the t yp e Y t ≤ i [ j ] and Y t ≥ i [ j ] (for i, j ∈ Z ) will b e called t -trunc ations of Y . 4. W e denote by X t = i the i -th cohomology of X with resp ect to t i.e. ( Y t ≤ i ) t ≥ 0 (cf. part 10 of §IV.4 of [GeM03]). 5. The follow ing statemen ts are ob vious (and w ell-kno wn): C t ≤ 0 = ⊥ C t ≥ 1 ; C t ≥ 0 = C t ≤− 1 ⊥ . No w we recall the notion of idemp oten t completion. Deﬁnition 1.1.4. An a dditiv e category B is said to b e idemp otent c omplete if for any X ∈ O bj B and a n y idempotent p ∈ B ( X, X ) there exists a decom- p osition X = Y L Z suc h that p = i ◦ j , where i is the inclusion Y → Y L Z , j is the pro jection Y L Z → Y . Recall that an y additiv e B can b e canonically idempotent completed. Its idempo t ent completion is (b y deﬁnition) the category B ′ whose ob jects are ( X , p ) for X ∈ O bj B and p ∈ B ( X , X ) : p 2 = p ; w e deﬁne A ′ (( X , p ) , ( X ′ , p ′ )) = { f ∈ B ( X , X ′ ) : p ′ f = f p = f } . It can b e easily c hec k ed that t his category is additiv e and idemp oten t com- plete, and for any idemp oten t complete C ⊃ B w e hav e a natural full em- b edding B ′ → C . The main result of [BaS01] (Theorem 1.5) states that an idemp oten t completion of a triangulated category C has a natural triangulation (with distinguished triangles b eing all retracts of distinguished triangles of C ). 15 Belo w we will need the notion of P ostnik ov to w er in a triangulated cate- gory sev eral times ( cf. §IV2 of [G eM03])). Deﬁnition 1.1.5. Let C b e a triangulated category . 1. Let l ≤ m ∈ Z . W e will call a b ounded P ostnik o v tow er f or X ∈ O bj C the follo wing data : a sequence of C -morphisms (0 =) Y l → Y l +1 → · · · → Y m = X along with distinguished triangles Y i → Y i +1 → X i (2) for some X i ∈ O bj C ; here l ≤ i < m . 2. An un b ounded P ostnik o v tow er for X is a collection of Y i for i ∈ Z that is equipp ed (for all i ∈ Z ) with: connecting arro ws Y i → Y i +1 (for i ∈ Z ), morphisms Y i → X suc h that all the corresp onding triangles comm ute, and distinguished triangles (2). In b oth cases, w e will denote X − p [ p ] by X p ; w e will call X p the factors of our P ostnik o v to w er. R emark 1.1 .6 . 1. Comp osing (and shifting) arrows from triangles (2) for t w o subseq uen t i one can construct a complex whose terms are X p (it is easily seen tha t this is a complex indeed, cf. Prop osition 2.2.2 of [Bon10]). This observ ation will b e imp ortan t for us below when w e will consid er certain w eigh t complex functors. 2. Certainly , a b ounded Postnik o v to w er could b e easily completed to an un b ounded one. F or example, one could tak e Y i = 0 for i < l , Y i = X for i > m ; then X i = 0 if i < l or i ≥ m . Lastly , w e recall the following (w ell-kno wn) result. Prop osition 1.1.7. F or any sma l l ab elian c ate gory A ther e exists a n exact faithful functor A → Ab . Pr o of. By the F reyd-Mitc hell’s embedding theorem, an y small A could b e fully faithfully em b edded into R − mo d for some (associative unital) ring R . It remains to apply the forgetful functor R − mo d → Ab . R emark 1.1.8 . 1. W e will need this statemen t b elo w in order to assume that ob jects of A ’ha v e elemen ts’; this will considerably simplify diagra m chase . Note that w e can assume t he existence of elemen ts for a not necessarily small A in the case when a reasoning deals only with a ﬁnite n um b er of ob jects o f A at a time. 16 2. In the pro of it suﬃces to hav e a f aithful em b edding A → R − mo d ; this w eak er assertion w as also prov ed b y Mitc hell. 1.2 Extending cohom ological functors from a triangu- lated sub category W e describe a metho d for extending cohomological functors fro m a full tri- angulated C ′ ⊂ C to C ( a fter H. Krause). Note that b elo w we will apply some o f the results o f [Kra0 0 ] in t he dual f o rm. The construction requires C ′ to b e sk eletally small i.e. there should exist a subset (not just a sub class!) D ⊂ O bj C ′ suc h that an y ob ject of C ′ is isomorphic to some elemen t of D . F or simplic ity , w e will sometimes (when writing sums o v er O bj C ′ ) assume that O bj C ′ is a set itself. Since the distinction b et w een small and sk eletally small categories will not aﬀect our arg uments and results, we will ignore it in the rest of the pap er. If A is an ab elian category , then AddF un( C ′ op , A ) is a belian a ls o; com- plexes in it are exact whenev er they are exact when a ppli ed to an y ob ject of C ′ . Supp ose that A satisﬁes AB5 i.e. it is closed with resp ect to all small copro ducts, and ﬁltered direct limits of exact sequences in A are exact. Let H ′ ∈ AddF un( C ′ op , A ) b e an additiv e functor (it will usually b e co- homological). Prop osition 1.2.1. I L et A , H ′ b e ﬁxe d. 1. Ther e exists an extension of H ′ to an additive functor H : C → A . It is c oh omolo gic al whene ver H i s . The c orr es p on denc e H ′ → H deﬁnes an additive functor AddF un( C ′ op , A ) → AddF un( C op , A ) . 2. Mor e o v er, supp o s e that in C we hav e a pr oje ctive system X l , l ∈ L , e quipp e d wi th a c omp atible system of morphi s m s X → X l , such that the latter system for an y Y ∈ O bj C ′ induc es an isomorphism C ( X , Y ) ∼ = lim − → C ( X l , Y ) . Then we have H ( X ) ∼ = lim − → H ( X l ) . II L e t X ∈ O bj C b e ﬁxe d. 1. One c an ch o ose a fa mily of X l ∈ O bj C ′ and f l ∈ C ( X , X l ) such that ( f l ) induc e a surje ction L H ′ ( X l ) → H ( X ) for a n y H ′ , A , and H as in assertion I1. 2. L et F ′ f ′ → G ′ g ′ → H ′ b e a (thr e e-term ) c omplex in AddF un( C ′ op , A ) that is exa c t in the midd le; supp ose that H ′ is c ohom olo gic al. Then the c omplex 17 F f → G g → H (her e F , G, H , f , g ar e the c orr esp ondin g extensi o ns) is exact in the midd le a l s o . Pr o of. I1. F ollow ing §1.2 o f [Kra00] (and dualizing it), w e consider the ab elian category C = C ′ ∗ = AddF un( C ′ , Ab ) (this is Mo d C ′ op in the no- tation of Krause). The deﬁnition easily implies that direct limits in C are exactly direct limits of functors (computed at each ob ject of C ′ ). W e ha v e the Y oneda’s functor i ′ : C op → C that sends X ∈ O bj C to the functor X ∗ = ( Y 7→ C ( X, Y ) , Y ∈ O bj C ′ ) ; it is ob viously cohomological. W e de- note by i the restriction of i ′ to C ′ ( i is opp osite to a full em b edding). By Lemma 2.2 of [Kra 00] (applied to the category C ′ op ) w e obtain that there exists an exact functor G : C → A that preserv es all small copro ducts and satisﬁes G ◦ i = H ′ . It is constructed in the follo wing wa y: if for X ∈ O bj C w e ha v e a n exact sequen ce ( in C ) M j ∈ J X ∗ j → M l ∈ L X ∗ l → X ∗ → 0 (3) for X j , X l ∈ C ′ , then w e set G ( X ) = Coke r M j ∈ J H ′ ( X j ) → M l ∈ L H ′ ( X l ) . (4) W e deﬁne H = G ◦ i ′ ; it w as prov ed in lo c. cit. that w e obtain a w ell- deﬁned functor this wa y . As was also prov ed in lo c. cit., the corresp o nde nce H ′ 7→ H yields a functor; H is cohomological if H ′ is. 2. The pro of of lo c. cit. sho ws (and men tions) that G resp ects (small) ﬁltered inv erse limits. Now note that our a ss ertions imply: X ∗ = lim − → X ∗ l in C . I I 1. This is immediate fro m (4). 2. Note that the assertion is ob viously v alid if X ∈ O bj C ′ . W e reduce the general statemen t to this case. Applying Y oneda’s lemma to (3) is w e o btain (canonically) some mor- phisms f l : X → X l for all l ∈ L and g lj : X l → X j for all l ∈ L , j ∈ J , suc h that: for any l ∈ L almost all g lj are 0 ; f or an y j ∈ J almost all g lj is 0 ; for an y j ∈ J we ha v e P l ∈ L g lj ◦ f l = 0 . No w, b y Prop osition 1.1.7, w e ma y assume that A = Ab (see Remark 1.1.8). W e should che c k: if for a ∈ G ( X ) we hav e g ∗ ( a ) = 0 , then a = f ∗ ( b ) for some b ∈ F ( X ) . 18 Using additivi ty of C ′ and C , w e can ga t her ﬁnite sets of X l and X j in to single ob jects. Hence we can assume that a = G ( f l 0 )( c ) fo r some c ∈ G ( X l ) (= G ′ ( X l )) , l 0 ∈ L and that g ∗ ( c ) ∈ H ( g l 0 j 0 )( H ( X j 0 )) for some j 0 ∈ J , whereas g l 0 j 0 ◦ f l 0 = 0 . W e complete X l 0 → X j 0 to a distinguished triangle Y α → X l 0 g l 0 j 0 → X j 0 ; w e can assume t hat B ∈ O bj C ′ . W e obtain that f l 0 could b e presen ted as α ◦ β for some β ∈ C ( X , Y ) . Since H ′ is cohomological, w e obtain that H ( α )( g ∗ ( c )) = 0 . Since Y ∈ O bj C , the complex F ( Y ) → G ( Y ) → H ( Y ) is exact in the middle; hence G ( α )( c ) = f ∗ ( d ) for some d ∈ F ( Y ) . Then we can t a k e b = F ( β )( d ) . 1.3 Some deﬁnitions o f V o ev o dsky: rem inder W e use mu c h notation from [V o e00a]. W e recall (some of ) it here for t he con v enience o f the reader, and introduce some nota tion of our o wn. V ar ⊃ S mV ar ⊃ S mP r V ar will denote the class of all v arieties ov er k , resp. of smo oth v arieties, resp. of smo oth pro jectiv e v a rietie s. W e recall that for categories of geometric orig in (in particular, for S mC or deﬁned b elo w) the addition of ob jects is deﬁned via the dis joint union of v arieties op eration. W e deﬁne the category S mC or of smo oth corresp ondence s. O bj S mC or = S mV ar , S mC or ( X , Y ) = L U Z for a ll in tegral closed U ⊂ X × Y that are ﬁnite o v er X and do minant o v er a connected comp onen t of X ; the comp osi- tion of corresp ondences is deﬁned in the usual w ay via in tersections (y et, w e do not need to consider corresp ondences up to a n equiv alence relation). W e will write · · · → X i − 1 → X i → X i +1 → . . . , fo r X l ∈ S mV ar , for the correspo ndin g complex o v er S mC or . P r eS hv ( S mC or ) will denote the (ab elian) category of additive cofunctors S mC or → Ab ; its ob j ec ts are usually called pr eshe aves with tr a nsfers . S hv ( S mC or ) = S hv ( S mC or ) N is ⊂ P r eS hv ( S mC or ) is the a belian cat- egory of additiv e cofunctors S mC o r → Ab that are shea v es in the Nisnevic h top ology (when restricted to the category of smo oth v arieties); these shea v es are usually called she aves with tr ansfers . D − ( S hv ( S mC or )) will b e the b ounded ab o v e deriv ed category of S hv ( S mC or ) . F or Y ∈ S mV ar (more generally , for Y ∈ V ar , see §4.1 of [V o e00a]) we consider L ( Y ) = S mC or ( − , Y ) ∈ S hv ( S mC or ) . F or a b ounded complex X = ( X i ) (a s ab o v e) we will denote b y L ( X ) the complex · · · → L ( X i − 1 ) → 19 L ( X i ) → L ( X i +1 ) → · · · ∈ C b ( S hv ( S mC or )) . S ∈ S hv ( S mC or ) is called homotop y inv ariant if f o r a ny X ∈ S mV ar the pro jection A 1 × X → X gives an isomorphism S ( X ) → S ( A 1 × X ) . W e will denote the category of homotop y in v ariant shea v es (with transfers) by H I ; it is an exact ab elian sub category of S mC or b y Prop osition 3.1.13 of [V o e00a]. D M ef f − ⊂ D − ( S hv ( S mC or )) is the full sub category of complexes whose cohomology sheav es a re homotop y in v arian t; it is triangulated b y lo c. cit. W e will need the homotopy t -structure on D M ef f − : it is the restriction of the canonical t -structure on D − ( S hv ( S mC or )) to D M ef f − . Belo w (when dealing with D M ef f − ) w e will denote it b y just b y t . W e hav e H t = H I . W e recall the fo llo wing results of [V o e00a]. Prop osition 1.3.1. 1. The r e exists an exact functor R C : D − ( S hv ( S mC or )) → D M ef f − right adjo int to the em b e dding D M ef f − → D − ( S hv ( S mC or )) . 2. DM ef f − ( M g m ( Y )[ − i ] , F ) = H i ( F )( Y ) ( the i -th Nisnevich hyp e r c oho- molo gy o f F c ompute d in Y ) for any Y ∈ S mV ar . 3. Denote R C ◦ L by M g m . Then the c orr esp o n ding functor K b ( S mC o r ) → D M ef f − c ould b e d escrib e d as a c ertain lo c alization of K b ( S mC o r ) . Pr o of. See §3 of [V o e00a]. R emark 1.3.2 . 1. In [V o e00a] (Deﬁnition 2.1 .1 ) t he triangulated category D M ef f g m (of eﬀe ctive g e o m etric motives ) was deﬁned as the idempotent com- pletion of a certain lo calization of K b ( S mC o r ) . This deﬁnition is compatible with a d iﬀer ential gr ade d enhanc ement fo r D M ef f g m ; cf. §5.3 b elo w. Y et in Theorem 3.2.6 of [V o e00a] it was sho wn that D M ef f g m is isomorphic to the idempo t ent comple tion of (the categorical image) M g m ( C b ( S mC o r )) ; this description of D M ef f g m will b e suﬃcien t fo r us till §5. 2. In fact, R C could b e describ ed in terms o f so-called Suslin complexes (see lo c. cit.). W e will not need this b elo w. Instead, w e will just note that RC sends D − ( S hv ( S mC or )) t ≤ 0 to D M ef f − t ≤ 0 . 1.4 Some prop erties of T ate t wists T a te t wisting in D M ef f − ⊃ D M ef f g m is giv en b y tensoring by the ob ject Z (1) (it is often denoted just by − (1) ). T ate t wist has sev eral descriptions and nice prop erties. W e will only need a few of them; our main source is §3.2 o f [V o e00a]; a more detailed exp osition could b e f o und in [MVW06] (see §§8–9). 20 In order to calculate the tensor pro duct of X , Y ∈ O bj D M ef f − one should tak e an y preimages X ′ , Y ′ of X, Y in O bj D − ( S hv ( S mC or )) with resp ect to RC (for example, one could tak e X ′ = X , Y ′ = Y ); next one should resolv e X , Y by direct sums o f L ( Z i ) f or Z i ∈ S mV ar ; lastly o ne should tensor these resolutions using the iden tit y L ( Z ) ⊗ L ( T ) = L ( Z × T ) for Z , T ∈ S mV ar , and apply R C to the result. This tensor pro duct is compatible with the natural tensor pro duct for K b ( S mC o r ) . W e note that any ob ject D − ( S hv ( S mC or )) t ≤ 0 has a resolution concen- trated in negativ e degrees (the canonical resolution of the b eginning of §3.2 of [V o e00a]). It follow s that D M ef f − t ≤ 0 ⊗ D M ef f − t ≤ 0 ⊂ D M ef f − t ≤ 0 (see Remark 1.3.2(2); in fact, there is an equalit y since Z ∈ O bj H I ). Next, w e denote A 1 \ { 0 } b y G m . The morphisms pt → G m → pt (the p oin t is mapp ed to 1 in G m ) induce a splitting M g m ( G m ) = Z ⊕ Z (1)[1] for a certain ( T ate ) motif Z (1) ; see Deﬁnition 3.1 of [MVW06]. F or X ∈ O bj D M ef f − w e denote X ⊗ Z (1) by X (1) . One could also presen t Z (1) as Cone( pt → G m )[ − 1] ; hence the T a t e t wist functor X 7→ X (1 ) is compatible with the functor − ⊗ (Cone( pt → G m )[ − 1]) on C b ( S mC o r ) via M g m . W e also obtain that D M ef f − t ≤ 0 (1) ⊂ D M ef f − t ≤ 1 . No w we deﬁne certain twis ts for functors. Deﬁnition 1.4.1. F or an G ∈ AddF un( D M ef f g m , Ab ) , n ≥ 0 , we deﬁne G − n ( X ) = G ( X ( n )[ n ]) . Note that this deﬁnition is compatible with those of §3.1 of [V o e00b]. In- deed, f or X ∈ S mV ar w e ha v e G − 1 ( M g m ( X )) = G ( M g m ( X × G m )) /G ( M g m ( X )) = Ker( G ( M g m ( X × G m )) → G ( M g m ( X ))) (with resp ect to natural morphisms X × pt → X × G m → X × pt ); G − n for larger n could b e deﬁned by iterating − − 1 . Belo w w e will extend this deﬁnition to (co)motiv es of pro- sc hemes. F or F ∈ O bj D M ef f − w e will denote by F ∗ the functor X 7→ D M ef f − ( X , F ) : D M ef f g m → Ab . Prop osition 1.4.2. L et X ∈ S mV ar , n ≥ 0 , i ∈ Z . 1. F or any F ∈ O bj DM ef f − we have: F ∗− n ( M g m ( X )[ − i ]) is a r etr act of H i ( F )( X × G × n m ) (wh ich c an b e desc rib e d explic i tly ). 2. Ther e exists a t -exact functor T n : D M ef f − → D M ef f − such that for any F ∈ O bj D M ef f − we have F ∗− n ∼ = ( T n ( F )) ∗ . Pr o of. 1. Prop osition 1.3.1 along with our description of Z (1) yields the result. 21 2. F or F represen ted b y a complex of F i ∈ O bj S hv ( S mC or ) ( i ∈ Z ) w e deﬁne T n ( F ) as the complex of T n ( F i ) , where T n : P r eS hv ( S mC or ) → P r eS hv ( S mC or ) is deﬁned similarly to − − n in Deﬁnition 1.4.1. T n ( F i ) are shea ves since T n ( F i )( X ) , X ∈ S mV ar , is a functorial retract of F i ( X × G n m ) . In order to c hec k t hat w e actually obtain a we ll-deﬁned a t -exact functor this wa y , it suﬃces to note that the restriction of T n to S hv ( S mC or ) is an exact functor b y Prop osition 3.4.3 of [Deg08a]. No w, it suﬃces to c hec k that T n deﬁned satisﬁes the assertion for n = 1 . In this case the statemen t follows easily from Prop osition 4.34 of [V o e00b] (note that it is not imp ortan t whether w e consider Zariski o r Nisnevic h top ol- ogy b y Theorem 5.7 of ibid.). 1.5 Pro-motiv es vs. comotiv es; the description of our strategy Belo w we will em b ed D M ef f g m in to a certain triangulated category D of c omo- tives . Its construction (and computations in it) is ra t her complicated; in fact, the author is not sure whether the main prop erties of D (describ ed b elo w) sp ec ify it up to an isomorphism. So, b efore w orking with co-motiv es we will (follo wing F. Deglise) describ e a simpler category of pr o-m otives . The latter is not needed for our main results (so the reader ma y skip t his subsection); y et t he comparison of the categories men tioned w ould clarify the nat ure of our metho ds. F ollow ing §3.1 of [Deg08a], w e deﬁne the category D naive as the additiv e category of naiv e i.e. formal (ﬁltered) pro-ob jects of D M ef f g m . This means that for an y X : L → D M ef f g m , Y : J → D M ef f g m w e deﬁne D naive (lim ← − l ∈ L X l , lim ← − j ∈ J Y j ) = lim ← − j ∈ J (lim − → l ∈ L D M ef f g m ( X l , Y j )) . (5) The main disadv antage of D naive is that it is not triangulated. Still, one has the ob vious shift for it; follo wing Deglise, one can deﬁne pro- dis tinguished triangles as (ﬁltered) in v erse limits of distinguished triangles in D M ef f g m . This allo ws to construct a certain motivic conive au exact couple for a motif of a smo oth v ariety in §4.2 of [Deg08b] (see also §5.3 o f [Deg08a]). This construc- tion is parallel to the classical construction of coniv eau sp ectral sequenc es (see §1 of [CHK97]). One starts with certain ’geometric’ P ostnik ov to w ers in D M ef f g m (Deglise calls them triangulate d exact c ouples ). F or Z ∈ S mV ar w e 22 consider ﬁltrations ∅ = Z d +1 ⊂ Z d ⊂ Z d − 1 ⊂ · · · ⊂ Z 0 = Z ; Z i is ev erywhere of co dimension ≥ i in Z for a ll i . Then w e hav e a system of distinguished triangles relating M g m ( Z \ Z i ) and M g m ( Z \ Z i → Z \ Z i +1 ) ; this yields a P ost- nik ov tow er. Then one passes to the inv erse limit of these to w ers in D naive (here the connecting morphisms are induced b y the corresp onding op en em- b eddings ). Lastly , the functorial form of the Gysin distinguished t ria ngle for motiv es allo ws Deglise to iden tify X i = lim ← − ( M g m ( Z \ Z i → Z \ Z i +1 )) with the pro duct o f shifted T a te t wists of pro-motive s of a ll p oin ts of Z of co dimension i . Using the results of see §5.2 of [Deg08a] (the relation of pro- motiv es with cycle mo dules of M. Rost, see [Ros96]) one can a lso compute the morphisms that connect X i with X i +1 . Next, for any cohomological H : D M ef f g m → A , where A is an ab elian category satisfying AB5, one can extend H t o D naive via the corresp onding direct limits. Applying H to the motivic coniv eau exact couple one gets the classical coniv eau sp ectral sequence (that con v erges to the H -cohomolog y of Z ). This allow s to extend the seminal results of §6 of [BOg 9 4 ] to a compre- hensiv e description o f the coniv eau sp ec tral sequence in the case when H is represen ted b y Y ∈ O bj D M ef f − (in terms of the homoto p y t -truncations of Y ; see Theorem 6.4 of [Deg09]). No w supp ose that one w ants to apply a similar pro cedure for an arbitra ry X ∈ O bj D M ef f g m ; say , X = M g m ( Z 1 f → Z 2 ) for Z 1 , Z 2 ∈ S mV ar , f ∈ S mC or ( Z 1 , Z 2 ) . One w o uld exp ect that the desired exact couple for X could b e constructed from those for Z j , j = 1 , 2 . This is indeed the case when f satisﬁes certain co dimension restrictions ; cf. §7.4 of [Bon10]. Y et for a general f it seems to b e quite diﬃcult to relate the ﬁltrations of distinct Z j (b y the corresp o ndin g Z j i ). On the other hand, the formalism of weigh t structures and w eigh t sp ectral sequences (dev elop ed in [Bon10]) a llows to ’glue’ certain weig h t P ostnik ov to w ers for ob jects of a triangulated categories equipped with a w eigh t structure; see Remark 4.1.2(3 ) b elo w. So, we construct a certain triangulated category D that is somewhat similar to D naive . Certainly , w e w ant distinguished triangles in D to b e com- patible with inv erse limits that come fro m ’geometry’. A w ell-kno wn recip e for this is: one should consider some category D ′ where (certain) cones of morphisms are functorial and pass to (in v erse) limits in D ′ ; D should b e a lo calization of D ′ . In fact, D ′ constructed in §5.3 b elo w could b e endo w ed with a certain (Quillen) mo del structure suc h that D is its homotopy cate- gory . W e will nev er use this fact b elo w; yet we will sometimes call inv erse 23 limits coming from D ′ homotop y limits (in D ). No w, in Prop osition 4.3.1 b elo w w e will pro v e that cohomological functors H : D M ef f g m → A could b e extended to D in a w ay t hat is compatible with homotop y limits (those coming from D ′ ). So one may say that ob jects of D ha v e the same cohomology as those of D naive . O n the other hand, w e hav e to pay the price for D b eing tria ngulat ed: (5) do es not compute morphisms b et we en homotopy limits in D . The ’diﬀerence’ could b e described in terms of certain higher pro jectiv e limits (of the corresponding morphism groups in D M ef f g m ). Unfortunately , the author do es not kno w ho w to con trol the corresp onding lim ← − 2 (and higher ones) in the general case; this do es not allow t o construct a w eigh t structure on a suﬃcien tly large triangulated sub category of D if k is uncoun table (y et see §6.5, esp ecially the last paragraph of it). In the case of a coun table k o nly lim ← − 1 is non-zero. In this case t he morphisms b et w een homotop y limits in D a re expressed by the fo rm ula (28) b elo w. This allows to pro v e that there are no morphisms of p ositiv e degrees b et w een ce rtain T a te twi sts of the comotiv es of function ﬁelds (o v er k ). This immediately yields that one can construct a certain w eigh t structure o n the triangulated sub catego r y D s of D generated by pro ducts of T ate tw ists of the comotiv es of function ﬁelds (in f a ct, w e also idemp oten t complete D s ). No w, in order to pro v e that D s con tains D M ef f g m it suﬃces to prov e that the motif of an y smo oth v a riet y X b elongs to D s . T o this end it clearly suﬃces to decomp ose M g m ( X ) in to a P ostnik o v to w er whose factors are pro ducts of T ate tw ists of the comotiv es of function ﬁelds. So, we lift the motivic coniv eau exact couple (constructed in [Deg08b]) fro m D naive to D . Since cones in D ′ are compatible with in v erse limits, we can construct a tow er whose terms are the homotop y limits of the corresp onding terms of the geometric to w ers mentione d. In fact, this could b e done for an uncoun table k also; the diﬃcult y is to iden tify the analogues of X i in D . If k is countable, the homotop y limits corresp onding to our tow er are coun table also. Hence (by an easy we ll-kno wn result) the isomorphism classes of these homotop y limits could b e computed in terms of the corresp onding o b jects and morphisms in D M ef f g m . This means: it suﬃces to compute X i in D naive (as w as done in [Deg08b]); this yields the result needed. Note that w e cannot (completely) compute the D -morphisms X i → X i +1 ; y et w e kno w ho w they act o n cohomology . The most in teresting application of the results described is the follo wing one. W e prov e that there are no p ositiv e D - morphis ms b et w een (certain) T at e t wists o f the comotiv es of smo oth semi-local sc hemes (o r primitive sc h emes , 24 see b elo w); this generalizes the corresponding result for function ﬁelds. It follo ws that these twis ts b elong to the he art o f the w eigh t structure on D s men tio ne d. Therefore the comotiv es of (connected) primitiv e sc hemes are retracts of the comotiv es o f their generic p oints . Hence the same is true for the cohomolog y of the comotives mentioned and also for the corresp onding pro-motiv es. Also, the comotif of a f unction ﬁeld contains as retracts the t wisted comotive s of its residue ﬁelds (fo r all geometric v aluations); this also implies the corresp onding results for cohomology and pro-motiv es. R emark 1.5.1 . In fact, Deglise mostly considers pro-ob jects for V o ev o dsky’s D M g m and of D M ef f − ; y et the distinctions are not imp ortan t since the full em b eddin gs D M ef f g m → D M g m and D M ef f g m → D M ef f − ob viously extend to full em b eddin g of the corresp onding categories of pro- ob jects. Still, the embed- dings mentione d allo w Deglise to extend sev eral nice results for V o ev o dsky’s motiv es to pro-motiv es. 2. One of the adv a n tages of the results of Deglise is that he nev er r equires k to b e coun table. Besides, our construction of w eigh t Postnik o v to w ers men- tioned hea vily relies on the functoriality of the Gysin distinguishe d triangle for motiv es (prov ed in [Deg08b]; see also Prop osition 2.4.5 of [Deg08a]). 2 W eigh t structures: reminder, truncations, w eigh t sp ectral sequences, and dualit y with t -structures In §2.1 w e recall basic deﬁnitions of the theory of w eigh t structures (it w as dev elop ed in [Bon10 ]; the concept was also inde p enden tly introduced in [P au08]). Note here that wei ght structures (usually denoted b y w ) are natural coun terparts of t -structures. W eigh t structures yield we ight tr unca- tions; those (v astly) generalize stupid truncations in K ( B ) : in particular, they a re not canonical, y et a ny morphism o f ob jects could b e extended (non- canonically) to a morphism of their w eigh t truncations. W e recall sev eral prop erties of w eigh t structures in §2.2. W e recall virtual t -trunc ations for a (cohomological) functor H : C → A (for C endow ed with a w eigh t structure) in §2.3 (these truncations are deﬁned in terms of we ight truncations). Virtual t -truncations we re in tro duced in §2.5 of [Bon10]; they yield a w ay to presen t H (canonically) as a n extension of a cohomological functor that is p ositiv e in a certain sense b y a ’negativ e’ one (as if H b elonged to some triangulated category of functors C → A 25 endo w ed with a t -structure). W e study this notion further here, a nd prov e that virtual t -truncations for a cohomological H could b e c haracterized up to a unique isomorphism b y their prop erties (see Theorem 2.3.1(I I I4)). In order to give some c haracterization also for the ’dimension shift’ (connecting the p ositiv e and the negativ e virtual t -truncations of H ), we introduce the notion of nic e (str on gly exact) complex of functors. W e prov e that complexes of represen ta ble functors coming from distinguished triangles in C are nice, as w ell a s those complexes that could b e obtained fro m nice strongly exact complexes of functors C ′ → A for some small triangulated C ′ ⊂ C (via the extension pro cedure giv en b y Prop osition 1.2.1). In §2.4 w e consider w eigh t sp ectral sequence s ( intro duced in §§2.3–2.4 of [Bon10]). W e prov e that the deriv ed ex act couple for the w eigh t sp ectral sequenc e T ( H ) (for H : C → A ) could b e naturally des crib ed in terms of virtual t -truncations of H . So, one can express T ( H ) starting from E 2 (as w ell as the corresp onding ﬁltration of H ∗ ) in these terms also. This is an imp ortan t result, since the basic deﬁnition of T ( H ) is given in terms of weight Postnikov towers for ob jects of C , whereas the latter are not canonical. In particular, this result yields canonical functorial spectral sequenc es in classical situations (considered b y Deligne; cf. Remark 2 .4 .3 of [Bon10]; note that w e do not need rational co eﬃcien ts here). In §2.5 we in tro duce the deﬁnition a ( nic e) duality Φ : C op × D → A , and of (left) o rtho go n al w eigh t and t -structures (with resp ect to Φ ). The latter deﬁnition generalizes the notion of ad j a c en t structures intro duced in §4.4 of [Bon10] (this is the case C = D , A = Ab , Φ = C ( − , ) ). If w is orthogona l to t then the virtual t -truncations (corresp onding to w ) o f functors of the t yp e Φ( − , Y ) , Y ∈ O bj D , are exactly the functors ’represen ted via Φ ’ b y the actual t - t runcations of Y (corresp onding to t ). W e also pro v e that (nice) dualities could b e extended f rom C ′ to C (using Prop osition 1.2.1). Note here that (to the knowle dge of the author) this pap er is the ﬁrst o ne whic h considers ’pairings’ of triangulated categories. In §2.6 w e pro v e: if w and t a re orthogonal with respect to a nice dualit y , the w eigh t sp ectral seque nce con v erging to Φ( X, Y ) (for X ∈ O bj C , Y ∈ O bj D ) is naturally isomorphic (starting from E 2 ) to the one coming from t -truncations of Y . Moreo ver ev en when the dualit y is not nice, all E pq r for r ≥ 2 and t he ﬁltrations corresp onding to these spectral sequences are still canonically isomorphic. Here nicenes s of a duality (deﬁned in §2.5) is a somewh at techn ical condition (deﬁ ned in terms of nice complexes of functors). Niceness generalizes to pairings ( C × D → A ) the axiom TR3 (of 26 triangulated catego r ies: an y comm utativ e square in C could b e completed to a morphism of distinguished triangles; no te that this axiom could b e described in terms of the functor C ( − , − ) : C × C → Ab ). W e also discuss some alternativ es and predeces sors o f our metho ds and results. In §2.7 w e compare w eigh t decompo sitions, virtual t -truncations, and w eigh t sp ec tral sequence s corresp onding to distinct w eigh t structures (in p os- sibly distinct triangulated categories, connected by an exact functor). 2.1 W e igh t structures: basi c deﬁnitions W e recall the deﬁn ition of a weigh t structure (see [Bon10]; in [P au08] D. P auksztello introduced w eight structures indep end en tly and called them co- t-structures). Deﬁnition 2.1.1 (Deﬁnition o f a w eigh t structure ) . A pair of subclasses C w ≤ 0 , C w ≥ 0 ⊂ O bj C for a triangulated category C will b e said to deﬁne a w eigh t structure w for C if they satisfy the follo wing conditions: (i) C w ≥ 0 , C w ≤ 0 are additiv e and Karoubi-closed (i.e. contain all retracts of their ob jects that b elong to O bj C ). (ii) Semi-in v ariance with resp ect to translations. C w ≥ 0 ⊂ C w ≥ 0 [1] ; C w ≤ 0 [1] ⊂ C w ≤ 0 . (iii) Ort hogonality . C w ≥ 0 ⊥ C w ≤ 0 [1] . (iv) W eigh t decomp osition . F or any X ∈ O bj C there exists a distinguished triangle B [ − 1] → X → A f → B (6) suc h t hat A ∈ C w ≤ 0 , B ∈ C w ≥ 0 . A simple example of a category with a w eigh t structure is K ( B ) for any additiv e B : p ositiv e ob jects are complexes that are homotopy equiv alen t to those concen trated in p ositiv e degrees; negativ e ob jects ar e complexes that are homotopy equiv alen t to those concen trated in negativ e degrees. Here one could also consider the sub categories of complexes that are b ounded from ab o v e, b elo w, or from b oth sides. The triangle (6) will b e called a weight de c omp osition of X . A weigh t decompo sition is (almost) nev er canonical; still w e will sometimes denote an y pair ( A, B ) as in (6) by X w ≤ 0 and X w ≥ 1 . Besides , we will call ob jects of the 27 t yp e ( X [ i ]) w ≤ 0 [ j ] and ( X [ i ]) w ≥ 0 [ j ] (for i, j ∈ Z ) w e ight trunc ations o f X . A shift of the distinguished triangle (6) b y [ i ] for an y i ∈ Z , X ∈ O bj C (as wel l as any its rotatio n) will sometimes b e called a shifte d weight de c omp osition . In K ( B ) (shifted) w eigh t decompositions come from stupid truncations of complexes. W e will also need the fo llowing deﬁnitions a nd notation. Deﬁnition 2.1.2. Let X ∈ O bj C . 1. The category H w ⊂ C whose ob j ec ts are C w =0 = C w ≥ 0 ∩ C w ≤ 0 , H w ( Z , T ) = C ( Z , T ) for Z , T ∈ C w =0 , will b e called the he art of the w eigh t structure w . 2. C w ≥ l (resp. C w ≤ l , resp. C w = l ) will denote C w ≥ 0 [ − l ] (resp. C w ≤ 0 [ − l ] , resp. C w =0 [ − l ] ). 3. W e denote C w ≥ l ∩ C w ≤ i b y C [ l,i ] . 4. X w ≤ l (resp. X w ≥ l ) will denote ( X [ l ]) w ≤ 0 (resp. ( X [ l − 1]) w ≥ 1 ). 5. w ≤ i X (resp. w ≥ i X ) will denote X w ≤ i [ − i ] (resp. X w ≥ i [ − i ] ). 6. w will b e called non-de gener ate if ∩ l C w ≥ l = ∩ l C w ≤ l = { 0 } . 7. W e consider C b = ( ∪ i ∈ Z C w ≤ i ) ∩ ( ∪ i ∈ Z C w ≥ i ) and call it the class o f b ounde d ob jects of C . F or X ∈ C b w e will usually tak e w ≤ i X = 0 for i small enough, w ≥ i X = 0 for i lar g e enough. W e will also denote b y C b the corresp onding full sub category of C . 8. W e will say that ( C , w ) is b ounded if C b = C . 9. W e will call a Postnik o v to w er for X (see Deﬁnition 1.1.5) a weight Postnikov tower if all Y i are some c hoices for w ≥ 1 − i X . In this case w e will call the complex whose terms are X p (see Remark 1.1.6) a weight c omplex fo r X . W e will call a w eigh t P ostnik o v to w er for X ne gative if X ∈ C w ≤ 0 and w e choose w ≥ j X to b e 0 for all j > 0 here. 28 10. D ⊂ O bj C will b e called extension-stable if fo r an y distinguished tri- angle A → B → C in C w e hav e: A, C ∈ D = ⇒ B ∈ D . W e will also sa y that the corresp onding f ull sub category is extension- stable. 11. D ⊂ O bj C will b e called ne gative if for any i > 0 w e hav e D ⊥ D [ i ] . R emark 2.1 .3 . 1. One could also dualize our deﬁnition of a weigh t P ostnik ov to w er i.e. build a to w er from w ≤ l X instead of w ≥ l X . Our deﬁnition of a w eigh t P ostnik o v to w er is more con v enien t fo r our purposes since in §3.6 b elo w we will consider Y i = j ( Z 0 \ Z i ) instead of = j ( Z 0 \ Z i → Z 0 )[ − 1] . Y et this do es not make m uch diﬀerence ; see §1.5 of [Bon10] and Theorem 2.2.1(12) b elo w. In particular, our deﬁnition of the w eigh t complex for X coincides with Deﬁnition 2.2 .1 o f ibid. Note also, that Deﬁnition 1.5.8 of ibid (of a weigh t Pos tnik ov to w er) con tained b oth ’our’ part of the data and the dual part. 2. W eigh t P ostnik o v tow ers fo r ob jects of C are far from b eing unique; their morphisms (pro vided b y Theorem 2.2 .1(15) b elo w) are not unique also (cf. Remark 1.5.9 of [Bon10]). Y et the corresp onding w eigh t sp ectral se- quences for cohomology are unique and functorial starting from E 2 ; see The- orem 2.4.2 of ibid. and Theorem 2.4 .2 b elo w fo r more detail. In particular, all p ossible c hoices of a w eigh t complex for X ar e homotop y equiv alen t (see Theorem 3.2.2( I I) and Remark 3.1.7(3 ) in [Bon10]). 2.2 Basic prop erties of w eigh t structures No w w e list some basic prop erties of notio ns deﬁned. In the theorem b elo w w e will a ssume that C is endo w ed with a ﬁxed w eigh t structure w ev erywhere except in assertions 18 – 20 . Theorem 2.2.1. 1. The axiomatics of weight structur es is self-dual: if D = C op (so O bj C = O bj D ) then one c an deﬁne the (o pp osite) weight structur e w ′ on D by taking D w ′ ≤ 0 = C w ≥ 0 and D w ′ ≥ 0 = C w ≤ 0 . 2. W e have C w ≤ 0 = C w ≥ 1 ⊥ (7) and C w ≥ 0 = ⊥ C w ≤− 1 . (8) 29 3. F or any i ∈ Z , X ∈ O bj C we hav e a distinguishe d triangle w ≥ i +1 X → X → w ≤ i X (given by a shifte d we ight de c omp osition). 4. C w ≤ 0 , C w ≥ 0 , and C w =0 ar e extension-stable . 5. A l l C w ≤ i ar e close d with r esp e ct to arbitr ary (smal l) pr o ducts (those, which exist in C ); al l C w ≥ i and C w = i ar e additive. 6. F or any weight de c omp osition of X ∈ C w ≥ 0 (se e (6)) we have A ∈ C w =0 . 7. If A → B → C → A [1] is a distinguishe d triang l e and A, C ∈ C w =0 , then B ∼ = A ⊕ C . 8. If we have a distinguishe d triangle A → B → C for B ∈ C w =0 , C ∈ C w ≤− 1 , then A ∼ = B L C [ − 1] . 9. If X ∈ C w =0 , X [ − 1] → A f → B is a weig h t d e c om p osition (of X [ − 1] ), then B ∈ C w =0 ; B ∼ = A ⊕ X . 10. L et l ≤ m ∈ Z , X , X ′ ∈ O bj C ; let weight de c omp ositions of X [ m ] an d X ′ [ l ] b e ﬁxe d. The n any morphism g : X → X ′ c an b e c omplete d to a morphism of distinguishe d triang les w ≥ m +1 X − − − → X c − − − → w ≤ m X   y a   y g   y b w ≥ l +1 X ′ − − − → X ′ d − − − → w ≤ l X ′ (9) This c ompletion is unique if l < m . 11. Consider so me c ompletion of a c omm ut ative triangl e w ≥ m +1 X → w ≥ l +1 X → X (that is uniquely determine d by the morphisms w ≥ m +1 X → X and w ≥ l +1 X → X c o ming fr om the c orr esp onding shifte d weight de c omp osi- 30 tions; se e the p r evious assertion) to a n o ctahe dr al diagr a m: w ≤ l X [1] & & ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ [1]   X o o w ≥ l +1 X 8 8 q q q q q q q q q q q q x x q q q q q q q q q q w [ l +1 ,m ] X [1] / / w ≥ m +1 X f f ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ O O w ≤ l X [1]   X y y r r r r r r r r r r r o o w ≤ m X f f ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ [1] % % ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ w [ l +1 ,m ] X 8 8 r r r r r r r r r r [1] / / w ≥ m +1 X O O Then w [ l +1 ,m ] X ∈ C [ l +1 ,m ] ; al l the distinguishe d triangles o f this o cta- he dr o n ar e shifte d weig h t de c omp ositions. 12. F or X , X ′ ∈ O bj C , l , l ′ , m, m ′ ∈ Z , l < m , l ′ < m ′ , l > l ′ , m > m ′ , c on - sider two o ctahe dr al diagr ams: ( 1 1) and a similar one c orr esp onding to the c ommutative triangl e w ≥ m +1 X → w ≥ l +1 X → X and w ≥ m ′ +1 X ′ → w ≥ l ′ +1 X → X (i.e. we ﬁx some choic es of these dia gr am s). Then any g ∈ C ( X , X ′ ) c ould b e uniquely extende d to a morphism of these di- agr ams. The c orr esp onding morphis m h : w [ l +1 ,m ] X → w [ l ′ +1 ,m ′ ] X ′ is char acterize d uniquely by any o f the f o l lowing c onditions: (i) ther e exists a C -morphism i that makes the sq uar es w ≥ l +1 X − − − → X   y i   y g w ≥ l ′ +1 X ′ − − − → X ′ (10) and w ≥ l +1 X − − − → w [ l +1 ,m ] X   y i   y h w ≥ l ′ +1 X ′ − − − → w [ l ′ +1 ,m ′ ] X ′ (11) 31 c ommutative. (ii) ther e exists a C -morphism j that m akes the squar es X − − − → w ≤ m X   y g   y j X ′ − − − → w ≤ m ′ X ′ (12) and w [ l +1 ,m ] X − − − → w ≤ m X   y h   y j w [ l ′ +1 ,m ′ ] X ′ − − − → w ≤ m ′ X ′ (13) c ommutative. 13. F or any choic e of w ≥ i X ther e exists a w e ight Postnikov tower for X (se e Deﬁnition 2.1.2(9)). F or any w eight Po s tnikov tower we have Cone( Y i → X ) ∈ C w ≤− i ; X i ∈ C w =0 . 14. Conversely, any b ounde d Postnikov towe r (for X ) with X i ∈ C w =0 is a weight Postnikov tower for it. 15. F or X, X ′ ∈ O bj C and arbitr ary weig ht Postnikov towe rs for them, any g ∈ C ( X , X ′ ) c an b e extend e d to a m orphism of Postnikov towers (i.e. ther e exist m o rphisms Y i → Y ′ i , X i → X ′ i , such that the c orr esp onding squar es c ommute). 16. F or X , X ′ ∈ C w ≤ 0 , s upp ose that f ∈ C ( X , X ′ ) c an b e extende d to a morphism of (some of ) their ne ga tive Postnikov towers that establishe s an iso m orphism X 0 → X ′ 0 . S upp ose also that X ′ ∈ C w =0 . T hen f yields a pr oje ction of X onto X ′ (i.e. X ′ is a r etr act of X vi a f ). 17. C b is a Kar o ubi- close d triangulate d sub c ate gory of C . w induc es a n on- de gener ate weight structur e fo r it, who se he art e quals H w . 18. F or a triangulate d idemp o ten t c omplete C let D ⊂ O bj C b e ne gative. Then ther e exists a unique weight structur e w on the Kar o ubi- c losur e T of h D i in C such tha t D ⊂ T w =0 . Its he art is the Kar oubi-clos ur e of the closur e of D in C wi th r e s p e ct to ( ﬁnite) dir e ct sums. 32 19. F or the weight structur e mentione d in the pr evious assertion, T w ≤ 0 is the smal lest Kar oubi-c lose d extension-s table sub class of O bj C c ontain- ing ∪ i ≥ 0 D [ i ] ; T w ≥ 0 is the sma l lest Kar oubi-c l o se d extension-stable sub- class of O bj C c ontaining ∪ i ≤ 0 D [ i ] . 20. F or the w e ight structur e mentione d in two pr evious as s ert ions we also have T w ≤ 0 = ( ∪ i< 0 D [ i ]) ⊥ ; T w ≥ 0 = ⊥ ( ∪ i> 0 D [ i ]) . Pr o of. 1. Ob vious; cf. Remark 1.1.3 of [Bon10 ] (and Remark 1.1.2 of ibid. for more detail). 2. Thes e are part s 1 and 2 of Prop osition 1.3.3 of ibid. 3. Ob vious (since [ i ] is exact up to c hange of signs of morphis ms); cf. Remark 1.2.2 of ibid. 4. This is part 3 of Prop osition 1.3.3 of ibid. 5. Ob vious from the deﬁnition and parts 4 of lo c. cit. 6. This is part 6 of Prop osition 1.3.3 of ibid. 7. This is part 7 of lo c. cit. 8. It suﬃces to note that C ( B , C ) = { 0 } , hence the triangle splits. 9. This is part 8 of lo c. cit. 10. This is Lemma 1.5.1 of ibid. 11. The only non- trivial statemen t here is that w [ l +1 ,m ] X ∈ C [ l +1 ,m ] (it easily implies: the left hand side of the low er cap in (11) also yields a shifted weigh t decomposition). (11) yie lds distinguishe d triangles: T 1 = ( w ≥ l +1 X → w [ l +1 ,m ] X → w ≥ m +1 X [1]) a nd T 2 = ( w ≤ l X → w [ l +1 ,m ] X [1] → w ≤ m X [1]) . Hence assertion 4 yields the result. 12. By a ss ertion 10, g extends uniquely to a morphism of the follo wing distinguished triangles: from T 3 = ( w ≥ m +1 X → X → w ≤ m X ) to T ′ 3 = ( w ≥ m ′ +1 X ′ → X ′ → w ≤ m ′ X ) , and from T 4 = ( w ≥ l +1 X → X → w ≤ l X ) to T ′ 4 = ( w ≥ l ′ +1 X ′ → X ′ → w ≤ l ′ X ) ; next w e also obtain a uniq ue morphism from T 1 (as deﬁned in the pro of of the previous assertion) 33 to its analogue T ′ 1 . Putting all of this together: w e obtain unique morphisms of all of the ve rtices of our o ctahedra, whic h are compatible with all the edges of the o ctahedra except (p ossibly) those that b elong to T 2 (as deﬁned ab o v e). W e also obtain that there exists unique i and h that complete (10) and (11 ) to comm utativ e squares. No w, the morphism w ≤ l X → w [ l +1 ,m ] X could b e decomp osed in to the comp osition of morphisms belonging to T 1 and T 3 . Hence in order to veri fy that w e ha v e actually constructed a morphism of o ctahedral diagrams, it remains to v erify the comm utativit y of the squares w ≤ m X − − − → w ≤ l X   y g   y j w ≤ m ′ X ′ − − − → w ≤ l ′ X ′ (14) and (13) i.e. w e should c hec k that the tw o p ossible comp ositions of arro ws for eac h of the squares are equal. Now , assertion 10 implies: the comp ositions in question fo r (14) b oth equal t he only morphism q that mak es the square X − − − → w ≤ m X   y g   y q X ′ − − − → w ≤ l ′ X ′ comm utativ e. Similarly , the comp ositions for (13) b oth equal the only morphism r that makes the square w ≥ l +1 X − − − → w [ l +1 ,m ] X   y   y r X ′ − − − → w ≤ m ′ X ′ comm utativ e. Here w e use the part of the o ctahedral a xiom that sa ys that the square w ≥ l +1 X − − − → w [ l +1 ,m ] X   y   y X − − − → w ≤ m X is comm utativ e (as w ell as the correspo nding square for ( X ′ , l ′ , m ′ ) ). 34 Lastly , as w e hav e already noted, the condition (i) c haracterizes h uniquely; f or similar (actually , exactly dual) reasons the same is true for (ii). Since the morphism w [ l +1 ,m ] X → w [ l ′ +1 ,m ′ ] X ′ coming from the morphism o f the o ctahedra constructed satisﬁes b oth of these condi- tions, it is c haracterized by an y of them uniquely . 13. Immediate f rom part 2 of (Prop osition 1.5.6) of lo c. cit. (and also from assertion 11). 14. Immediate from Remark 1.5.9(2) of ibid. 15. Immediate from part 1 (of Remark 1.5.9 ) of lo c. cit. 16. It suﬃces to pro v e that Cone f ∈ C w ≤− 1 . Indeed, then the distin- guished triangle X f → X ′ → Cone f necessarily splits. W e complete the comm utativ e triangle X w ≤− 1 → X ′ w ≤− 1 → X 0 (= X ′ 0 ) to an o ctahedral diagram. Then w e obtain Cone f ∼ = Cone( X w ≤− 1 → X ′ w ≤− 1 )[1] ; hence Cone f ∈ C w ≤− 1 indeed. 17. This is Prop osition 1.3.6 of ibid. 18. By Theorem 4.3.2(I I1) of ibid., there exists a unique w eigh t structure on h D i suc h that D ⊂ h D i w =0 . Nex t, Prop osition 5.2.2 of ibid. yields that w can b e extended to the whole T ; along with part Theorem 4.3.2(I I2) of lo c. cit. it a lso allows to calculate T w =0 in this case. 19. Immediate from Prop osition 5.2.2 o f ibid. and the description of h H i w ≤ 0 and h H i w ≥ 0 in the pro of of Theorem 4.3.2(I I1) of ibid. 20. If X ∈ T w ≤ 0 then the orthogonality condition f or w immediately yields: Y ⊥ X for any Y ∈ ∪ i< 0 D [ i ] . Con v ersely , supp ose that for some X ∈ O bj T w e hav e Y ⊥ X for all Y ∈ ∪ i< 0 D [ i ] . Then Y ⊥ X also for all Y b elonging to the smallest extension-stable sub class of O bj C con taining ∪ i< 0 D [ i ] . Hence this is also true for all Y ∈ T w ≥ 1 (see the previous assertion). Hence (7) yields: X ∈ T w ≤ 0 . W e obtain t he ﬁrst part o f the a ss ertion. The second pa r t of the assertion is dual to the ﬁrst one (and easy from (8)). 35 R emark 2.2.2 . 1 . In the notatio n o f assertion 10, for an y a (resp. b ) suc h that the left (resp. right) hand square in (9) comm utes there exists some b (resp. some a ) that mak es (9) a morphism of distinguis hed triangles (this is just axiom TR3 of triangulated categories). Hence for l < m the left (resp. righ t) hand side of (9) c haracterizes a (resp. b ) uniquely . 2. Asse rtions 10 a nd 12 yield might y to ols for pro ving that a construction described in terms of w eigh t decomp o siti ons is functorial (in a certain sense). In particular, the pro ofs of functorialit y of w eigh t ﬁltra t io n and virtual t -truncations for cohomology (w e will cons ider these notions b elo w) in [Bon10] w ere based on assertion 1 0. No w w e explain what kind of functoriality could b e obtained us ing assertion lo c. cit. A ctually , suc h an argumen t w as already used in the pro of of assertion 12. In the notation of assertion 10 w e will sa y that a and b are compatible with g (with resp ec t to the correspo ndin g we ight decomp ositions). No w suppo se t ha t for some X ′′ ∈ O bj C , some n ≤ l , g ′ ∈ C ( X ′ , X ′′ ) , and a distinguished triangle w ≥ n +1 X ′′ → X ′ → w ≤ n X ′ w e ha v e morphisms a ′ : w ≥ l +1 X ′ → w ≥ n +1 X ′′ and b ′ : w ≤ l X ′ → w ≤ n X ′′ compatible with g ′ . Then a ′ ◦ a and b ′ ◦ b ar e compatible with g ′ ◦ g (with resp ect to the corresp onding w eigh t decomp ositions )! Moreo v er, if n < m then ( a ′ ◦ a, b ′ ◦ b ) is exactly the (unique!) pair of morphisms compatible with g ′ ◦ g . 3. In the notation of assertion 12 we will (also) sa y that h : w [ l +1 ,m ] X → w [ l ′ +1 ,m ′ ] X ′ is compatible with g . Note that h is uniquely c haracterized b y (i) (or (ii)) of lo c. cit.; hence in order to c hara cterize it uniquely it suﬃces to ﬁx g and all the row s in (10 ) and (11) (or in (12) and (13)). Besides, w e obtain that h is functorial in a certain sense (cf. the reasoning ab o v e). 4. Asse rtion 11 immediately implies: for any l < m the class o f all p ossible w ≤ l X coincides with the class of p ossible w ≤ l ( w ≤ m X ) , whereas the class of p ossible w ≥ m X coincides with those of w ≥ m ( w ≥ l X ) . Besides, assertion 11 also a llo ws to construct w eigh t P ostnik ov tow ers (cf. §1.5 of [Bon10]). Hence w [ i,i ] X is just X i [ − i ] (for an y i ∈ Z , X ∈ 36 O bj C ), and a wei ght complex for any w [ l +1 ,m ] X can b e a ss umed to b e the corresp onding stupid truncation of the we ight complex of X . 5. Asse rtions 10 and 15 will b e generalized in §2.7 b elo w to the situation when there are t w o distinc t w eigh t structures; this will also clarify the pro ofs of these statemen ts. Besid es, note that our remarks on functorialit y are also actual for this setting. Some of the pro ofs in §2.7 may also help to understand the concept of virtual t -truncations (that we will start to study just now) b etter. 2.3 Virtual t -truncations of (cohomol o gical) functors Till the end of the section C will b e endo w ed with a ﬁxed weigh t structure w ; H : C → A ( A is an ab elian category) will b e a con tra v ariant (usually , co- homological) f unc tor. W e will not consider cov ariant (homological) functors here; y et certainly , dualization is absolutely no problem. No w w e recall the results of §2.5 of [Bon10] and dev elop the theory further. Theorem 2.3.1. L et H : C → A b e a c ontr a variant functor, k ∈ Z , j > 0 . I The assignments H 1 = H k j 1 : X 7→ Im( H ( w ≤ k X ) → H ( w ≤ k + j X )) and H 2 = H k j 2 : X 7→ Im( H ( w ≥ k X ) → H ( w ≥ k + j X )) deﬁne c on tr avariant functors C → A that do not dep e nd (up to a c anonic al isomorph i s m) fr o m the choic e of weight de c omp ositions. W e h a ve natur al tr ansformations H 1 → H → H 2 . II L et k ′ ∈ Z , j ′ > 0 . Then ther e exist the fol low i n g natur al isom orphisms. 1. ( H k j 1 ) k ′ j ′ 1 ∼ = H min( k,k ′ ) , max( k + j,k ′ + j ′ ) − min( k, k ′ ) 1 . 2. ( H k j 2 ) k ′ j ′ 2 ∼ = H min( k,k ′ ) , max( k + j,k ′ + j ′ ) − min( k, k ′ ) 2 . 3. ( H k j 1 ) k ′ j ′ 2 ∼ = ( H k ′ j ′ 2 ) k j 1 ∼ = Im( H ( w [ k ,k ′ ] X ) → H ( w [ k + j,k ′ + j ′ ] X )) . Her e the last term is d e ﬁne d using the c onne ction morphism w [ k + j,k ′ + j ′ ] X → w [ k ,k ′ ] X that is c omp atible with id X in the sense of R em ark 2.2.2(3); the last isomor- phism is functorial in the s e nse de scrib e d in lo c. cit. III L et H b e c oh omolo gic al, j = 1 ; let k b e ﬁxe d. 1. H l ( l = 1 , 2 ) ar e als o c ohom olo gic al; the tr ansfo rmations H 1 → H → H 2 extend c a n onic al ly to a long exa c t se q uenc e of functors · · · → H 2 ◦ [1] → H 1 → H → H 2 → H 1 ◦ [ − 1] → . . . (15) (i.e. the se q uenc e is exact when applie d to any X ∈ O bj C ). 37 2. H 1 ∼ = H whe n ever H vanishes on C w ≥ k +1 . 3. H ∼ = H 2 whenever H vanishes on C w ≤ k . 4. L et H ′ f → H g → H ′′ b e a (thr e e-term) c omplex of functors exact in the midd l e such that: (i) H ′ , H ′′ ar e c ohomolo gic al. (ii) fo r an y X ∈ O bj C we ha v e Coker g ( X ) ∼ = Ker f ( X [ − 1]) (we do n o t ﬁx these isomorphis m s). (iii) H ′ vanishes on C w ≥ k +1 ; H ′′ vanishes on C w ≤ k . Then H ′ f → H is c a nonic al ly isomorph ic to H 1 → H ; H g → H ′′ is c an o n- ic al ly isom orphic to H → H 2 . Pr o of. I This is Prop osition 2.5.1(I II1) of [Bon10]. I I Easily follo ws f rom Theorem 2.2.1, parts 11 and 12; see Remark 2.2.2. I II1. This is Prop osition 2.5.1(I I I2) of [Bon10]. 2. If H v a nis hes on C w ≥ k +1 then for an y X w e ha v e w ≥ k +1 X = 0 ; hence H 2 v anishes. Therefore in the long exact sequence · · · → H 2 ( X [1]) → H 1 → H → H 2 ( X ) → . . . give n by assertion I I1 w e hav e H 2 ( X [1]) ∼ = 0 ∼ = H 2 ( X ) ; w e obtain H 1 ∼ = H . Con v ersely , supp ose that H 1 ∼ = H . Let X ∈ O bj C w ≥ k +1 ; w e can as- sume that w ≤ k X = 0 . Then we hav e H ( X ) ∼ = H 1 ( X ) = Im H ( w ≤ k X ) → H ( w ≤ k +1 X )) = 0 . 3. It suﬃces to apply assertion I I1 to the dual functor C op → A op ; note that the axiomatics of ab elian categories, triangulated ca t ego r ies, and we ight structures are self-dual (see Remark 1.1.3( 1 ) and Theorem 2.2 .1 ( 1 )). 4. W e should c hec k that in the diagram H ′ 1 g − − − → H 1   y h   y H ′ − − − → H g and h are isomorphisms. Then g ◦ h − 1 will yield the ﬁrst isomorphism desired, whereas dualization will yield the remaining half of the statemen t. No w, assertion I I I2 yields that g in isomorphism. Next, for an X ∈ O bj C w e c ho ose some weigh t decomp ositions for X [ k ] and X [ k + 1 ] and consider the diagram H ′′ (( w ≤ k X )[1]) − − − − → H ′ ( w ≤ k X ) l − − − − → H ( w ≤ k X ) − − − − → H ′′ ( w ≤ k X )   y a   y b H ′′ (( w ≤ k +1 X )[1]) − − − − → H ′ ( w ≤ k +1 X ) m − − − − → H ( w ≤ k +1 X ) − − − − → H ′′ ( w ≤ k +1 X ) . 38 By our assumptions, H ′′ (( w ≤ k X )[1]) ∼ = H ′′ ( w ≤ k X ) ∼ = H ′′ (( w ≤ k +1 X )[1]) ∼ = 0 ; hence l is an isomorphism and m is a monomorphism. Hence the induced map Im a → Im b is an isomorphism; so h is an isomorphism (since its appli- cation to an y X ∈ O bj C is an isomorphism). Deﬁnition 2.3.2. [virtual t -t runcations of H ] Let k , m ∈ Z . F or a (co)homological H w e will call H k 1 l , l = 1 , 2 , k ∈ Z , virtual t -trunc ations of H . W e will often denote them simply b y H l ; in this case w e will a ss ume k = 0 unless k is sp eciﬁed explicitly . W e denote the following functors C → A : H k 1 1 , H k − 1 , 1 2 , ( H m 1 2 ) k 1 1 , and X 7→ ( H 01 1 ) − 11 2 ( X [ k ]) by τ ≤ k H , τ ≥ k H , τ [ m +1 ,k ] H , and H τ = k , resp ectiv ely . Note that all of these functors are cohomological if H is. R emark 2.3.3 . 1 . Note that H o ften lies in a certain triangulated ’category of functors’ D (whose ob jects are certain cohomological functors C → A ). W e will axiomatize this b elo w b y in tro ducing the notion of duality Φ : C op × D → A : if Φ is a duality then for a ny Y ∈ O bj D w e ha v e a cohomological functor Φ( − , Y ) : C → A . It is also often the case when the virtual t -truncations deﬁned are compatible with actual t -truncations with respect to some t - structure t on D (see b elo w). Still, it is v ery amusin g that these t -truncated functors as well as their transformations corresp onding to t -decomp ositions (see Deﬁnition 1.1.1) can b e describ ed without sp ec ifying any D and Φ ! 2. Belo w w e will need an explicit description o f the connecting morphisms in (15). W e giv e it here (following the pro of of Propo si tion 2.5.1 of [Bon10]). The transformation H 1 → H (resp. H → H 2 ) for a n y k , j can b e cal- culated b y applying H to any p ossible c hoice either of X → w ≤ k X or of X → w ≤ k + j X (resp. of w ≥ k X → X or of w ≥ k + j X → X ) that comes f r o m an y p ossible c hoice the corresp onding weigh t decomposition. The transfor- mation H 2 → H 1 ◦ [ − 1] for j = 1 is given by applying H to any p ossible c hoice either of the morphism w ≤ k +1 X → w ≥ k +2 X [1] or of the morphism w ≤ k X → w ≥ k +1 X [1] that comes from an y p ossible c hoice of a w eigh t decom- p osition of X [ k ] . Here w e use the f o llo wing trivial observ ation: for A -morphisms X 1 f 1 → Y 1 and X 2 f 2 → Y 2 an y g : X 1 → X 2 (resp. h : Y 1 → Y 2 ) is compatible with a t most one morphism i : Im f 1 → Im f 2 ; if suc h an i exists, we will say that it is induced b y g (resp. by h ). Certainly , here f 1 could b e equal to id X 1 or f 2 could b e equal to id X 2 . 39 3. F or any k , j , and an y C -morphism g : X → Y the morphism H 1 ( X ) → H 1 ( Y ) (resp. H 2 ( X ) → H 2 ( Y ) ) is induced b y a n y c hoice of either of the morphism w ≤ k X → w ≤ k Y or of w ≤ k + j X → w ≤ k + j Y (resp. of the morphism w ≥ k X → w ≥ k Y or of w ≥ k + j X → w ≥ k + j Y ) that is compatible with g with respect to the corresp onding w eigh t decomp osition (in the sense of R em ark 2.2.2(2)); see the pro of o f Prop osition 2.5 .1 of [Bon10]. W e w ould lik e to extend assertion I I I4 of Theorem 2.3.1 to a statemen t on a (canonical) isomorphism of lo ng exact sequences of functors. T o this end w e need the followi ng deﬁnition. Deﬁnition 2.3.4. 1 . W e will call a sequence of functors C = · · · → H ′′ ◦ [1] [1]( h ) → H ′ f → H g → H ′′ h → H ′ ◦ [ − 1] → . . . of contra v a r ia n t functors C → Ab a str ongly exact c ompl e x if H ′ , H , H ′′ are cohomological and C ( X ) is a long exact sequenc e for an y X ∈ O bj C ; here [1]( h ) is the transformation induced b y h . 2. W e will also sa y that a strongly exact complex C is ni c e in H if the follo wing condition is fulﬁlled: F or any distinguished triangle T = A l → B m → C n → A [1] in C the natural morphism p : Ker(( H ′ ( A ) M H ( B ) M H ′′ ( C ))      f ( A ) − H ( l ) 0 0 g ( B ) − H ′′ ( m ) − H ′ ([ − 1]( n )) 0 h ( C )      − − − − − − − − − − − − − − − − − − − − − − − − − − − → ( H ( A ) M H ′′ ( B ) M H ′ ( C [ − 1]))) p → Ker(( H ′ ( A ) M H ( B )) f ( A ) ⊕− H ( l ) − − − − − − − → H ( A )) is epimorphic. (16) No w w e describe the connection of (16) with truncated realizations; our argumen ts will also somewhat clarify the meaning of this condition. Theorem 2.3.5. 1. L et C b e a str ongly ex act c omplex of functors that is nic e in H ; let H ′ f → H g → H ′′ (a ’pie c e’ of C ) satisfy the c ond itions of assertion III4 of The or em 2.3.1. Then C is c anonic al l y isomorphic to (15). 2. L et X → Y → Z b e a distinguishe d triangle in C . Then C = · · · → C ( − , X ) → C ( − , Y ) → C ( − , Z ) → . . . is a str ongly exact c omplex of func- tors C → Ab ; it is ni c e in C ( − , Y ) . 40 3. L et ther e exist a (skele tal ly) smal l ful l trian g ula te d C ′ ⊂ C such that the r estriction of a str ongly exact c ompl e x C to C ′ is nic e in H . F or D ∈ O bj C we c onsider the pr oje ctive system L ( D ) whose elements ar e ( E , i ) : E ∈ O bj C ′ , i ∈ C ( D , E ) ; we set ( E , i ) ≥ ( E ′ , i ′ ) if ( E , i ) = ( E ′ L E ′′ , i ′ ⊕ i ′′ ) for some ( E ′′ , i ′′ ) ∈ L ( D ) . Supp ose tha t for any D ∈ C and for G = H ′ and G = H we have lim − → L ( D ) (Im G ( i ) : G ( E ) → G ( D )) = G ( D ); (17) her e we also assume that these limits exist. Then C is nic e on C also. 4. L et C ′ ⊂ C b e a (skeletal ly) sm al l triangulate d sub c ate g o ry, let A satisfy AB5. L et C ′ = · · · → H ′ → H → H ′′ → . . . b e a str ongl y exact c omplex of func tors C ′ → A . W e e xtend al l its terms fr om C ′ to C by the metho d of Pr op osition 1 . 2 .1 and d enote the c omplex obtaine d by C ; we c arry on the notation for the terms an d arr ows fr om C ′ to C . T h en C i s a str ong l y exact c om plex a l s o (and its terms ar e c oho m olo gic al functors). It is nic e in H wh e never C ′ is. Pr o of. 1. It suﬃces to che c k that the isomorphism pro vided b y Theorem 2.3.1(I I I4) is compatible with the cob oundaries if (16) is fulﬁlled. W e can assume A = Ab ; see Remark 1.1.8. Then (16) transfers into: for an y ( x, y ) : x ∈ H ′ ( A ) , y ∈ H ( B ) , f ( A )( x ) = H ( l ) ( y ) there exists a z ∈ H ′′ ( C ) suc h that g ( B )( y ) = H ′′ ( z ) and H ([ − 1]( n ))( x ) = h ( C )( z ) . (1 8) W e should prov e: if the images of x ∈ H 2 ( X ) and o f y ∈ H ′′ ( X ) in H ′′ 2 ( X ) coincide, w ∈ H 1 ( X [ − 1]) a nd t = H ( X )( y ) ∈ H ′ ( X [ − 1]) a re their cob oundaries, then w and t come from some (single) u ∈ H ′ 1 ( X [ − 1]) . W e lift x to some x ′ ∈ H ( w ≥ k +1 X ) . Then (16) (if w e substitute w ≥ k +1 for A and X for B in it) implies the existence of some v ∈ H ′ (( w ≤ k X )[ − 1]) whose image in H ′ ( X [ − 1]) (resp. in H ( w ≤ k X [ − 1]) ) coincides with t (resp. with the cob oundary of x ′ ). Hence we can tak e u b eing t he image of v (in H ′ 1 ( X [ − 1]) ). 2. Since the bi-functor C ( − , − ) is (co)homological with resp ec t to b oth argumen ts, C is a strongly exact complex indeed. It remains to note: (1 6) in this case just means that an y comm utative square can b e completed to a morphism of distinguished triangles; so it follows from the corresp onding axiom (TR3) of triangulated categories. 41 3. First suppose that A = Ab (or a n y other a belian catego ry equipped with an exact faithful functor A → Ab that resp ects small direct limits; note that b elo w we will only need A = Ab ). Then w e should che c k (18). No w note: it suﬃce s to prov e that there exist A ′ , B ′ ∈ O bj C ′ , l ′ ∈ C ( A ′ , B ′ ) , α ∈ C ( A, A ′ ) , β ∈ C ( B , B ′ ) , x ′ ∈ H ′ ( A ′ ) , g ′ ∈ H ( B ′ ) such that: x = H ′ ( α )( x ′ ) , y = H ( β )( y ′ ) , l ′ ◦ α = β ◦ l , f ( A ′ )( x ′ ) = H ( l ′ )( y ′ ) . (19) Indeed, denote C ′ = Cone( l ′ ) ; denote b y γ some elemen t of C ( C , C ′ ) that completes A − − − → B   y   y A ′ − − − → B ′ to a morphism of triangles. Let z ′ ∈ H ′′ ( C ′ ) b e some elemen t satisfying the ob vious analogue of (18). Then h = H ′′ ( γ )( h ′ ) is easily seen to satisfy (18). No w w e construct A ′ , B ′ , . . . as desired. No t e that in this case the as- sumption (17) is equiv alen t to: for any t ∈ G ( D ) there exist E ∈ O bj C ′ , s ∈ G ( D ) , and r ∈ C ( D , E ) , suc h that t = G ( r )( s ) (since C ′ is addi- tiv e). So, we can c ho ose A ′ ∈ O bj C ′ , α ∈ C ( A, A ′ ) , x ′ ∈ H ′ ( A ′ ) suc h that x = H ′ ( α )( x ′ ) . W e complete q = α ⊕ l ∈ C ( A, A ′ L B ) to a distinguished triangle A → A ′ L B p = p 1 ⊕ p 2 → D . Since H ( q )(( − H ′ ( f ( A ′ )( x ′ ) , y )) = 0 , there exists an s ∈ H ( D ) suc h that H ( p )( s ) = ( − H ′ ( f ( A ′ )( x ′ ) , y ) (recall that H is cohomological on C ). So, w e hav e H ( p 2 )( s ) = y , − H ( p 1 )( s ) = f ( A ′ )( X ′ ) , p 2 ◦ l = − p 1 ◦ α . D ﬁts for B ′ if it lies in O bj C ′ . In the general case using (17) again, w e c ho ose B ′ ∈ O bj C ′ , δ ∈ C ( D , B ′ ) , g ′ ∈ H ( Y ) , suc h that s = H ( δ )( g ′ ) . Then it is easily seen that taking l ′ = − δ ◦ p 1 , β = δ ◦ p 2 , w e complete the choic e of a set of data satisfying (19). This argumen t can b e mo diﬁed to work for a general A . T o this end w e separate those par t s of the reasoning where w e used the fact that H is cohomological fr o m those where w e deal with limits; this allow s us to ’work as if A = Ab ’. W e denote Ker( H ′ ( A ) L H ( B )) → H ( A )) (with resp ect to the morphism in (16) b y S ( A, B ) , and K er ( H ′ ( A ) L H ( B ) L H ′′ ( C )) → H ( A ) L H ′′ ( B ) L H ′ ( C [ − 1]) b y T ( A, B , C ) . 42 Then we ha v e a comm utativ e diagram lim − → (Im( T ( A ′ , B ′ , C ′ ) → T ( A, B , C ))) t ′ − − − → lim − → (Im( S ( A ′ , B ′ ) → S ( A, B )))   y   y i T ( A, B , C ) t − − − → S ( A, B ) here the ﬁrst direct limit ab o v e is tak en with resp ect to morphisms of triangles ( A → B → C ) → ( A ′ → B ′ → C ′ ) for A ′ , B ′ , C ′ ∈ O bj C ′ (the o rderin g is similar to those of (17)); the second limit is tak en similarly with r esp ect to morphisms ( A → B ) → ( A ′ → B ′ ) for A ′ , B ′ ∈ O bj C ′ . Since the restriction of C to C ′ is nice in H , for all A ′ , B ′ , C ′ the morphism T ( A ′ , B ′ , C ′ ) → S ( A ′ , B ′ ) is epimorphic; hence t ′ is epimorphic. Therefore, it suﬃces to pro v e that i is epimorphic. No w let us ﬁx A ′ = A 0 and α = α 0 . W e use the notation in tro duced ab o v e; denote the preimage of Im( H ′ ( α ) : H ′ ( A ′ ) → H ′ ( A )) with resp ect to the natural morphism S ( A, B ) → H ′ ( A ) by J . Then J equals Im( H ′ ( A ′ ) × H ( D ) → S ( A, B )) . Indeed, here we can apply Prop osition 1.1.7 (see Remark 1.1.8) and then apply the r easoning ’with elemen ts’ used ab o v e. In an y A w e obtain: since Φ( D , Y ) = lim − → (Im(Φ( B ′ , Y ) → Φ( D , Y ))) , w e obtain that G = lim − → (Im( S ( A 0 , B ′ , X , Y ) → S ( A, B , X , Y ))) . Here w e use the follo wing fact (v alid in any ab elian A ): if J i ⊂ J ′ ∈ O bj A , lim − → J i = J (f o r some pro jectiv e system), u : J ′ → J is an A -epimorphism, then lim − → u ( J i ) = J . No w, passing to the limit with resp ect to ( A 0 , α 0 ) (using (17)) ﬁnishes the pro of. 4. C is a complex indeed since the extension pro cedure is functorial. By Prop osition 1.2.1(I1), all the terms o f C are cohomological on C . Also, part I I2 of lo c. cit. immediately implies that C is exact (i.e. C ( X ) is exact for any X ∈ O bj C ). Hence C is a strongly exact complex. Ob viously , if C is nice in H then C ′ also is. Con v ersely , let C ′ b e nice in H . Then Prop osition 1.2.1 (II1) implies that H ′ and H satisfy (1 7) (for all D ). Hence C is nice in H b y assertion 3. 43 2.4 W e igh t sp ectral sequences and ﬁltrations; relation with virtual t -truncations Deﬁnition 2.4.1. F o r an arbitrary ( C , w ) let H : C → A b e a cohomological functor ( A is any ab elian category). W e deﬁne W i ( H ) : C → A as X → Im( H ( w ≤ i X ) → H ( X )) . By Prop osition 2.1.2(2) of [Bon10], W i ( H )( X ) do es not depend on the c hoice of a w eigh t decomp osition for X [ i ] ; it also deﬁnes a (canonical) sub- functor of H ( X ) . No w recall that Postnik o v tow ers yield sp ectral sequences for cohomology . W e will denote H ( X [ − i ]) b y H i ( X ) (for X ∈ O bj C ). W e will also use the notation of Deﬁnition 2.3.2 . Theorem 2.4.2. L et k , m ∈ Z . I1. F or an y weight Postnikov tower for X (se e De ﬁnition 2.1.2(9)) ther e exists a sp e ctr al se q uenc e T = T ( H , X ) with E pq 1 ( T ) = H q ( X − p ) such that the map E pq 1 → E p +1 q 1 is induc e d by the morphism X − p − 1 → X − p (c oming fr om the tower). W e have T ( H , X ) = ⇒ H p + q ( X ) for any X ∈ C b . One c an c onstruct it using the fo l lowing exact c ouple: E pq 1 = H q ( X − p ) , D pq 1 = H q ( X w ≥ 1 − p ) . 2. T is (c o v a riantly) functoria l in H ; it is c ontr avaria ntly C -functorial in X starting fr om E 2 . 3. De note the step o f ﬁltr ation given by ( E l,m − l ∞ : l ≥ − k ) on H m ( X ) by F − k H m ( X ) . T hen F − k H m ( X ) = ( W k H m )( X ) . II T h e de rive d exact c ouple fo r T ( H, X ) c an b e n atur al ly c alc ulate d in terms of virtual t -trunc ations of H in the fol lowi n g way: E pq 2 ∼ = E ′ pq 2 = ( H q ) τ = − p ( X ) , D pq 2 = D ′ pq 2 = ( τ ≥ q H )( X [1 − p ]) ; the c onne cting morphisms of the c ouple (( E ′ 2 , D ′ 2 )) c om e fr om (1 5 ). III1. F − k H m ( X ) = Im(( τ ≤ k H m )( X ) → H m ( X )) (with r esp e ct to the c onne cting morphism m e ntione d in The or em 2.3.1(I)). 2. F o r an y r ≥ 2 , p, q ∈ Z ther e exists a functorial isomorph i s m E pq r ∼ = ( F p ( τ [ − p +2 − r, − p + r − 2] H ) q ) p /F p +1 ( τ [ − p +2 − r, − p + r − 2] H ) q ) p . Pr o of. I This is Theorem 2 .4.2 of [Bon10]; see also Remark 2.4.1 of ibid. for the discussion of exact couples. In fact, assertion 1 follo ws easily from w ell kno wn prop erties of P ostnik o v to w ers and of related sp ectral sequences. 44 I I Since virtual t -truncations are functorial, the exact couple ( D ′ 2 , E ′ 2 ) is functorial also. The deﬁnitions o f the deriv ed exact couple and of the virtual t - t runcations imply immediately that D pq 2 and their connecting maps a r e exactly D ′ pq 2 (and their connecting morphisms) sp eciﬁed in the assertion. It remains to compare E 2 with E ′ 2 , and also the connecting maps of exact couples starting a nd ending in E 2 with those for E ′ 2 . It suﬃces to consider p = q = 0 . Our strategy is the follo wing one. First w e construct an isomorphism E 00 2 → E ′ 00 2 ; our construction dep ends on some c hoices. Then w e prov e that the isomorphism constructed is actually natural (in particular, it do es not dep end o n the c hoices made). Lastly w e v erify that the isomorph isms of the terms of the exact couples constructed is compatible with the connectin g morphisms of these couples. Note that in this (last) part of the argumen t w e can make those c hoices (of certain w eigh t decomp ositions) tha t w e lik e. By the deﬁnition o f the deriv ed exact couple we ha v e: E 00 2 is the 0 - th cohomology of the complex ( H ( X − j )) (for any choice of the w eigh t complex ( X i ) ). E ′ 00 2 is the image of H ( k ) where k ∈ C ( w [0 , 1] X , w [ − 1 , 0] X ) is any mor- phism that is compatible with id X with resp ec t to the correspo nding w eigh t decompo sitions (see see Theorem 2.3.1(I I3) and Remark 2.2.2(3)). So, w e should compare a subfactor of H ( X 0 ) with a sub ob ject of H ( w [0 , 1] X ) . No w suppo se that w e are giv en a n o ctahedral diagram con taining a com- m utativ e triangle w [1 , 1] X → w [0 , 1] X → w [ − 1 , 1] X (see Theorem 2.2.1(11)). W e could obtain it as follows: ﬁx some w [ − 1 , 1] X ; then c ho ose certain w [0 , 1] X = w ≥ 0 ( w [ − 1 , 1] X ) and w [1 , 1] X = w ≥ 1 ( w [ − 1 , 1] X ) (see Remark 2 .2 .2 (4)). F or any p ossible completion of the comm utativ e triangle w [1 , 1] X → w [0 , 1] X → w [ − 1 , 1] X to an o ctahedral diag r a m, the remaining vertic es of the o ctahedron a re cer- tain w [ − 1 , 0] X , w [0 , 0] X = X 0 , and w [ − 1 , − 1] X = X − 1 [1] (by Theorem 2.2.1(11)). W e obtain morphisms w [0 , 1] X i → X 0 j → w [ − 1 , 0] X suc h that k = j ◦ i . Moreo v er, Im( H ( X 1 ) → H ( X 0 )) = Ker H ( i ) . Hence H ( i ) induces some monomorphism α : H ( X 0 ) / Im( H ( X 1 ) → H ( X 0 )) to H ( w [0 , 1] X ) . Besides, Ker( H ( X 0 ) → H ( X − 1 )) = Im H ( j ) ; therefore the restriction of α to α − 1 (Im H ( k )) yields an isomorphism β : E 00 2 → E ′ 00 2 . No w we v erify that the isomorphism constructed is natural. Note that it actually dep end s only on w [0 , 1] X i → X 0 and Im H ( k ) (we used the remaining data only in order to v erify that w e actually obtain an isomorphism). So, suppose that we ha v e X ′ ∈ O bj C , g ∈ C ( X , X ′ ) , a nd some c hoice of w ≥ 0 X ′ , w ≥ 1 X ′ , and w ≥ 2 X ′ . W e hav e canonical connecting 45 morphisms w ≥ 0 X ′ → w ≥ 1 X ′ → w ≥ 2 X ′ that are compatible with id X ′ with respect to the morphisms w ≥ l X ′ → X ′ ( l = 0 , 1 , 2 ) . Applying Theorem 2.2.1(11), we o btain a ch oice of w [0 , 1] X ′ i ′ → X ′ 0 . W e also ﬁx some c hoice of H ( k ′ ) (in order to do this we ﬁx some c hoice of w ≤− 1 X and o f w [ − 1 , 0] X ). Note that all o f these c hoices are necessarily compatible with some choice of the isomorphism β ′ : E 00 2 ( X ′ ) → E ′ 00 2 ( X ′ ) constructed as ab o v e (see 2 .2.2(2)). No w w e c ho ose some morphisms g l : w ≥ l X → w ≥ l X ′ , for − 1 ≤ l ≤ 2 , compatible with g (see Remark 2.2.2(2)). These choices could b e extended to some morphisms a : w [0 , 1] X → w [0 , 1] X ′ and b : X 0 → X ′ 0 (b y extending morphisms of arro ws to morphism of distinguished triangles). No w we v erify the commutativit y of t he diagram w [0 , 1] X i − − − → X 0   y a   y b w [0 , 1] X ′ i ′ − − − → X ′ 0 It follows f r o m Theorem 2.2.1(10) applied to the morphism g 0 : w ≥ 0 X → w ≥ 0 X ′ , l = 1 , m = 2 (since b oth b ◦ i and i ′ ◦ a are compatible with g 0 ). Moreo v er, Remark 2.2 .2(3) yields that H ( a ) sends H ( k ) to H ( k ′ ) . W e o bta in a comm utativ e diag r a m E 00 2 β − − − → E ′ 00 2   y   y E 00 2 ( H , X ′ ) β ′ − − − → E ′ 00 2 ( H , X ′ ) Since E 00 2 ( H , − ) and E ′ 00 2 ( H , − ) are C op -functorial (and t he vertic al arrows in the diagram a r e exactly those that yield this functorialit y; see Remark 2.3.3(3)), w e obtain the naturalit y in question. No w it remains to prov e that the isomorphisms of terms of exact cou- ples constructed ab ov e is compatible with the (tw o remaining) connecting morphisms of these couples. First consider the morphisms E 00 2 → D 10 2 . Recall (b y the deﬁnition of the derive d exact couple) that it is induced b y an y morphism w ≥ 0 X → X 0 that extends to a we ight decomp osition of w ≥ 0 X (here w e consider E 00 2 as a subfactor of H ( X 0 ) ). On the other hand, the morphism E ′ 00 2 → D ′ 10 2 = Im( H ( w ≥− 1 X ) → H ( w ≥ 0 X )) is induced b y an y p ossible c hoice of a morphism 46 w ≥ 0 X → w [0 , 1] X that yields a w eigh t decomp osition of w ≥ 0 X [1] (b y Remark 2.3.3(2); see also R em ark 2.2.2(3)). Hence it suﬃces to note that the t r ia ng le w ≥ 0 X → w [0 , 1] X i → X 0 is necessarily comm utativ e b y Remark 2.2.2. It remains consider the morphism D 1 , − 1 2 → E 00 2 . It is induced b y the morphism X 0 → w ≥ 1 X (that yields a we ight decomp osition of w ≥ 0 X ). The morphism D ′ 1 , − 1 2 (= Im( H ( w ≥ 1 X )[1]) → H ( w ≥ 2 X )[1])) → E ′ 00 2 is induced by the morphism w [0 , 1] X → w ≥ 2 X [1] . Hence it suﬃces to construct a commu - tativ e square w [0 , 1] X i − − − → X 0   y   y w ≥ 2 X [1] − − − → w ≥ 1 X [1] By applying Theorem 2.2.1( 1 1) to the comm utative triangle w ≥ 2 X → w ≥ 1 X → w ≥ 0 X w e obtain that there exists suc h a comm utativ e square with a certain i 0 instead of i . Note that (b y loc. cit.) i 0 yields a w eigh t decomp osi- tion of w [0 , 1] X . It suﬃces to v erify that w e ma y tak e i 0 for i i.e. that i 0 could b e completed to an o ctahedral diagram one of whose faces yields some c hoice of the comm utativ e triangle w [1 , 1] X → w [0 , 1] X → w [ − 1 , 1] X . W e take w [1 , 1] X = Cone i 0 [ − 1] , c ho ose some w [ − 1 , 1] X (coming from the same w ≤ 1 X as w [0 , 1] X ). By Remark 2.2.2(2) w e obtain a uniq ue commutativ e trian- gle w [1 , 1] X → w [0 , 1] X → w [ − 1 , 1] X tha t is compatible with id w ≤ 1 X respect to the corresp onding we ight decomp ositions. It remains to apply Theorem 2.2.1(11). I II W e can assume k = m = 0 . 1. In the notation of Theorem 2 .3 .1 we consider the morphism of sp ectral sequenc es M : T ( H 1 , X ) → T ( H , X ) (induced by H 1 → H ). Part I I of lo c. cit. implies: M is an isomorphism on E pq 2 for p ≥ − k and E pq 2 ( T ( H 1 , X )) = 0 otherwise. The assertion follow s immediately . 2. Similarly to the previous reasoning, w e ha v e natural isomorphisms: E pq 2 ( T ( τ [2 − r,r − 2] H , X ) ∼ = E pq 2 ( T ( H, X )) for 2 − r ≤ p ≤ r − 2 and = 0 other- wise. It easily fo llo ws that E pq ∞ ( T ( τ [2 − r,r − 2] H , X ) ∼ = E pq r ( T ( τ [ − p +2 − r, − p + r − 2] H , X ) . The result follo ws immediately . R emark 2.4.3 . 1. The dual of assertion I I is: if w e consider the alternativ e exact couple fo r our w eigh t sp ectral seque nce (see Remark 2 .1.3) then the 47 deriv ed exact couple can also b e describ ed in terms of virtual t -truncations (in a w a y that is dual in an appropriate sense to that of Theorem 2.4.2). 2. Poss ibly , at least a part of (a ss ertion I I of ) t he theorem could be pro v ed b y studying the functorialit y of the deriv ed exact couple (and applying Theorem 2.3.5( 1 )). 2.5 Dualities of triangul ated categories; orthogo na l w eigh t and t -structures Let C , D b e triangulated categories. W e study certain pairings of triangu- lated categories C op × D → A . In the fo llo wing deﬁnition w e consider a general A , y et b elo w we will mainly need A = Ab . Deﬁnition 2.5.1. 1 . W e will call a (cov ariant) bi-functor Φ : C op × D → A a duality if it is bi-additive , homological with resp ec t to b oth argumen ts; and is equipped with a (bi)natural bi-additiv e transformation Φ( X , Y ) ∼ = Φ( X [1] , Y [1]) . 2. W e will say that Φ is nic e if for an y distinguished triangle X → Y → Z the corresp onding (strongly exact) complex of functors · · · → Φ( − , X ) → Φ( − , Y ) → Φ( − , Z ) f → Φ([ − 1]( − ) , X ) → . . . (20) is nice in Φ( − , Y ) (see Deﬁnition 2.3.4); here f is obtained from the natural morphism Φ( − , Z ) → Φ( − , X [1]) by applying the (bi)natural transformation men tio ne d ab ov e. 3. Supp ose that C is endow ed with a w eigh t structure w , D is endo w ed with a t -structure t . Then w e will sa y that w is (left) ortho gona l to t with respect to Φ if the f o llo wing ortho gonal i ty c ondition is fulﬁlled: Φ( X, Y ) = 0 if: X ∈ C w ≤ 0 and Y ∈ D t ≥ 1 , or X ∈ C w ≥ 0 and Y ∈ D t ≤− 1 . (21) 4. If w is deﬁned o n C op , t is deﬁned on D op , w is left or t hog onal to t (with resp ect to some dualit y); then w e will sa y that the corresp onding opp osite w eigh t structure on C is right ortho gonal to the opp osite t -structure for D . R emark 2.5.2 . 1. The axioms of Φ immediately imply that (2 0) is a strongly exact complex of functors indeed (whether Φ is nice or not). 2. Certainly , if Φ is nice then (20) is nice at an y term (sin ce w e can ’rotate’ distinguished triangles in D ). 48 First we prov e a statemen t that will simplify ch ec king the orthogonality of w eigh t and t -structures. Prop osition 2.5.3. L e t Φ : C op × D → A b e som e duality; let ( C , w ) b e b ounde d. Then w is (left) ortho gonal to t whe never ther e exists a D ⊂ C w =0 such that any obje ct of C w =0 is a r etr act of a ﬁnite dir e ct sum of elements of D and Φ( X, Y ) = 0 ∀ X ∈ D , Y ∈ D t ≥ 1 ∪ D t ≤− 1 . (22) Pr o of. If w is is left o rthogonal to t , then (22 ) for D = C w =0 follo ws imme- diately from the orthogonality condition. Con v ersely , let D satisfy the assumptions of our assertion. Hence w e hav e: Φ( X, Y ) = 0 if X ∈ D [ i ] , i ≥ 0 , Y ∈ D t ≥ 1 , or if X ∈ D [ i ] , i ≤ 0 , Y ∈ D t ≤− 1 . No w w e note: if for some E , F ⊂ O bj C w e hav e Φ( X , Y ) = 0 if X ∈ E and Y ∈ D t ≥ 1 (resp. X ∈ F and Y ∈ D t ≤− 1 ), then w e also ha v e Φ( X , Y ) = 0 if X ∈ E ′ and Y ∈ D t ≥ 1 (resp. X ∈ F ′ and Y ∈ D t ≤− 1 ) where E ′ (resp. F ′ ) is the smallest Karoubi-closed extension-stable sub class of O bj C con taining E (resp. F ). No w b y Theorem 2.2.1 ( 1 9), f o r E = ∪ i ≥ 0 D [ i ] , F = ∪ i ≤ 0 D [ i ] , we ha v e E ′ = C w ≤ 0 , F ′ = C w ≥ 0 . Hence w e obtain the orthogo na lity desire d. When (we ight and t - ) structures are orthog o nal, virtual t -truncations of Φ( − , Y ) are giv en b y t -truncations in D . W e use the notation o f Deﬁnition 2.3.2. Prop osition 2.5.4. 1. L e t t b e ortho gona l to w with r esp e ct to Φ , k ∈ Z . F or Y ∈ O bj D denote the functor Φ( − , Y ) : C → A b y H . Then we have an isomorphism of c omplexes ( τ ≤ k H → H → τ ≥ k +1 H ) ∼ = (Φ( − , t ≤ k Y ) → H → Φ( − , t ≥ k +1 Y )) (wh er e the c onne cting maps of the se c ond c o mplex ar e in duc e d by t -trunc ations ); this isomorph ism is natur al in Y . 2. S upp os e also that Φ is nic e. T h en the (str ong ly exact) c om plex of functors that sends X to · · · → Φ( X, t ≤ k Y ) → Φ( X , Y ) → Φ( X , t ≥ k +1 Y ) → Φ( X [ − 1] , t ≤ k Y ) → . . . (23) (c onstructe d as in the deﬁnition of a nic e duality) is natur al ly is o morphic to (15). 49 Pr o of. 1. Since t a nd w orthogonal, Φ( − , t ≤ k Y ) v anishes on C w ≥ k +1 , whereas Φ( − , t ≥ k +1 Y ) v anishes on C w ≤ k . Moreo v er, (23) yields that H ′ = Φ( − , t ≤ k Y ) and H ′′ = Φ( − , t ≥ k +1 Y ) a ls o satisfy the condition (iii) of Theorem 2.3 .1 (II I4). Hence the theorem yields the claim. 2. Immediate from the previous assertion and Theorem 2.3.5(1). R emark 2.5 .5 . Note tha t w e actually need quite a partial case of the ’niceness condition’ fo r Φ in order to pro v e assertion 2. Hence here (and so, in all the applications b elo w) we will not need the niceness condition in its full generalit y . P ossibly , the corresp onding partial case of the condition is w eak er than the whole assertion; y et c hec king it do es not seem to b e m uc h easier. Also, it seems quite p ossible that for an arbitrary (not necess arily nice) dualit y there exists some isomorphism o f (15) with (23 ) if we mo dify the b oundary maps of the se cond complex. Y et there seems to b e no wa y t o c ho ose suc h a mo diﬁcation canonically . ’Natural’ dualities are nice; w e will justify this thesis now . Prop osition 2.5.6. 1. If A = Ab , D = C , then Φ : ( X , Y ) 7→ C ( X , Y ) i s a nic e d uali ty. 2. F or some d uali ty Φ : C op × D → A let ther e exi s t a (skeletal ly) smal l ful l triangulate d C ′ ⊂ C such that: the r estriction of Φ to C ′ op × D is a nic e d uality (of C ′ with D ); for any X ∈ O bj D the func tor G = Φ( − , X ) , C op → A , satisﬁes (17). Then Φ is nic e also. 3. F or D , C ′ ⊂ C as ab ove, A satisfying AB5, let Φ ′ : C ′ op × D → A b e a duality. F or an y Y ∈ O bj D we extend the functor Φ ′ ( − , Y ) fr om C ′ to C by the m e tho d of Pr op os ition 1.2.1; we denote the functor obtaine d by Φ( − , Y ) . Then the c orr esp ondin g bi-functor Φ is a duality ( C op × D → A ). It is nic e whenever Φ ′ is. Pr o of. Immediate from parts 2– 4 of Theorem 2.3.5. R emark 2.5.7 . 1. Prop osition 2 .5.6(1) yields an imp ortan t family of nice dualities; this case w as thoroughly studied in [Bon10] (in sections 4 a nd 7 ) . W e will sa y that w is left (resp. righ t) adja c en t to t if it is left (resp. righ t) orthogonal to it with respect to Φ( X , Y ) = C ( X , Y ) . Note that for w left (resp. righ t) adjacen t to t with resp ec t to this deﬁnition w e necessarily ha v e 50 C w ≤ 0 = C t ≤ 0 (resp. C w ≥ 0 = C t ≥ 0 ) b y Theorem 2.2.1(2) a nd Remark 1 .1.3(2); so this deﬁnition is actually compatible with Deﬁnition 4.4.1 of [Bon10]. One can generalize this fa mily as in §8.3 of ibid.: f or A = Ab and an exact F : D → C w e deﬁne Φ( X , Y ) = C ( X , F ( Y )) . Certainly , one could also dualize this construction (in a certain sense) and consider F : C → D and Φ( X , Y ) = C ( F ( X ) , Y ) . 2. Another (general) family o f dualities is men tioned in Remark 6.4.1(2) of ibid. All the families of dualities men tioned can b e expanded using part 3 of the prop osition. 3. It is also easy to construct a dualit y that is not nice. T o this end one can start with C = D , Φ = C ( − , − ) and then mo dify the c hoice of distinguished triangles in D (without c hanging the shift in D , and c hanging nothing in C ) in a w ay that would not aﬀect the prop erties of functors to b e cohomological. The simplest w ay to do this is to pro claim a triangle X f → Y g → Z h → X [1 ] to b e distinguished in D if X − f → Y − g → Z − h → X [1] is distinguished in C . Certainly , suc h a mo diﬁcation is not ve ry ’serious’; in par ticular, one can ’ﬁx the problem’ b y mu ltiplying the isomorphism Φ( X , Y ) ∼ = Φ( X [1] , Y [1]) b y − 1 . The author does not kno w whether an y dualit y can b e made nice b y mo difying the c hoice of the class of distingui shed triangles (in D ), or by mo difying the isomorphism mentione d. Note also that the question whether there exists a D for whic h suc h a mo diﬁcation can change the ’equiv alence class’ of triangulations is w ell-kno wn to b e op en. 2.6 Comparison of w eigh t sp ectr al sequences wi th those coming from (orthogo nal) t -truncations No w w e describe the relation of w eigh t sp ectral sequences with orthogonal structures. Theorem 2.6.1. L et w for C and t for D b e o rtho gon al with r esp e c t to a duality Φ ; let i, j ∈ Z , X ∈ O bj C , Y ∈ O bj D . 1. Consider the sp e ctr al se quenc e S c oming fr om the fol lowing exact c ou- ple: D pq 2 ( S ) = Φ( X , Y t ≥ q [ p − 1 ]) , E pq 2 ( S ) = Φ( X , Y t = q [ p ]) (we start S fr om E 2 ). It natur al ly c onver ges to Φ( X, Y [ p + q ]) if X ∈ C b . 51 2. L et T b e the weight sp e ctr a l se quenc e given by T he or em 2.4.2 for the functor H : Z 7→ Φ( Z , Y ) . Then for al l r ≥ 2 we have natu- r al isomorphisms E pq r ( T ( H, X )) ∼ = E pq r ( S ) . Ther e is als o an e quality F − k H m ( X ) = Im(Φ( X, t ≤ k Y [ m ]) → H m ( X )) (her e we use the nota- tion of p art I4 of lo c. cit.) c omp atible with this isomorphi sm. 3. Supp ose that Φ is also nic e. Then the is omorphism mentione d in the pr evious assertion extends natur al ly to the isomorp h ism of of T with S (starting fr om E 2 ). 4. L et · · · → X − j − 1 → X − j → X 1 − j → . . . denote a n arbitr ary c h oic e of the weig h t c o m plex for X . Then we hav e a functorial is o morphism Φ( X, Y t = i [ j ]) ∼ = (Ker(Φ( X − j , Y [ i ]) → Φ( X − 1 − j , Y [ i ])) / Im (Φ( X 1 − j , Y [ i ]) → Φ( X − j , Y [ i ])) . (24) Pr o of. 1. The theory of t - struc tures easily yields: Y t ≥ q and Y t = q can b e functorially organized in to a certain P ostnik o v tow er for Y . Hence the usual results on sp ectral sequences coming from P ostnik ov to w ers (see §IV2, Exercise 2, of [GeM03]) yield the assertion easily . 2. Immed iate from Prop osition 2.5.4(1) and Theorem 2.4.2(I I I). Note that the latter assertion do es not use the ’dimension shift’ in (1 5 ). 3. Proposition 2.5.4(2) and Theorem 2.4.2(I I) imply: there is a natural isomorphism of the deriv ed exact couple for T with the exact couple of S (’at lev el 2’). The result follows immediately . 4. This is just assertion 2 for E 2 -terms. R emark 2.6.2 . 1 . So, w e justiﬁed parts 4 and 5 of R emark 4.4.3 of [Bon10]. 2. Note that the sp ectral sequence denoted b y S in (Remark 4.4.3(4) and §6.4 of ) ibid. started fro m E 1 ; so it diﬀered from our S and T by a certain shift of indices. 52 3. So, we dev elop ed an ’abstract triangulated alternativ e’ to the metho d of comparing similar spectral sequences that w as dev elop ed b y Deligne and P aranjap e. The latter method used ﬁltered complexes; it w as applied in [P ar96], [Deg09], and in §6.4 of [Bon10]. The disadv an tage of this approach is that one needs extra information in order to construct the correspo ndin g ﬁltered complexes; this mak es diﬃcult to study the naturalit y of the isomorphism constructed. Moreo v er, in some cases the complexes required cannot exist a t all; this is the case for the spheric al weight structur e and its adjacen t P ostnik ov t -structure for C = D = S H (the top ological stable homotopy category; see §4.6 of [Bon10]; y et in this case one can compare the corresponding sp ec tral sequenc es using top ology). 4. One could mo dify our reasoning to pro v e a v ersion of the theorem that do es not men tion we ight a nd t -structures. T o this end instead of considering a w eigh t P ostnik o v tow er for X and the P ostnik o v tow er coming fro m t -tr uncations of Y one should just compare sp ec tral se- quences coming from some Postnik o v tow ers for X and Y in the case when these P ostnik ov to wers satisfy those ’o r tho g onalit y’ conditions (with resp ect to a (nice) dualit y Φ ) that are implied b y the orthogo - nalit y of structures condition in our situation. Y et it seems diﬃcult to obtain the naturalit y of the isomorphisms in Theorem 2.6.1(3) using this approach. 5. Ev en more generally , it suﬃces t o ha v e an inductiv e system of P ostnik o v to w ers in D and a pro jectiv e system of P ostnik ov to w ers in C suc h that the orthogonalit y conditions required are satisﬁed in the (double) limit. Then the comparison statemen ts for the double limits of the correspo ndin g spectral seq uences are v alid also. A v ery partial (y et rather imp ortan t) example of a reasoning of this sort is describ ed in §7.4 of [Bon10]. Besides, this approac h could p ossibly yield the comparison result of §6 of [Deg09] (ev en without assuming k to b e coun table as w e do here). 6. A simple (y et imp ortan t) case of (24) is: fo r an y i ∈ Z X ∈ C w = i = ⇒ ∀ Y ∈ O bj D w e ha v e Φ( X , Y ) ∼ = Φ( X, Y t = i ) . (25) 53 2.7 ’Change of w ei gh t structures’; comparing w eigh t sp ectral sequences No w w e compare weigh t decompositions, virtual t -truncations, and weigh t sp ec tral sequence s corresp onding to distinct w eigh t structures. In order mak e our results more general (and to apply them b elo w) w e will assume that these structures are deﬁned on distinct triangulated categories; ye t the case when b oth are deﬁned on C is also in teresting. So, till the end of the section w e will assume: C , D are triangulated categories endo w ed with weigh t structures w and v , resp ectiv ely; F : C → D is an exact functor. Deﬁnition 2.7.1. 1. W e will sa y that F is right weight-exact if F ( C w ≥ 0 ) ⊂ D v ≥ 0 . 2. If F is fully faithful and right we ight-ex act, w e will say that v dominates w . 3. W e will sa y that F is left weight-exa ct if F ( C w ≤ 0 ) ⊂ D v ≤ 0 . 4. F will b e called weight-exact if it is b oth right and left w eigh t-exact. W e will say that w induc es v (via F ) if F is a weigh t-exact lo calization functor. Prop osition 2.7.2. L et F b e a right weight-exac t functor; let l ≥ m ∈ Z , X ∈ O bj D , X ′ ∈ O bj C , g ∈ D ( F ( X ′ ) , X ) . 1. L et w eight d e c omp o sitions of X [ m ] with r esp e c t to v and X ′ [ l ] with r esp e ct to w b e ﬁxe d. T hen g c a n b e c omplete d to a m o rphism of distinguishe d triangles F ( w ≥ l +1 X ′ ) − − − → F ( X ′ ) − − − → F ( w ≤ l X ′ )   y a   y g   y b v ≥ m +1 X − − − → X − − − → v ≤ m X (26) This c ompletion is unique if l > m . 2. F or arbitr ary weight Postnikov towers P o v ( X ) for X (with r esp e ct to v ) and P o w X ′ for X ′ (with r esp e ct to w ), g c an b e extende d to a morphism F ∗ ( P o w X ′ ) → P o v ( X ) . 3. L et H : D → A b e any functor, k ∈ Z , j > 0 . Denote H ◦ F by G . 54 Then (26) al lo ws to extend H ( g ) natur al ly to a diagr am H v 1 ( X ) − − − → H ( X ) − − − → H v 2 ( X )   y   y H ( g )   y G w 1 ( X ′ ) − − − → G ( X ′ ) − − − → G w 2 ( X ′ ) her e we add the w e ight structur e chosen a s an index to the notation of The- or em 2. 3 .1(I). Pr o of. 1. Since F is right weigh t-exact, D ( F ( w ≥ n +1 X ′ ) , v ≤ m X ) = { 0 } for an y n ≥ m . Hence the comp osition morphism F ( w ≥ l +1 X ′ ) → v ≤ m X is zero; if l > m then D ( F ( w ≥ l +1 X ′ ) , ( v ≤ m X )[ − 1]) = { 0 } . The result follows easily; see Prop osition 1.1.9 of [BBD82]. 2. Assertion 1 (in the case l = m ) yields that there exists a system of morphisms f i ∈ D ( F ( w ≥ i X ′ ) , v ≥ i X ) compatible with g ; we ﬁx suc h a system. Applying the same assertion for any pair of l , m : l > m , w e obtain that f l is compatible with f m (here w e use argumen ts similar to those describ ed in R emark 2.2.2) . F inally , since an y comm utativ e square can b e extended to a morphism of the corresp onding distinguished triangles (an axiom of triangulated categories), w e obta in that w e can comple te (unique ly up to a non-canonical isomorphism) the data chose n to a morphism o f P ostnik o v to w ers (i.e. c ho ose a compatible system of morphisms F ( X ′ i ) → X i ). 3. Easy fro m a ssertion 1; note that for any comm uta tive square in A X f − − − → Y   y h   y Z g − − − → T if we ﬁx the row s then the morphism g ◦ h : X → T completely determines the morphism Im f → Im g induced b y h . W e easily obtain a comparison morphism of weigh t sp ectral sequenc es. Prop osition 2.7.3. I L et F , X ′ , G b e as in the pr evious pr op osi tion ; supp ose also that H is c oh o molo gic al. Set X = F ( X ′ ) , g = id X . 1. Ther e exists some c om p a rison morphism o f the c orr esp onding w eight sp e ctr al se quen c es M : T v ( H , X ) → T w ( G, X ′ ) . Mor e over, this morphism is unique and additively functorial (in g ) starting fr o m E 2 . 55 2. L et ther e exist D ⊂ C w =0 such that any Y ∈ C w =0 is a r etr act of some Z ∈ D , and that for any Z ∈ D ther e exists a choic e of Z w ≥ 1 such that E pq 2 T v ( H , F ( Z w ≥ 1 )) = { 0 } for al l p, q ∈ Z . Then (any choic e of ) M yields an isomorph i s m of the sp e ctr a l se quenc e functors starting fr om E 2 . 3. L et E b e a triangulate d c ate gory en dowe d with a weig h t structur e u , F ′ : D → E a right weight-exact functor; s upp os e that H = E ◦ F ′ for som e c ohomol o gic a l functor E : E → A . Then we h ave the fol lowing asso cia- tivity pr op erty f o r c om p ari s o n of weight sp e ctr al se quenc es: the c omp os i tion of M with (an y choic e of ) a c omp ariso n morphism s M ′ : T u ( E , F ′ ( X )) → T v ( H , X ) c onstructe d as in assertion 1, starting fr om E 2 is c anonic al ly iso- morphic to (any choic e of a similarly c onstructe d) c omp arison morphism T u ( E , F ′ ( X )) → T w ( G, X ′ ) . II L et H , X ′ , X , G b e as ab ove, but supp ose that F : C → D is le ft w eight- exact. Th en a m e tho d dual to the on e for assertion I1 yields a tr ansfo rm ation N : T w ( G, X ′ ) → T v ( H , X ) ; this c onstruction satisﬁes the duals for al l pr op- erties of M describ e d in assertion I. Pr o of. I 1. In order to construct some comparison morphism, it suﬃces to construct a morphism of the corresp onding exact couples that is compatible with id X . Hence it suﬃces to use Prop osition 2.7.2(2) to obtain a morphism of the corresp onding P ostnik o v tow ers, and then apply H to it. Theorem 2.4.2 ( I I) yields that w eight sp ectral sequences could b e de- scribed in terms of the corresponding virtual t -truncations. Hence Prop o- sition 2.7.2(3 ) implies all the functoriality prop erties of M listed. 2. It suﬃces to pro v e that M is an isomorphism on E ∗∗ 2 T w ( G, Y ) for all Y ∈ O bj C . Since D ⊂ C w ≥ 0 , this assertion is true for an y Y ∈ D . Since Z 7→ E 2 ( T ( G , Z )) is a cohomological functor for any we ight structure (see Theorem 2.4.2 a nd the remark at Deﬁnition 2.3.2), the assertion is also true for an y Y ∈ O bj C b . T o conclude it suﬃces to note that fo r any H , an y Y ∈ O bj C , an y ﬁnite ’piece’ of E ∗∗ 2 T w ( G, Y ) coincides with the corresp onding piece of E ∗∗ 2 T w ( G, w [ i,j ] Y ) (for an y choice of w [ i,j ] Y ) if i is small enough and j is large enough, and this isomorphism is compatible with M . 3. W e recall that comparison morphisms for w eigh t sp ectral sequen ces w ere constructed using Prop osition 2 .7 .2 (1). It easily follo ws that M ′ ◦ M is one of the p ossible c hoices for a comparison morphism T u ( E , F ′ ◦ F ( X )) → T w ( G, X ′ ) . It suﬃces to apply assertion I1 to conclude that this ﬁxed c hoice of a comparison morphism coincides with a ny other p ossible c hoice starting 56 from E 2 . I I W e obtain the assertion from a ssertion I immediate ly by dualization (see Theorem 2.2.1(1 ) ) ; here one should consider the duals of C , D , and A (and also ’dualize’ the connecting functors). R emark 2.7 .4 . M is an isomorphism (starting from E 2 ) also if: t here exists a lo calization of D suc h that H f actorizes through it, v induces a w eigh t structure v ′ on it, w induce s a w eigh t structure on the categorical image of C that coincides with the restriction of v ′ to it (since b oth w eigh t spectral sequenc es a re isomorphic to the sp ectral sequence corresp onding to this new w eigh t structure). Y et this conditions are somewhat restrictiv e since w eight structures do not ’descend’ to lo calizations in general (since for an exact F ′ : C → E the classes F ′ ∗ ( C w ≥ 1 ) a nd F ′ ∗ ( C w ≤ 0 ) are not necessarily orthogonal in E ). In order to simplify chec king righ t and left weigh t-exactness of functors, w e will need the followi ng easy statemen t. Lemma 2.7.5. L et w b e b ounde d. 1. A n exact J : C → D is a right weight-exa c t whenever ther e exi s ts a D ⊂ C w =0 such that any Y ∈ C w =0 is a r etr act of some X ∈ D , and that for any X ∈ D we have J ( Y ) ∈ D v ≥ 0 . 2. A n exact J : C → D is a left weight-ex a ct whenever ther e exists a D ⊂ C w =0 such that any Y ∈ C w =0 is a r etr act of some X ∈ D , and that for any X ∈ D we have J ( Y ) ∈ D v ≤ 0 . Pr o of. It suﬃces to pro v e assertion 1, since assertion 2 is exactly its dual. If J is right w eigh t-exact functor, then w e can take D = C w =0 . No w w e pro v e the con v erse statemen t. Since D v ≥ 0 is Karoubi-closed and extension-stable in D , Theorem 2.2.1(13) yields that J ( C w ≥ 0 ) indeed b elongs to D v ≥ 0 . 3 Categories of comotiv es (main prop erties) W e em b ed D M ef f g m in to a certain big triangulated motivic category D ; w e will call it ob j ects c omotives . W e will need sev eral pro p erties of D ; y et w e will nev er use its description directly . F or this reason in §3.1 we only list the main prop erties of D . 57 In §3.2 w e associate certain comotive s to (disjoin t unions of ) ’inﬁnite in tersections’ of smo oth v arieties ov er k (w e call those pro-sc hemes). W e also in tro duce certain T ate tw ists for these comotiv es. In §3.3 w e recall the deﬁnition of a primitiv e sc heme (note that in the case of a ﬁnite k we call a sc heme primitiv e whenev er it is smo oth semi-lo cal). The main motivic prop ert y of primitiv e sch emes (prov ed b y M. W alk er) is: F ( S ) injects in to F ( S 0 ) if S is primitiv e connected, S 0 is its generic p oin t, and F is a homotopy in v arian t presheaf with transfers. In §3.4 we study the relation of (the comotiv es o f ) primitiv e sc hemes with the homotopy t -structure for D M ef f − . In §3.5 w e prov e that there are no D -morphisms of p ositiv e degrees b e- t w een the comotiv es of primitiv e sc hemes (and also certain T at e twis ts of those); this is also true f o r pro ducts of the comotives men tioned. In §3.6 w e prov e that one can pass to coun table homotop y limits in Gysin distinguished triangles; this yields Gysin distinguished triangles for the co- motiv es of pro-sc hemes. This allow s to construct certain P ostnik o v tow ers for the comotive s o f pro-sc hemes (and their T ate twi sts), whose factors are t wisted pro ducts of the comotives of function ﬁelds (ov er k ). The construc- tion of the to w er is parallel to the classic al construction of coniv eau sp ectral sequenc es (see §1 of [CHK97]). 3.1 Comotiv es: an ’a xi o matic description’ W e will deﬁne D b elo w as the deriv ed category of diﬀeren tial graded functors J → B ( Ab ) ; here J yie lds a diﬀeren tial g raded enhancemen t of D M ef f g m (cf. [Be V08 ] or [Lev98]), B ( Ab ) is the diﬀerential graded category of complexes o v er Ab . W e will also need some category D ′ that pro jects to D (a certain mo del of D ). Deriv ed categories of diﬀeren tial graded functors were studied in detail in [Dri04] and [Kel06]. W e will deﬁne and study them in §5 b elo w; no w we will only list their prop erties that are needed for the pro ofs of main statemen ts. Belo w w e will also need certain (ﬁltered) in v erse limits sev eral times. D is a triangulated category; so it is no w onder that there a re no nice limits in it. So we consider a certain additiv e D ′ endo w ed with an additiv e functor p : D ′ → D . W e will call (the images of ) inv erse limits from D ′ homotop y limits in D . The relation of D ′ with D is similar to the relation of C ( A ) with D ( A ) . In particular, D ′ is closed with resp ect to all (small ﬁltered) in v erse limits; 58 w e ha v e functorial cones o f morphisms in D ′ that are compatible with in v erse limits. W e will need some con v en tions and deﬁnitions in tro duced in Notation; in particular, I , L will b e pro jectiv e systems; I is countable. Prop osition 3.1.1. 1. D is a triangulate d c ate g o ry; D ⊃ D M ef f g m , a nd al l o bje cts of D M ef f g m ar e c o c o mp act in D . 2. Ther e is an additive c ate gory D ′ close d with r esp e ct to arbitr ary (smal l ﬁlter e d) inverse limits, and an additive functor p : D ′ → D that pr e - serves (smal l) pr o ducts. Mor e over, p is surje ctive on obje cts. 3. D ′ is endowe d with a c ertain inv ertible shift functor [1] that is c omp at- ible with the shift on D and r esp e cts inve rs e limits. 4. Ther e is a func toria l c one of morphi s ms in D ′ deﬁne d; it is c omp atible with [1] and r e s p e cts in verse li m its. 5. A ny triangle of the form X f → Y → Cone( f ) → X [1] in D ′ b e c omes distinguishe d in D . 6. The c omp osition functor M g m : C b ( S mC o r ) → D M ef f g m → D c an b e c anonic al l y f a ctorize d thr ough an add i tive functor j : C b ( S mC o r ) → D ′ . Shifts and c ones of morphisms in C b ( S mC o r ) ar e c o m p a tible with those in D ′ via j . 7. F or any X ∈ M g m ( C b ( S mC o r )) ⊂ O bj D , a ny Y : L → D ′ we have D ( p (lim ← − l ∈ L Y l ) , X ) = lim − → l ∈ L D ( p ( Y l ) , X ) . 8. D M ef f g m we akly c o gener ates D (i.e . we ha v e ⊥ D M ef f g m = { 0 } , se e Nota- tion). 9. L et a se quenc e i n ∈ I , n > 0 , b e incr e asing (i.e. i n +1 > i n for any n > 0 ) unb ounde d (se e Notation). Then for al l f unc tors X : I → D ′ , we have f unctorial distinguishe d triangles in D : p (lim ← − i ∈ I X i ) → p ( Y X i n ) e → p ( Y X i n ); (27) e i s the pr o duct of id X i n ⊕ − φ n : X i n +1 → X i n ; her e φ n ar e the mor- phisms c omi ng fr om I via X . 59 10. Ther e exists a di ﬀ er en tial gr ade d en hanc ement for D ; s e e §5.1 b elow . R emark 3.1 .2 . 1. Since b elo w w e will prov e some statemen ts for D only using its ’axiomatics’ (i.e. the prop erties listed in Prop osition 3.1.1), these results w ould also b e v alid in any other category that fulﬁlls these prop erties . This could b e useful, since the author is not sure a t all that all p ossible D are isomorphic. 2. Moreo v er, one could mo dify the axiomatics of D and consider instead a category that w ould only con tain the triangulated sub category of D M ef f g m generated by motiv es of smo oth v arieties of dimension ≤ n (f o r a ﬁxed n > 0 ). Our results a nd arguments b elo w can b e easily carried o v er to this setting (with minor mo diﬁcations; it is also useful here to w eak en condition 8 in the Prop osition). This mak es sense since these ’geometric pieces’ of D M ef f g m are self-dual with resp ect to P oincare duality (at least, if c har k = 0 ) ; see §6.4 b elo w. See also Remark 4.5.2(2). Alternativ ely , w e can w eak en the condition f or the functor D M ef f g m → D to b e a full em b edding. F or example, it could b e in teresting to consider the v ersion o f D for whic h this functor kills D M ef f g m ( n ) (for some ﬁxed n > 0 ). Lastly note that w e do not really need condition 2 in its full generalit y (b elo w). No w we deriv e some consequences from the axiomatics listed. Corollary 3.1.3. 1. F or any Z ∈ O bj DM ef f g m ⊂ O bj D , any X : L → D ′ we have D ( p (lim ← − l ∈ L X l ) , Z ) = lim − → l ∈ L D ( p ( X l ) , Z ) . 2. F or any T ∈ O bj D , al l functors Y : I → D ′ we ha v e functorial sho rt exact se q uen c es { 0 } → lim ← − 1 D ( T , p ( Y i )[ − 1]) → D ( T , p ( lim ← − Y i )) → lim ← − D ( T , p ( Y i )) → { 0 } ; her e lim ← − 1 is the (ﬁrst) de riv e d functor of lim ← − = lim ← − I . 3. F or al l functors X : L → C b ( S mC o r ) , Y : I → C b ( S mC o r ) , we have functorial short exact se quenc es { 0 } → lim ← − 1 i ∈ I (lim − → l ∈ L D ( M g m ( X l ) , M g m ( Y i )[ − 1])) → D ( p (lim ← − l ∈ L j ( X l )) , p (lim ← − i ∈ I j ( Y i ))) → lim ← − i ∈ I (lim − → l ∈ L D ( M g m ( X l ) , M g m ( Y i ))) → { 0 } . (28) 60 4. D is idem p o tent c o m plete. Pr o of. 1. If Z ∈ M g m ( C b ( S mC o r )) , then the assertion is exactly Prop o- sition 3.1.1(7 ) . It remains to note that a ny Z ∈ O bj D M ef f g m is a retract of some o b ject coming from C b ( S mC o r ) . 2. Since in ve rse limits and their deriv ed functors do not c hange when w e replace a pro jectiv e sys tem by any un b ounded subsystem, w e can assume that L consists of some i n as in (27). No w, (2 7 ) yields a long exact sequen ce · · · → Y i ∈ I D ( T , p ( Y i )[ − 1]) f → Y i ∈ I D ( T , p ( Y i )[ − 1]) → D ( T , p ( lim ← − i ∈ I Y i )) → Y i ∈ I D ( T , p ( Y i )) g → Y i ∈ I D ( T , p ( Y i )) → . . . , here f and g are induced b y e in (27). It is easily seen that Ker g ∼ = lim ← − D ( T , M g m ( Y m )) . Lastly , Remark A.3.6 of [Nee01] allows to iden tify Cok er f with lim ← − 1 D ( T , M g m ( Y m )[ − 1]) . 3. Immed iate from the previous assertions. 4. Since D ′ is closed with respect to all in v erse limits, it is closed with respect to all (small) pro ducts. Then Prop osition 3.1.1(2) yields tha t D is also closed with resp ec t to a ll pro ducts. No w, Remark 1.6.9 o f [Nee01] yields the result (in fact, the pro of uses o nly coun ta ble pro ducts). W e will often call the ob jects of D c o m otives . 3.2 Pro-sc hem es and their com otiv es No w w e ha v e certain inv erse limits f or ob jects (coming from) C b ( S mC o r ) ; this allo ws to deﬁne (reasonable) comotiv es f or certain sc hemes that are not (necessarily ) o f ﬁnite t yp e ov er k (and for their disjoint unions). W e also deﬁne certain T ate tw ists of those. W e will call certain formal disjoint unions of pro jectiv e limits of smo oth v arieties o v er k pr o-schem es . Suc h a union V /k is a pro-sc heme if it is a 61 coun table disjoin t union of pro jectiv e limits of smo oth v arieties of dimension ≤ c V for some ﬁxed c V ≥ 0 (in the category of sc hemes) with connecting morphisms b eing op en dense em b eddings (note that t hese pro jectiv e limits are oft en k -sc hemes). One may sa y that a pro-sc heme is a coun table disjoin t union of in tersections of smo oth v a rietie s. Note that (the sp ectrum of ) an y function ﬁeld ov er k yiel ds a pro -sc heme; any smo oth k -v ariet y or an y its (semi)-lo calizations is a pro-sc heme also. W e hav e the op eration of countable disjoin t union for pro-sc hemes of b ounded dimension. No w, w e would like to presen t a (not necessarily connected) pro-sc heme V as pro jectiv e limits of smo oth v arieties V i . This is easy if V is connected (cf. Lemma 3.2 .9 of [Deg08a]). In the general case one should allow (for mally) zero morphism s b et w een connected components of V i (for distin ct i ). So w e consider a new category S mV ar ′ con taining the catego r y of all smo oth v arieties as a (non-full!) sub category . W e tak e O bj S mV ar ′ = S mV ar ; fo r an y smooth connected v a rietie s X, Y ∈ S mV ar w e ha v e S mV ar ′ ( X , Y ) = M or V ar ( X , Y ) ∪ { 0 } ; the comp osition of a zero morphism with an y other one is zero; S mV ar ′ ( ⊔ i X i , ⊔ j Y j ) = ⊔ i,j S mV ar ′ ( X i , Y j ) . S mV ar ′ can b e em b edde d into S mC o r (certainly , this em b edding is not full). W e will write V = lim ← − V i . Note that the set of connected comp onen ts of V is the inductiv e limit o f the corresp onding sets for V i . No w, for an y pro - sc heme V = lim ← − V i , an y s ≥ 0 , we introduce the following notation: M g m ( V )( s ) = p (lim ← − ( j ( V i )( s ))) ∈ O bj D (see Prop osition 3.1.1); we will denote M g m ( V ) ( 0 ) b y M g m ( V ) and call M g m ( V ) the comotif of V . This notation should b e considered as formal i.e. w e do not deﬁne T ate t wists o n D (till §5.4.3). Ob viously , if V ∈ S mV ar , its comotif (and its twis ts) coincides with its motif (and its twis ts), so w e can use the same notation for them. If A is a category closed with resp ect to ﬁltered direct limits, H ′ : D M ef f g m → A is a functor, we can (formally) extend it to co-motive s in ques- tion; w e set: H ( M g m ( V ) ( s ) [ n ]) = lim − → H ′ ( M g m ( V i )( s )[ n ]) . (29) R emark 3.2.1 . 1. F or a general H ′ this notation should be cons idered as formal. Y et in the case H ′ = ( − , Y ) : D → Ab , Y ∈ O bj D M ef f g m ⊂ O bj D , w e hav e H ( M g m ( V )( i )[ n ]) = D ( M g m ( V ) ( i )[ n ] , X ) ; see Corollary 3.1.3(1), i.e. (29) yields the v alue of a w ell-deﬁned functor D → Ab at M g m ( V ) ( s ) [ n ] . W e will only need H ′ of this sort till §4.3. 62 More generally , there exists suc h an H if A satisﬁes AB5 and H ′ is coho- mological; we will call the corresp onding H an extende d c oh o molo gy the ory , see Remark 4.3.2 b elo w. 2. Let V j b e a coun table set of pro-sc hemes (of b ounded dimensions). Then M g m ( ⊔ V j ) = Q M g m ( V j ) b y Prop osition 3.1.1(2). Besides, for an y H ′ as in (29) w e ha v e H ( M g m ( ⊔ V j )( s )[ n ]) = L H ( M g m ( V j )( s )[ n ]) . Belo w w e will need some con v en tions for pro-sc hemes. F or pro-sc hemes U = lim ← − U i and V = lim ← − V j w e will call an elemen t of lim ← − i ∈ I (lim − → j ∈ J S mC or ( U i , V j )) an o pen em b eddin g if it can b e obtained as a double limit of op en em b eddings U i → V j (as v arieties). If U = V \ W for some pro-sc heme W , w e will sa y that W is a closed sub-pro-sc heme of V . Note that in t his case any connected comp onen t of W is a closed subsc heme of some connected comp onen t of V ; y et some comp onen ts of V could con tain an inﬁnite set of connected comp onen ts of W . F or V = ⊔ V j , V j are connected pro-sc hemes, w e will call the maxim um of the transcendence degrees of the function ﬁelds of V j the dimension of V (note that this is ﬁnite). W e will sa y that a sub-pro-sc heme U = ⊔ U m , U m are connected, is ev erywhere of co dime nsion r (resp. ≥ r , for some ﬁxed r ≥ 0 ) in V = ⊔ V j if for ev ery induced em b edding U m → V j the diﬀerence of their dimensions (deﬁned as ab ov e) is r (resp. ≥ r ). W e will call the inv erse limit of the sets of p oin ts of V i of a ﬁxed co dimen- sion s ≥ 0 the set of p oin ts of V of co dimension s (note that any elemen t of this set indeed deﬁnes a p oin t of some connected comp onen t of V ). 3.3 Primitiv e sc hemes: remi nde r In [W al98] M. W alk er prov ed that primitiv e sc hemes in the case of a n inﬁnite k hav e ’motivic’ prop erties similar to those of smo oth semi-lo cal sc hemes (in the sense of §4.4 of [V o e00b]). Since we don’t w an t to discriminate the case of a ﬁnite k , w e will mo dify sligh tly the standard deﬁnition of primitiv e sc hemes. Deﬁnition 3.3.1. If k is inﬁnite then a (pro-)sc heme is called primitiv e if all of its connected comp onen ts a re aﬃne k -sc hemes and their coordinate rings R j satisfy the follo wing primitivit y criterion: for a n y n > 0 eve ry p olynomial in R j [ X 1 , . . . , X n ] whose co eﬃcien ts g ene rate R j as an ideal o v er itself, represen t s an R j -unit. 63 If k is ﬁnite, then w e will call a pro- sc heme primitiv e whenev er all of its connected comp onen ts are semi-lo cal (in the sense of §4.4 of [V o e00b]). R emark 3.3.2 . Recall tha t in the case of inﬁnite k all semi-lo cal k -algebras satisfy the primitivit y criterion (see Example 2.1 of [W al98]). Belo w w e will mostly use the follow ing basic prop ert y of primitiv e sc hemes. Prop osition 3.3.3. L e t S b e a primitive p r o-scheme, let S 0 b e the c ol le ction of al l of its generic p oints; F is a homotopy in variant pr eshe af with tr a nsfers. Then F ( S ) ⊂ F ( S 0 ) ; her e we deﬁne F on pr o-s chemes as in ( 2 9). Pr o of. W e can assume that S is connected (so it is a smo oth primitiv e sc heme). Hence in the case of inﬁnite k our assertion follow s from Theorem 4.19 o f [W al98]. No w, if k is ﬁnite, then S 0 is semi-lo cal (by our con v en tion); so w e ma y apply Corollary 4 .18 of [V o e00b] instead. 3.4 Basic motivic prop erties o f primitiv e s c hemes W e will call a primitiv e pro-sc heme just a primitiv e sc heme. W e prov e certain motivic prop erties of primitiv e sc hemes (in the for m in whic h we will need them b elo w). Prop osition 3.4.1. F or F ∈ O bj DM ef f − we de ﬁ ne H ′ ( − ) = D M ef f − ( − , F ) on D M ef f g m ; w e also deﬁne H ( M g m ( V )( i )[ n ]) as in (29). L et S b e a primitive scheme, m ≥ 0 , i ∈ Z . 1. L et F ∈ D M ef f − t ≤− 1 ( t is the homotopy t -structur e, that we c onsid e r e d in §1.3). Then H ( M g m ( S )( m )[ m ]) = { 0 } . 2. Mor e ge ner a l ly, for any F ∈ O bj D M ef f − we have H ( M g m ( S )( m )[ m ]) ∼ = F 0 − m ( S ) wher e F 0 = F t =0 , F 0 − m is the m -th T ate twist of F 0 (se e D eﬁnition 1.4.1). Pr o of. 1. W e consider the homotop y inv ariant presheaf with tra ns fers F − m : X 7→ D M ef f − ( M g m ( X )( m )[ m ] , F ) . W e should prov e that F − m ( S ) = 0 (here w e extend F − m to pro-sc hemes in the usual w a y i.e. as in (29)). (29) also yields that F − m ( ⊔ S i ) = L F − m ( S i ) . Hence by Prop osition 3.3.3, it suﬃces to consider the case o f S b eing (the sp ectrum o f ) a f unc tion ﬁeld o v er k . Since F − m is represen ted b y a n ob ject of D M ef f − t ≤− 1 (see Prop osition 64 1.4.2(2)), it suﬃces to note that a n y ﬁeld is a Henselian sc heme i.e. a p oin t in the Nisnevic h top ology . 2. By Prop osition 1.4.2, for any X ∈ S mV ar w e ha v e M g m ( X )( m )[ m ] ⊥ D M ef f − t ≥ 1 . Hence w e can assume F ∈ D M ef f − t ≤ 0 . Next, using assertion 1, w e can easily reduce the situation to the case F = F t =0 ∈ O bj H I (by considering the t -decomp osition of F [ − 1] ). In this case the statemen t is immediate from Prop osition 1.4.2( 1 ). Lemma 3.4.2. L e t U → U ′ b e an op en dense emb e d ding of smo oth varieties. 1. W e have Cone( M g m ( U ) → M g m ( U ′ )) ∈ D M ef f − t ≤− 1 . 2. L et S b e primitive. Then for any n, m, i ≥ 0 the map D ( M g m ( S )( m )[ m ] , M g m ( U )( n )[ n + i ]) → D ( M g m ( S )( m )[ m ] , M g m ( U ′ )( n )[ n + i ]) is surje ctive. Pr o of. 1. W e denote Cone( M g m ( U ) → M g m ( U ′ )) by C . Ob viously , C ∈ D M ef f − t ≤ 0 . Let H denote C t =0 ( H ∈ O bj H I ). By Corollary 4.19 of [V o e00b], w e ha v e H ( U ) ⊂ H ( U ′ ) . Next, from the long exact sequence { 0 } (= D M ef f − ( M g m ( U )[1] , H )) → D M ef f − ( C , H ) → D M ef f − ( U ′ , H ) → D M ef f − ( U, H ) → . . . w e obtain C ⊥ H . Then the long exact sequence · · · → D M ef f − ( C t ≤− 1 [2] , H ) → D M ef f − ( H , H ) → D M ef f − ( C , H ) → . . . yields H = 0 . 2. It suﬃces to che c k that M g m ( S )( m )[ m ] ⊥ C ( n )[ n + i ] . Since M g m ( U )( n )[ n ] is canonically a retract of M g m ( U × G n m ) , we can assume that n = 0 . No w the claim follo ws immediately fr o m assertion 1 and Prop osition 3.4.1(1). 3.5 On morphisms b et w een the comotiv es of prim itiv e sc hem e s W e will need the fact tha t certain ’p ositiv e’ morphism g roups are zero. Let n, m, ≥ 0 , i > 0 , Y = lim ← − Y l ( l ∈ L ), b e any pro-sc heme, X b e a primitiv e sch eme. Prop osition 3.5.1. 1. The natur al homomo rp h ism D ( M g m ( X )( m )[ m ] , M g m ( Y )[ n ]( n )) → → lim ← − l (lim − → X ⊂ Z ,Z ∈ S mV ar D M ef f g m ( Z ( m )[ m ] , M g m ( Y l )( n )[ n ])) 65 is surje ctive. 2. M g m ( X )( m )[ m ] ⊥ M g m ( Y )[ n + i ]( n ) . Pr o of. Note ﬁrst that by the deﬁnition of t he T a t e t wist (1) , it can b e lifted to C b ( S mC o r ) . 1. This is immediate from the short exact sequence (28). 2. By Remark 3.2.1(2), w e ma y supp ose that Y is connected. Then w e apply (28) again. The corresp onding lim ← − -term is zero b y Prop osition 3.4.1(1). Lastly , the surjectivit y prov ed in Lemma 3.4.2(2) yields that the corresponding lim ← − 1 -term is zero. Indeed, the groups D ( M g m ( X )( m )[ m ] , M g m ( Y l )[ n + i − 1]( n )) ob viously satisfy the Mittag- L eﬄer condition; see §A.3 o f [Nee01]. In fa ct, one could easily deduce the assertion from the results of the previous subsec tion and (27) directly (w e do not need muc h of the theory of higher limits in this pap er). R emark 3.5 .2 . In fact, this statemen t, as we ll as all other prop erties of (prim- itiv e) pro-sc hemes that w e need, are also true for not necess ary countable disjoin t unions of (primitiv e) pro-sc hemes. This observ ation could b e used to extend the main results of the pap er to a somewhat larger category; y et suc h a n extension do es not seem to b e imp ortan t. 3.6 The Gysin distingui shed tria ng le for pro-sc hemes; ’Gersten’ P ostnik ov to w e rs for the comotiv es of pro- sc hem e s W e prov e that w e can pass to countable homotop y limits in Gysin distin- guished triangles. Prop osition 3.6.1. L et Z , X b e pr o-schem es, Z a close d subscheme of X (everywher e) of c o dimension r . Then for any s ≥ 0 the natur al morphism M g m ( X \ Z )( s ) → M g m ( X )( s ) extends to a distinguishe d triangle (in D ): M g m ( X \ Z )( s ) → M g m ( X )( s ) → M g m ( Z )( r + s )[2 r ] . 66 Pr o of. First assume s = 0 . W e can assume X = lim ← − X i , Z = lim ← − Z i for i ∈ I , where X i , Z i ∈ S mV ar , Z i is closed ev erywhere of co dimension r in X i for all i ∈ I . W e ta k e Y i = j ( X i \ Z i → X i ) , Y = p (lim ← − i ∈ I Y i ) . By parts 4 and 5 o f Prop osition 3.1.1 we hav e a distinguished triangle M g m ( X \ Z ) → M g m ( X ) → Y . It remains to pro v e that Y ∼ = M g m ( Z )( r )[2 r ] . Prop osition 2.4.5 of [Deg08a] (a functorial form of the Gysin distinguished triangle for V o ev o dsky’s mo- tiv es) yields that p ( Y i ) ∼ = M g m ( Z i )( r )[2 r ] ; moreo ver, the connecting mor- phisms p ( Y i ) → p ( Y i +1 ) are obtained from the corr esp onding morphism s M g m ( Z i ) → M g m ( Z i +1 ) b y tensoring by Z ( r )[2 r ] . It remains to recall: by Prop osition 3.1.1(9), the isomorphism class of a homotopy limit in D can b e completely described in terms of (ob jects and morphisms) of D (i.e. w e don’t ha v e to consider the lifts of ob jects and morphisms to D ′ ). This yields the result. No w, since M g m ( X × G m ) = M g m ( X ) L M g m ( X )(1)[1] for an y X ∈ S mV ar (hence this is also true for pro-sc hemes), the a ss ertion for the case s = 0 yields the g ene ral case easily . No w we will construct a certain P ostnik o v to w er P o ( X ) for X b eing the (t wisted) comotif of a pro-sc heme Z that will b e related to the coniv eau sp ec tral sequences (fo r cohomology) of Z ; o ur metho d w as described in §1.5 ab o v e. Note tha t w e consider the general case of an arbitra r y pro-sc heme Z (since in this pap er pro-sc hemes pla y an imp o rtan t role); yet this case is not m uc h distinct f ro m the (partial) case of Z ∈ S mV ar . Corollary 3.6.2. W e denote the dimension of Z by d (r e c al l the c onv e ntions of §3.2). F or al l i ≥ 0 we denote by Z i the set of p oints of Z of c o dimension i . F or any s ≥ 0 ther e e xists a Postnikov tower for X = M g m ( Z )( s )[ s ] such that l = 0 , m = d + 1 , X i ∼ = Q z ∈ Z i M g m ( z )( i + s )[2 i + s ] . Pr o of. As ab o v e, it suﬃces to pro v e the statemen t for s = 0 . Since any pro duct o f distinguished triangles is distinguished, we can assume Z to b e connected. W e consider a pro jectiv e system L whose elemen ts are sequences of closed subsc hemes ∅ = Z d +1 ⊂ Z d ⊂ Z d − 1 ⊂ · · · ⊂ Z 0 . Here Z 0 ∈ S mV ar , Z l ∈ V ar for l > 0 , Z is op en in Z 0 (see §3.2; Z 0 is connected; in the case when Z ∈ S mV ar w e only tak e Z 0 = Z ); for all j > 0 w e hav e: Z j is 67 ev erywhere of co dimension ≥ j in Z 0 ; all irr educible comp onen ts of all Z j are ev erywhere of co dimension ≥ j in Z 0 ; and Z j +1 con tains the singular lo cus of Z j (for j ≤ d ) . The ordering in L is giv en b y op en em b eddings of v arieties U j = Z 0 \ Z j for j > 0 . F o r l ∈ L w e will denote the corresp onding sequenc e b y ∅ = Z l d +1 ⊂ Z l d ⊂ Z l d − 1 ⊂ · · · ⊂ Z l 0 . Note that L is coun table! By the previous prop osition, for a n y j w e hav e a distinguished triangle M g m (lim ← − ( Z l 0 \ Z l j )) → M g m (lim ← − ( Z l 0 \ Z l j +1 )) → M g m (lim ← − ( Z l j \ Z l j +1 )( j )[2 j ]) . It remains to compute the la st term; w e ﬁx some j . W e hav e lim ← − l ∈ L ′ ( Z l j \ Z l j +1 )) ∼ = Q z ∈ Z i M g m ( z ) . Inde ed, for all l ∈ L the v ariet y Z l j \ Z l j +1 is the disjoin t union of some lo cally clos ed smo oth subsc hemes of Z l 0 of co dimension j ; for any z 0 ∈ Z j for l ∈ L large enough z 0 is con tained in Z l j \ Z l j +1 as an op en sub-pro-sc heme, and the in v erse limit of connected comp onen ts o f Z l j \ Z l j +1 con taining z 0 is exactly z 0 . Now, we can apply the functor X 7→ M g m ( X )( j )[2 j ] to this isomorphism. W e obtain M g m (lim ← − ( Z l j \ Z l j +1 )( j )[2 j ]) ∼ = Q z ∈ Z i M g m ( z )( i ) . This yields the result. R emark 3.6.3 . 1. Alternative ly , one could construct P o ( X ) for the (t wisted) comotif of a pro-sc heme T = lim ← − T l as the in v erse limit of the P ostnik o v to w ers for T l (constructed as ab o v e y et with ﬁxed Z l 0 = T l ); certainly , to this end one should pass to the limit in D ′ . It is easily seen that one w ould get the same to w er this wa y . 2. Certainly , if w e shift a P ostnik o v to w er fo r M g m ( Z )( s )[ s ] by [ j ] for some j ∈ Z , w e obtain a Pos tnik ov to w er for M g m ( Z )( s )[ s + j ] . W e didn’t form ulate assertion 2 for these shifts only b ecause w e wan ted X p to b elong to D w =0 s (see Prop osition 4.1.1 b elo w). 3. Since the calculation of X i used Prop osition 3.1.1 (9), our metho d cannot describ e connecting morphisms b et wee n them (in D ). Y et one can calculate the ’images’ of the connecting morphisms in D naive ; see §1.5 and §6.1. 4 Main motivic results The results of the previous section com bined with those of §2.2 allow us to construct (in §4.1) a certain Gersten w eight structur e w on a certain trian- gulated D s : D M ef f g m ⊂ D s ⊂ D . Its main prop ert y is t hat the comotive s 68 of function ﬁelds ov er k (and their pro ducts) b elong to H w . It f o llo ws im- mediately that the P ostnik o v to w er P o ( X ) pro vided b y Corollary 3 .6.2 is a weight Postnikov towe r with resp ect to w . Using this, in §4.2 w e pro v e: if S is a primitiv e sch eme, S 0 is its dense sub-pro-sc heme, then M g m ( S ) is a direct summand of M g m ( S 0 ) ; M g m ( K ) (fo r a function ﬁeld K /k ) con tains (as retracts) the comotiv es of primitiv e sc hemes whose g en eric p oin t is K , as w ell as the t wisted comotiv es of residue ﬁelds of K (fo r all geometric v a luations). In §4.3 we (easily) translate these results to cohomology; in particular, the cohomology of (the sp ectrum of ) K con tains direct summands corresp onding to the cohomology of primitiv e sche mes whose g ene ric p oin t is K , as wel l as t wisted cohomology o f residue ﬁeld s of K . Here one can consider an y cohomology theory H : D s → A ; one can obtain such an H by extending to D s an y cohomological H ′ : D M ef f g m → A if A satisﬁes AB5 (by means of Prop osition 1.2.1). Note: in this case the cohomology o f pro-sc hemes men tio ne d is calculated in the ’usual’ w ay . In §4.4 w e consider w eigh t sp ectral sequenc es corresp onding to (the Ger- sten w eigh t structure) w . W e observ e that these sp ectral sequen ces general- ize naturally the classical coniv eau sp ectral sequenc es. Besides, for a ﬁxed H : D s → A our (generalized) conive au spectral sequence con v erging to H ∗ ( X ) (where X could b e a motif or just an ob ject of D s ) is D s -functorial in X (i.e. it is motivically functorial fo r ob jects of D M ef f g m ); this fact is non-trivial ev en when restricted to mot ives of smo oth v arieties. In §4.5 w e prov e that there exists a nice dualit y D op × D M ef f − → Ab (ex- tending the bi-functor D M ef f − ( − , − ) : D M ef f g m op × D M ef f − → Ab ); the G erste n w eigh t structure w (on D s ) is left orthogonal to the homotopy t - struc ture t on D M ef f − with respect to it. This allow s to apply Theorem 2.6.1: in the case when H comes fr o m Y ∈ O bj D M ef f − w e pro v e the isomorphism ( starting from E 2 ) of (the coniv eau) T ( H , X ) with the sp ectral sequence corresp ond- ing to the t -truncations of Y . W e describ e O bj DM ef f g m ∩ D w ≤ i s in terms of t (for D M ef f − ). W e also note that our results a llo w to describ e torsion motivic cohomology in terms of (torsion) étale cohomology (see Remark 4.5.4(4)). In §4.6 w e deﬁne the coniv eau sp ectral sequence (starting from E 2 ) for cohomology of a motif X ov er a not ( necessarily) coun table p erfect base ﬁeld l as the limit of the corresp onding conive au sp ectral sequenc es o v er coun table p erfect subﬁelds of deﬁnition f or X . This deﬁnition is compatible with the classical one (fo r X b eing the motif of a smo oth v ariet y); so we obtain motivic functorialit y of classical coniv eau sp ectral sequences ov er a general base ﬁeld. 69 In §4.7 w e pro v e that the Chow w eight structure for D M ef f g m (in tro duced in §6 of [Bon10]) could b e extended to D (certainly , the corresp onding w eigh t structure w C how diﬀers fro m w ). W e will call the corresp onding w eigh t sp ec- tral sequences C h ow-weight ones; note that they are isomorphic to classical (i.e. Deligne’s) we ight sp ectral sequences when the latter are deﬁned. In §4.8 w e use the results §2.7 t o compare Cho w-w eigh t sp ectral sequenc es with coniv eau ones. W e alw a ys ha v e a comparison morphism; it is an iso- morphism if H is a b i r ational cohomolog y theory . In §4.9 w e consider the category of birational comotiv es D bir (a certain ’completion’ of birational mot ives o f [KaS02]) i.e. the lo calization of D b y D (1) . It turns o ur that w and w C how induce the same w eigh t structure w ′ bir on D bir . Con v ersely , starting from w ′ bir one can glue ’from slices’ the weigh t structures induced by w and w C how on D / D ( n ) fo r all n > 0 . F urthermore, these structures b elong to an in teresting family of w eigh t structures indexed b y a single in tegral parameter; other terms of this family could b e a lso in t er- esting! 4.1 The Gersten w eigh t structure for D s ⊃ D M ef f g m No w w e describ e the main w eigh t structure of this pap er. Unfortunately , the author do es not know whether it is p ossible to deﬁne the G ers ten we ight structure (see b elo w) on the whole D . Y et for our purp oses it is quite suﬃcien t to deﬁne the corresp onding w eigh t structure on a certain trian- gulated sub catego r y D s ⊂ D con taining D M ef f g m (and the comotiv es of all pro-sc hemes). In order t o mak e the ch oice of D s ⊂ D compatible with extensions of scalars, w e b ound certain dimensions of ob jects of H w . W e will den ote by H the full subcategor y of D whose ob jects are all coun table pro ducts Q l ∈ L M g m (Sp ec K l )( n l )[ n l ] ; here K l are (the sp ectra o f ) function ﬁelds ov er k , n l ≥ 0 ; we assume that the transcendence degrees of K l /k a nd n l are b ounded. Prop osition 4.1.1. 1. L et D s b e the Kar oubi-c losur e of h H i in D . Then C = D s c an b e endow e d with a unique weight structur e w such that H w c ontains H . 2. H w is the idemp otent c om pletion of H . 3. D s c ontains D M ef f g m as wel l as al l M g m ( Z )( l ) for Z b eing a pr o-scheme, l ≥ 0 . 70 4. F or any primitive S , i ≥ 0 , we h ave M g m ( S )( i )[ i ] ∈ D w =0 s . 5. L et Z b e a pr o-sch eme, s ≥ 0 . Then M g m ( Z )( s )[ s ] ∈ D w ≤ 0 s ; the Post- nikov tower for M g m ( Z )( s )[ s ] given by Cor ol la ry 3.6.2 is a weig h t Postnikov tower for it. Pr o of. 1. By Prop osition 3.5.1(2), H is negativ e (since an y ob ject of H is a ﬁnite sum of M g m ( X i )( m i ) for some primitiv e pro-sc hemes X i , m i ∈ Z ). Besides, D is idemp oten t complete (see Corollary 3.1.3( 4 )); hence D s and D w =0 s also are. Hence w e can apply Theorem 2.2.1(1 8) (for D = H ). 2. Also immediate from Theorem 2.2.1(18). 3. M g m ( Z )( l ) ∈ O bj D s b y Corollary 3.6.2; in particular, this is true for Z ∈ S mV ar . It remains to note that D M ef f g m is the Karoubi-closure of h M g m ( U ) : U ∈ S mV ar i in D . 4. It suﬃces to note t hat M g m ( S )( i )[ i ] b elongs b oth to D w ≤ 0 s and to D w ≥ 0 s b y Theorem 2.2.1(20). Here w e use Prop osition 3.5.1(2) again. 5. W e hav e X i ∈ D w =0 s . Hence Theorem 2.2.1(14) yields the result. Note here that we hav e Y 0 = 0 in the notation of Deﬁnition 2 .1.2(9). W e will call w the Gersten w eigh t structure, since it is closely connected with Gersten resolutions of cohomology (cf. §4.5 b elo w). By default, b elo w w will denote the Gersten we ight structure. R emark 4.1.2 . 1. H w is idemp oten t complete since D s is. 2. In fact, one could easily prov e similar statemen ts fo r C b eing just h H i (instead of its Karoubi-closure in D ). Certainly , for this vers ion of C w e will only hav e C ⊃ M g m ( K b ( S mC o r )) . Besides, note that f or an y function ﬁeld K ′ /k , an y r ≥ 0 , there exists a function ﬁeld K /k suc h that M g m (Sp ec K ′ )( r )[ r ] is a retract of M g m (Sp ec K ) (see Corollary 4.2.2 b elo w). Hence it suﬃces take H b eing the full sub cate- gory of D whose ob jects are Q l ∈ L M g m (Sp ec K l ) (for b ounded transcendence degrees of K l /k ). 3. The prop osition implies that D s is exactly the Karoubi-closure in D of the triangulated category generated by the comotive s of all pro- sc hemes. 4. The author do es not know whether one can describe w eigh t de- comp ositions for arbitrary ob jects of D M ef f g m explicitly . Still, one can sa y something ab out these weigh t decomp ositions and w eight complexes using their functorialit y prop erties. In particular, knowin g w eigh t complexes for X , Y ∈ O bj D M ef f g m (or just ∈ O bj D s ) and f ∈ D s ( X , Y ) o ne can describe 71 the weigh t complex of Cone( f ) up to a homotopy equiv alence as the cor- respo ndin g cone (see Lemma 6.1.1 b elo w). Besides, let X → Y → Z b e a distinguished triangle (in D ). Then for any c hoice of ( X w ≤ 0 , X w ≥ 1 ) and ( Z w ≤ 0 , Z w ≥ 1 ) there exists a c hoice of ( Y w ≤ 0 , Y w ≥ 1 ) suc h that there exist dis- tinguished triangles X w ≤ 0 → Y w ≤ 0 → Z w ≤ 0 and X w ≥ 1 → Y w ≥ 1 → Z w ≥ 1 ; see Lemma 1.5.4 of [Bon10 ]. In particular, w e obtain that j maps com- plexes (ov er S mC or ) concen trated in degrees ≤ j in to D w ≤ j s (w e will pro v e a stronger statemen t in Remark 4.5.4(4) b elo w). If X ∈ O bj D M ef f g m comes from a complex o v er S mC or whose connecting morphisms satisfy certain co dimens ion restrictions, these observ ations could b e extended to an explicit description of a w eigh t decomp osition for it; cf. §7.4 of [Bon10]. 4.2 Direct summa nd resul ts for comoti v es Prop osition 4.1.1 easily implies the follo wing in teresting result. Theorem 4.2.1. 1. L et S b e a primitive schem e ; le t S 0 b e its dense sub- pr o-schem e. Then M g m ( S ) is a dir e ct summand of M g m ( S 0 ) . 2. Supp ose mor e over that S 0 = S \ T wher e T is a close d subsche m e of S everywher e of c o d i m ension r > 0 . Then we have M g m ( S 0 ) ∼ = M g m ( S ) L M g m ( T ) ( r )[2 r − 1] . Pr o of. W e can assume that S and S 0 are connected. 1. By Prop osition 4.1.1(5), w e hav e: M g m ( S 0 ) , M g m ( S ) ∈ D w ≤ 0 s ; M g m (Sp ec ( k ( S ))) could b e assumed to b e the zeroth term of their w eigh t complexes for a c hoice of w eigh t complexes compatible with some negativ e P ostnik o v w eigh t tow ers for them; the em b edding S 0 → S is compatible with id M gm (Spec ( k ( S ))) (since we ha v e a comm utativ e triangle Sp ec k ( S ) → S 0 → S of pro- sche mes). Hence Theorem 2.2.1( 1 6) yields the result. 2. By Prop osition 3.6.1 w e ha v e a distinguished triangle M g m ( S 0 ) → M g m ( S ) → M g m ( T ) ( r )[2 r ] . By parts 4 and 5 of Prop osition 4.1.1 w e ha v e M g m ( S 0 ) ∈ D w ≤ 0 s , M g m ( S ) ∈ D w =0 s , M g m ( T ) ( r )[2 r ] ∈ D w ≤− r s ⊂ D w ≤− 1 s . Hence Theorem 2 .2.1(8) yields the result. Corollary 4.2.2. 1. L e t S b e a c onne cte d primitive scheme, let S 0 b e its generic p oint. Then M g m ( S ) is a r etr a ct o f M g m ( S 0 ) . 2. L et K b e a function ﬁeld over k . L et K ′ b e the r esidue ﬁeld for a ge ometric valuation v of K of r ank r . Then M g m (Sp ec K ′ )( r )[ r ] is a r etr act of M g m (Sp ec K ) . 72 Pr o of. 1. This is just a partial case of part 1 of the previous theorem. 2. Ob viously , it suﬃces to prov e the statemen t in the case r = 1 . Next, K is the function ﬁeld of some normal pro jectiv e v ariet y ov er k . Hence there exists a U ∈ S mV ar suc h that: k ( U ) = K , v yield s a non- empty closed subsc heme of U ( since the singular lo cus has co dimension ≥ 2 in a normal v ariet y). It easily f o llo ws that there exists a pro-sche me S (i.e. an in v erse limit of smo oth v arieties) whose only p oin ts are the sp ectra of K and K 0 . So, S is lo cal, hence it is primitiv e. By part 2 o f the theorem, we ha v e M g m (Sp ec K ) = M g m ( S ) M M g m (Sp ec K 0 )(1)[1]; this concludes t he pro of. R emark 4.2 .3 . 1. Note that w e do not construct any explicit splitting mor- phisms in the decomp ositions ab o v e. Probably , one cannot c ho ose any canon- ical splittings here (in the general case); so there is no (automatic) com- patibilit y fo r a ny pair of related decomp ositions. R es p ectiv ely , thoug h the comotiv es of (sp ectra of ) function ﬁelds contain tons of direct summands, there seems to b e no general w ay to decomp ose them into indecompo sable summands. 2. Y et Prop osition 3.6.1 easily yields that M g m (Sp ec k ( t )) ∼ = Z L Q E M g m ( E )(1)[1] ; here E runs through all closed p oin ts of A 1 (considered as a sc heme o v er k ). 4.3 On cohomolog y of pro-sc hemes, and its direct sum- mands The results prov ed ab o v e immediately imply similar assertions for cohomol- ogy . W e a lso construct a class of cohomology theories that resp ect homotopy limits. Prop osition 4.3.1. L et H : D s → A b e c ohomolo gic al, S b e a p rimitive scheme. 1. L et S 0 b e a de nse sub-pr o-sc h eme of S . Then H ( M g m ( S )) is a dir e ct summand of H ( M g m ( S 0 )) . 2. Supp ose m or e over tha t S 0 = S \ T w h er e T is a close d subsc heme of S of c o d i m ension r > 0 . Then we hav e H ( M g m ( S 0 )) ∼ = H ( M g m ( S )) L H ( M g m ( T ) ( r )[2 r − 1]) . 73 3. L et S b e c on ne cte d, S 0 b e the generic p oint of S . Then H ( M g m ( S )) is a r etr ac t of H ( M g m ( S 0 )) in A . 4. L et K b e a function ﬁeld over k . L et K ′ b e the r esidue ﬁeld for a ge ometric valuation v of K of r ank r . The n H ( M g m (Sp ec K ′ )( r )[ r ]) is a r etr act of H ( M g m (Sp ec K )) in A . 5. L et H ′ : D M ef f g m → A b e a c ohomolo gic al functor, let A satisfy AB5. Then Pr op osition 1.2.1 al lows to extend H ′ to a c ohom olo gic al functor H : D → A that c onverts inverse limits in D ′ to the c orr e s p o n ding dir e ct limits in A . Pr o of. 1. Immediate from Theorem 4.2.1 ( 1 ). 2. Immediate from Theorem 4.2.1 (2). 3. Immediate from Corollary 4.2 .2 ( 1 ). 4. Immediate from Corollary 4.2 .2 ( 2 ). 5. Immediate from Prop osition 1.2.1; note that D M ef f g m is sk eletally small. Here in order to prov e that H con v erts ho moto p y limits in to direct limits we use part I2 of lo c. cit. and Prop osition 3.1.1 ( 7 ). R emark 4.3.2 . 1. In the setting of assertion 5 we will call H an extende d cohomology theory . Note that f or H ′ = D M ef f g m ( − , Y ) , Y ∈ O bj DM ef f g m , w e ha v e H = D ( − , Y ) ; see ( 4 ). 2. Now recall that for any pro-sc heme Z , any i ≥ 0 , M g m ( Z )( i ) (b y deﬁnition) could b e presen ted as a coun table homoto py limit of g eome tric motiv es. Moreo v er, the same is true for all small coun table products of M g m ( Z l )( i ) . Hence if H is extended, then the cohomology of Q M g m ( Z l )( i ) is the corresp onding direct limit; this coincides with the deﬁnition giv en b y (29) (cf. Remark 3.2.1). In particular, one can a pply the results of Prop osition 4 .3 .1 to the usual étale cohomology of pro-sc hemes men tioned (with v alues in Ab or in some category of Galois mo dules). 3. If H ′ is also a tensor functor (i.e. it conv erts tensor pro duct in D M ef f g m in to tensor pro ducts in D ( A ) ), then certainly the cohomology of M g m (Sp ec K ′ )( r )[ r ] is the correspo ndin g tensor pro duct of H ∗ ( M g m (Sp ec K ′ )) with H ∗ ( Z ( r )[ r ]) . Note that the latter one is a retract of H ∗ ( G r m ) ; we obtain the T ate t wist f o r cohomology this w a y . 74 4.4 Coniv eau sp ectral se quenc es for cohom ology of (co)motiv es Let H : D op s → A b e a cohomological functor, X ∈ O bj D s . Prop osition 4.4.1. 1. Any choic e of a weight s p e ctr al se quenc e T ( H , X ) (se e The or em 2.4.2) c orr es p o n ding to the Gersten weight structur e w is c anonic al and D s -functorial in X s tarting fr o m E 2 . 2. T ( H , X ) c onv e r ges to H ( X ) . 3. L et H b e an extende d the ory (se e R emark 4.3.2), X = M g m ( Z ) for Z ∈ S mV ar . Then any choic e of T ( H , X ) starting fr om E 2 is c anonic al ly isomorphic to the class i c al c onive au sp e ctr a l se quenc e (c onver ging to the H - c ohomol o gy of Z ; s e e §1 of [CHK97]). Pr o of. 1. This is just a partial case of Theorem 2.4.2 ( I). 2. Immediate since w is b ounded; see part I2 of lo c. cit. 3. Recall that in the pro of of Corollary 3.6.2 a P ostnik ov to we r P o ( X ) fo r X w as obtained from certain ’geometric’ P ostnik o v tow ers (in j ( C b ( S mC o r )) ) b y passing to the homotop y limit. No w, the coniv eau sp ectral sequence (for the H -cohomology of Z ) in §1.2 of [CHK97] was constructed by applying H to the same geometric tow ers and then passing to the inductiv e limit (in A ). F urthermore, Remark 4.3.2(2) yields that the latter limit is (naturally) isomorphic to the sp ectral sequenc e obtained via H from P o ( X ) . Next, since P o ( X ) is a weigh t P ostnik ov tow er for X (see Prop osition 4.1.1(5)), we obtain that the latter sp ectral sequence is one of the p ossible choic es for T ( H , X ) . Lastly , asse rtion 1 yields that all other p ossible T ( H , X ) (they dep end on the ch oice of a w eigh t Pos tnik ov tow er fo r X ) starting from E 2 are also canonically isomorphic to the classical coniv eau sp ectral sequence mentione d. R emark 4.4.2 . 1. Hence w e prov ed ( in particular) that classical coniv eau sp ec tral sequences (f or cohomology theories that can b e f actorized through motiv es; this includes étale and singular cohomology o f smo oth v ar ieties) are D M ef f g m -functorial (starting f rom E 2 ); w e also obtain suc h a functoriality for the coniv eau ﬁltration for cohomology! These facts are far f rom b eing o b vious from the usual deﬁnition of the coniv eau ﬁltration and sp ectral sequ ences, and seem to b e new (in the general case). So, w e justiﬁed the title of the pap er. W e also obtain certain coniv eau sp ectral sequenc es for cohomology of singular v arieties (for cohomology theories that can b e factorized through D M ef f g m ; in the case c har k > 0 one also needs rational co eﬃcie n ts here). 75 2. Assertion 3 of the prop osition yields a nice reason to call (an y c hoice of ) T ( H , X ) a c oni v e au sp e ctr al s e q uen c e (for a g ene ral H , A , and X ∈ O bj D s ); this will also distinguish (this vers ion of ) T from w eigh t sp ectral sequences correspo ndin g to other w eigh t structures. W e will give more justiﬁcation for this term in Remark 4.5.4 b elo w. So, the corresp onding ﬁltration could b e called the (generalized) coniv eau ﬁltration. 4.5 An extension of results of Blo c h and Ogus No w w e wan t to relate coniv eau spectral sequences with the homotopy t - structure (in D M ef f − ). This w o uld b e a v ast extension of the seminal results of §6 of [BOg94] (i.e. of the calculation b y Blo c h and Ogus of the E 2 -terms of coniv eau sp ectral sequences ) and of §6 of [Deg09]. W e should relate t (for D M ef f − ) and w ; it turns out that they a r e orthog- onal with respect to a certain quite natural nice dualit y . Prop osition 4.5.1. F or any Y ∈ O bj D M ef f − we extend H ′ = D M ef f − ( − , Y ) fr om D M ef f g m to D ⊃ D s by the metho d of Pr op osition 1.2 . 1 ; we deﬁne Φ( X, Y ) = H ( X ) . The n the fo l lowing statements a r e valid. 1. Φ is a nic e d uality (se e Deﬁnition 2.5.1). 2 w i s left ortho gonal to the homotopy t -stru ctur e t (on D M ef f − ) with r esp e ct to Φ . 3. Φ( − , Y ) c onve rts homotopy limits (in D ′ ) into dir e c t li m its in Ab . Pr o of. 1. By Prop osition 2.5.6(1), the restriction of Φ to D M ef f g m op × D M ef f − is a nice duality . It remains to apply part 3 o f lo c. cit. 2. In the notation of Prop osition 2.5.3, w e tak e fo r D the set of all small pro ducts Q l ∈ L M g m (Sp ec K l )( n l )[ n l ] ∈ O bj D s ; here M g m (Sp ec K l ) denote the comotive s of (sp ectra of ) some function ﬁelds o v er k , n l ≥ 0 and the transcendenc e degrees of K l /k are b ounded (cf. §4.1). Then D , Φ satisfy the assumptions of the prop osition by Prop osition 3.4 .1 (2) (see also R emark 4.3.2(2)). 3. Immediate from Prop osition 4.3.1(3). R emark 4.5.2 . 1. Supp ose that we ha v e an inductiv e family Y i ∈ O bj DM ef f − connected b y a compatible family of morphisms with some Y ∈ D M ef f − suc h that: fo r an y Z ∈ O bj D M ef f g m w e ha v e DM ef f − ( Z , Y ) ∼ = lim − → D M ef f − ( Z , Y i ) 76 (via these morphisms Y i → Y ). In suc h a situation it is reasonable to call Y a homotopy colimit of Y i . The deﬁnition of Φ in the pro p osition easily implies: for any X ∈ O bj D w e ha ve Φ( X , Y ) = lim − → Φ( X, Y i ) . So, one may sa y that all ob jects of D are ’compact with resp ec t to Φ ’, whereas part 3 of the prop osition yields that all ob jects of D M ef f − are ’co compact with resp ect to Φ ’. No t e that no analogues of these nice prop erties can hold in the case of an a djacen t w eigh t and t -structure (deﬁned on a single triangulated category). 2. Now , w e could ha ve replaced D M ef f g m b y DM g m ev erywhere in the ’axiomatics’ of D (in Prop osition 3.1.1). Then the corresp onding categor y D g m could b e used fo r our purp oses (instead of D ), since our argumen ts work for it a lso. Note that w e can extend Φ to a nice duality D op g m × D M ef f − → Ab ; to this end it suﬃc es for Y ∈ O bj D M ef f − to extend H ′ to D M g m in the fo llowing wa y: H ′ ( X ( − n )) = D M ef f − ( X , Y ( n )) for X ∈ O bj DM ef f g m ⊂ O bj D M g m , n ≥ 0 . Moreo v er, the metho ds of §5.4.3 allow to deﬁne a n in v ertible T ate twis t functor on D g m . Corollary 4.5.3. 1. If H is r epr esente d by a Y ∈ O bj D M ef f − (via our Φ ) then for a (c o)motif X our c onive au sp e ctr al se quenc e T ( H , X ) s tarting fr om E 2 c ould b e natur al ly expr esse d in terms of the c oho m olo gy of X with c o e ﬃ c ients in t -trunc ations of Y (as in The or em 2.6.1) . In p articular, the c onive au ﬁltr ation for H ∗ ( X ) c ould b e desc rib e d a s in p art 2 of lo c. cit. 2. F or U ∈ O bj D M ef f g m , i ∈ Z , we have U ∈ D w ≤ i s ⇐ ⇒ U ∈ D M ef f − t ≤ i . Pr o of. 1. Immediate from Prop osition 4.5.1. 2. By Theorem 2.2.1(20), we should ch ec k whether Z ⊥ U for any Z = Q l ∈ L M g m (Sp ec K l )( n l )[ n l + r ] , where K l are function ﬁelds o v er k , n l ≥ 0 and the transcendence degrees of K l /k are b ounded, r > 0 (see Prop o- sition 4.1.1(2 ) ). Moreo v er, since U is co compact in D , it suﬃces to consider Z = M g m (Sp ec K ′ )( n )[ n + r ] ( K ′ /k is a function ﬁeld, n ≥ 0 ). Lastly , Corol- lary 4.2.2(2) reduces the situation to the case Z = M g m (Sp ec K ) ( K /k is a function ﬁeld). Hence (25) implies: U ∈ D w ≤ i s whenev er f o r an y j > i , an y f unc tion ﬁeld K/ k , the stalk of U t = j at K is zero. No w, if U ∈ D M ef f − t ≤ i then U t = j = 0 for all j > i ; hence all stalks o f U t = j are zero. Conv ersely , if all stalks of U t = j at function ﬁelds a r e zero, then Corollary 4.19 of [V o e00b] yields U t = j = 0 (see also Corollary 4.20 of lo c. cit.); if U t = j = 0 for a ll j > i then U ∈ D M ef f − t ≤ i . 77 R emark 4.5.4 . 1. Our comparison statemen t is true for Y -cohomology of a n arbitrary X ∈ O bj DM ef f g m ; this extends t o motiv es Theorem 6.4 of [Deg09] (whereas the latter essen tially extends the results o f §6 of [BOg94]). W e obtain one more reason to call T (in this case) the coniv eau sp ectral sequence for (cohomology o f ) motiv es. Note also that the metho ds of Deglise do not (seem to) yield the motivic functorialit y of the isomorphism in question (cf. Remark 2.6.2). 2. If Y ∈ O bj H I , then E 2 ( T ) yields the Gersten resolution for Y (when X v aries); this is why w e called w the G ersten w eigh t structure. 3. Now , let Y represen t étale cohomolog y with co eﬃcien ts in Z /l Z , l is prime to c har k ( Y is actually unbounded from ab o v e, y et this is not imp or- tan t). Then the t -truncations of Y represen t Z /l Z -motivic cohomology by the (recen tly pro v ed) Beilinson-Lic h ten baum conjecture (see [V o e08]; this pap er is not published at the momen t). Hence Prop osition 2.5.4(1) yields some new form ulae for Z /l Z -motivic cohomology of X and for the ’diﬀerence’ b et w een étale and motivic cohomology . Note also that the virtual t -truncations (men- tioned in lo c. cit.) ar e exactly t he D 2 -terms of the alternativ e exact couple for T ( H , X ) and for the v ersion of the exact couple used in the curren t pap er respectiv ely (i.e. w e consider exact couples coming from the t w o p ossible v ersions for a w eigh t P ostnik ov tow er for X , as describ ed in Remark 2.1.3 ). See also §7.5 of [Bon10] fo r more explicit results of this sort. It could also b e in teresting to study conive au sp ectral sequenc es for singular cohomology; this could yield a certain t heory of ’motiv es up to algebraic equiv alence’; see Remark 7.5.3(3 ) o f lo c. cit. for more details. 5. Assertion 2 of the corollary yields that D w ≤ 0 s ∩ O bj D M ef f g m is large enough to reco v er w (in a certain sense); in particular, this assertion is similar to the deﬁnition of adjacen t structures (see Remark 2.5.7). In con trast, D w ≥ 0 s ∩ O bj D M ef f g m seems to b e to o small. 4.6 Base ﬁeld c hange for coniv eau sp ectral sequences; functorialit y for an uncoun table k It can b e easily seen (and w ell-kno wn) that for any p erfect ﬁeld extension l /k there exist an extension of scalars functor D M ef f g m k → D M ef f g m l compatible with the extension of scalars for smo oth v arieties (a nd for K b ( S mC o r ) ). In 5.4.2 b elo w w e will pro v e that this functor could b e expanded to a functor 78 Ext l/k : D k → D l that sends M g m ,k ( X ) to M g m ,l ( X l ) for a pro-sch eme X/k ; this extension pro cedure is functorial with resp ec t to embeddings of base ﬁelds. Moreo v er, Ext l/k maps D s k in to D s l . Note the existence of base c hange for comotiv es do es not follo w from the prop erties of D listed in Prop osition 3.1.1; yet one can deﬁne base c hange fo r our mo del of comotives (described in §5 b elo w) and (probably) for an y o ther p ossibl e reasonable v ersion o f D . No w w e pro v e that ba se c hange for comotiv es yields base c hange for coniv eau sp ectral sequences; it also allo ws to prov e that these sp ectral se- quences are motivically functorial for not necessary countable base ﬁelds. In order to mak e the limit in Prop osition 4.6.1(2) b elo w we ll-deﬁned, we assume that f or any X ∈ O bj DM ef f g m there is a ﬁxed represen tativ e Y , Z, p c hosen, where: Z , Y ∈ C b ( S mC o r ) , M g m ( Y ) ∼ = M g m ( Z ) , p ∈ C b ( S mC o r )( Y , Z ) yields a direct summand of M g m ( Y ) in D M ef f g m that is isomorphic to X . W e also assume that all the comp onen ts of ( X, Y , p ) hav e ﬁxed expressions in terms of algebraic equations ov er k ; so one may sp eak ab out ﬁelds of deﬁni- tion for X . Prop osition 4.6.1. L et l b e a p erfe ct ﬁeld, H : D l → A b e any c ohomolo gic al functor (for an ab elian A ). F or any p erfe ct k ⊂ l we denote H ◦ Ex t l/k : D k → A by H k . 1. L et l b e c ountable. Then for any X ∈ O bj D k the metho d of Pr op osition 2.7.3(II) yield s some morphis m N l/k : T w k ( H k , X ) → T w l ( H , Ext l/k ( X )) ; this morphism is unique a nd D k -functorial in X starting f r om E 2 . The c orr esp ondenc e ( l , k ) 7→ N l/k is asso ciative with r esp e ct to extension s of c ountable ﬁeld s (starting fr o m E 2 ); cf. p art I3 of lo c. cit. 2. L et l b e a not (ne c essarily) c ountable p erfe ct ﬁeld, let A satisfy AB5. F or X ∈ O bj D M ef f g m l we deﬁ ne T w ( H , X ) = lim − → k T w k ( H k , X k ) . Her e we take the lim it with r esp e ct to al l p erfe c t k ⊂ l such that k is c ountable, X is deﬁne d over k ; the c onne cting morphisms ar e given by the maps N − / − mentione d in asse rtion 1; we start our sp e ctr al se quenc es fr om E 2 . Then T w ( H , X ) is a wel l-de ﬁ ne d sp e ctr al se quenc e that is D M ef f g m l -functorial in X . 3. If X = M g m ,l ( Z ) , Z ∈ S mV ar , H is as an extende d the ory, and A sat- isﬁes AB5, the sp e ctr al se q uen c e given by the pr evious assertion is c anonic al l y isomorphic to the classic al c onive au sp e ctr al se quenc e (for ( H , Z ) ; c o n sider e d starting fr om E 2 ). Pr o of. 1. By Prop osition 2.7.3(I I) it suﬃc es to c heck that Ext l/k is left w eigh t-exact (with resp ect to w eigh t structures in question). W e tak e D b e- ing the class of all small pro ducts Q l ∈ L M g m (Sp ec K l ) , where M g m (Sp ec K l ) 79 denote t he comotives of (sp ec tra o f ) function ﬁelds ov er k of b ounded tran- scendenc e degree. Proposition 4.1.1 and Corollary 4.2.2(2) yield that any X ∈ D s w =0 k is a retract of some eleme nt of D . It suﬃces to ch ec k that for an y X = Q l ∈ L M g m ,k ( K l ) w e ha v e Ext l/k X ∈ D s w l ≤ 0 l ; here w e recall that w k is b ounded and apply Lemma 2.7.5. No w, X is the comotif of a certain pro-sch eme, hence the same is true for Ext l/k X . It remains to apply Prop osition 4 .1 .1(5). 2. By the asso ciativit y statemen t in the previous assertion, the limit is w ell-deﬁned. Since A satisﬁes AB5, w e obtain a spectral sequence indeed. Since w e ha v e k -motivic functoriality of coniv eau sp ectral sequences o v er each k , w e obtain l -motivic functoriality in the limit. 3. Again (as in the pro of of Prop osition 4 .4.1(3)) w e recall that the classical coniv eau sp ectral sequence for this case is deﬁned b y applying H to ’geometric’ P ostnik ov tow ers (coming from elemen ts of L as in the pro of of Corollary 3.6.2 ) and then passing to the limit (in A ) with resp ect to L . Our assertion follows easily , sinc e eac h l ∈ L is deﬁned ov er some p erfect coun table k ⊂ l ; the limit of the sp ectral sequences with respect to the subset of L deﬁned o v er a ﬁxed k is exactly T w k ( H k , X k ) since H sends ho moto p y limits to inductiv e limits in A (b eing an extended theory). Here w e certainly use the functorialit y o f T starting from E 2 . R emark 4.6.2 . 1. F or a general X ∈ O bj D M ef f g m w e only ha v e a canonical c hoice of base c hange maps (for T ( H k l , X ) ) starting from E 2 ; this is wh y w e start our sp ectral sequence from the E 2 -lev el. 2. Assertion 2 of the pro position is also v alid f or any comotif deﬁned o v er a (p erfect) coun t a ble subﬁeld of l . Unfortunately , this do es not seem to include the comotiv es of function ﬁelds ov er l (of p ositiv e transcendence degrees, if l is not coun table). 4.7 The Cho w w eigh t structure for D Till the end of the section, w e will either assume that c har k = 0 , or that w e deal with motive s, comotiv es, and cohomology with rational co eﬃcien ts (w e will use the same notation fo r motive s with integral and rational co eﬃcie nts ; cf. §6.3 b elo w). W e pro v e that D supports a w eigh t structure that extends the Cho w w eigh t structure of D M ef f g m (see §6.5 and Remark 6 .6 .1 of [Bon10], and a ls o 80 [Bon09]). In this subsection we do not require k to b e coun table. Prop osition 4.7.1. 1. Ther e exists a Chow weight structur e on D M ef f g m that is uniquely char acterize d by the c ondition that al l M g m ( P ) for P ∈ S mP r V ar b elong to its he art; it c ould b e extende d to a weigh t structur e w C how on D . 2. The he art of w C how is the c ate gory H C how of arbitr ary smal l pr o ducts of (eﬀe ctive) Chow motives. 3. W e h a ve X ∈ D w C how ≥ 0 if and only if D ( X , Y [ i ]) = { 0 } for any Y ∈ O bj C how ef f , i > 0 . 4. Ther e ex ists a t -structur e t C how on D that is right adjac ent to w C how (se e R em a rk 2.5.7). Its he art is the opp osite c ate gory to C how ef f ∗ (i.e. it is e quivalent to (AddF un( C how ef f , Ab )) op ). 5. w C how r esp e cts pr o ducts i.e. X i ∈ D w C how ≤ 0 = ⇒ Q X i ∈ D w C how ≤ 0 and X i ∈ D w C how ≥ 0 = ⇒ Q X i ∈ D w C how ≥ 0 . 6. F or Q X i ther e exists a weight de c omp osition: Q X i → Q X w ≤ 0 i → Q X w ≥ 1 i . 7. If H : D → A is an extend e d the o ry, then the functor that send s X to the derive d exact c ouple for T w C how ( H , X ) (se e The o r em 2.4.2 ) c on verts al l smal l pr o ducts into di r e ct sums. Pr o of. 1. It was pro v ed in (Prop osition 6.5.3 and Remark 6.6.1 of ) [Bon10] that there exists a unique w eigh t structure w ′ C how on D M ef f g m suc h that M g m ( P ) ∈ D w ′ C how =0 for all P ∈ S mP r V ar . Moreo ver, the heart of this structure is exactly C how ef f ⊂ D M ef f g m . No w, D M ef f g m is g ene rated b y C how ef f . It easily fo llows that { M g m ( P ) , P ∈ S mP r V ar } w eakly cogenerates D . Then the dual (see Theorem 2.2.1(1)) of Theorem 4.5.2(I2) of [Bon10] yields that w ′ C how could b e extended to a w eigh t structure w C how for D . Moreov er, the dual to part I I1 of lo c. cit. yields that for this extension w e ha v e: H w C how is the idemp oten t completion of H C how . 2. It remains to prov e that H C how is idemp oten t complete. This is ob vious since C how ef f is. 3. This is just the dual of (27) in lo c. cit. 4. The dual statemen t to part I2 of lo c. cit. (cf. Remark 1.1.3(1 )) yields the existence of t C how . Applying the dual of Theorem 4.5.2(I I1) of [Bon10] w e obtain for the heart o f t : H t C how ∼ = ( C how ef f ∗ ) op . 5. Theorem 2.2.1(2) easily yields that D w C how ≤ 0 is stable with respect to pro ducts. The stabilit y of D w C how ≥ 0 with resp ect to pro ducts follows from assertion 3; here we recall that all ob jects of C how ef f are co compact in D . 81 6. Immediate from the previous assertion; note that an y small pro duct of distinguished triangles is distinguished (see Remark 1 .2 .2 of [Nee01]). 7. Since H is extended, it con v erts pro ducts in D in to direct sums in A . Hen ce for an y X i ∈ O bj D there exist a c hoice of exact couples for the correspo nding w eigh t sp ectral sequen ces for X i and Q X i that re- sp ec ts pro duc ts i.e such that D pq 1 T w C how ( H , Q X i ) ∼ = L i D pq 1 T w C how ( H , X i ) and E pq 1 T w C how ( H , Q X i ) ∼ = L i E pq 1 T w C how ( H , X i ) (for all p.q ∈ Z ; this iso- morphism is also compatible with the connecting morphisms of couples). Since A satisﬁes AB5, w e obtain t he isomorphism desired for D 2 and E 2 - terms (note that those are uniquely determined b y H and X ). R emark 4.7.2 . 1. In Remark 2.4.3 of [Bon10] it w as sho wn that we ight sp ec- tral seque nces corresp onding to the Cho w w eigh t structure are isomorphic to the class ical (i.e. Deligne’s) w eigh t sp ectral sequences when the latter are deﬁned (i.e. for singular or étale cohomology of v arieties). Y et in order to sp ecify the c hoice of a we ight structure here we will call these sp ectral sequenc es Ch o w-weight o ne s. 2. All the assertions of the Prop osition could b e extended to arbitrary triangulated categories with negativ e families of co compact we ak cogenera- tors (sometimes one should also demand all pro ducts to exist; in a ss ertion 7 w e only need H to con v ert all pro ducts in to direct sums). 3. Since (eﬀectiv e) Cho w motive s are co compact in D , H w C how is the cat- egory of ’f o rmal pro ducts’ of eﬀectiv e Cho w motiv es i.e. D ( Q l ∈ L X l , Q i ∈ I Y i ) = Q i ∈ I ( ⊕ l ∈ L C how ef f ( X l , Y i )) for X l , Y l ∈ O bj C how ef f ⊂ O bj D (cf. Remark 4.5.3(2) of [Bon1 0 ]). 4. Recall (see §7.1 of ibid.) that D M ef f − suppo rts (adjacen t) Chow wei ght and t -structures (w e will denote them b y w ′ C how and t ′ C how , resp ectiv ely). O ne could also c hec k that these structures are righ t orthog onal to the corresp ond- ing Cho w structures for D . Hence, applying Prop osition 2.5.4(1) rep eatedly one could relate the comp ositions of truncations (on D s ⊂ D ) via w and via t C how (resp. via w and via w C how ) with truncations via t and via w ′ C how (resp. via t a nd via t ′ C how ) on D M ef f − ; cf. §8.3 of [Bon10]. One could also a pply w C how -truncations and then w -t runcations (i.e. comp ose truncations in the opp osite order) when starting from an ob ject of D M ef f g m . Recall also that truncations via t C how (and their comp ositions with t -truncations) are related with unramiﬁed cohomology; see Remark 7.6.2 of ibid. 82 4.8 Comparing Cho w-w eigh t and coniv eau sp ectral se- quences No w we prov e that Cho w-w eigh t and coniv eau sp ectral sequenc es are natu- rally isomorphic for birational cohomology theories. Prop osition 4.8.1. 1. w C how for D do minates w (for D s ) in the sense of §2.7. 2. L et H : D M ef f g m → A b e an extende d c oho m olo gy the ory in the sense of R emark 4.3.2; supp ose that H is bir ational i . e . that H ( M g m ( P )(1)[ i ]) = 0 for al l P ∈ S mP r V ar , i ∈ Z . Then for any X ∈ O bj D s the Chow- weight s p e ctr al se quenc e T w C how ( H , X ) (c orr esp onding to w C how ) is natur al ly isomorphic starting fr om E 2 to (our) c onive au sp e ctr al s e quenc e T w ( H , X ) via the c omp arison morphism M giv en by Pr op osition 2.7.3 (I1). Pr o of. 1. Let D b e the class of all coun table pro ducts Q l ∈ L M g m (Sp ec K l ) , where M g m (Sp ec K l ) denote the comotives of (sp ectra of ) function ﬁelds ov er k of b ounded transcendenc e degree. Prop osition 4.1.1 and Corollary 4.2.2(2 ) yield that an y X ∈ D w =0 s is a retract of some elemen t of D . It suﬃces to c hec k that any X = Q l ∈ L M g m (Sp ec K l ) b elongs to D w C how ≥ 0 ; here we recall that w is b ounded a nd apply Lemma 2.7.5. By Prop osition 4.7.1(5), w e can assume that L consists of a single elemen t. In this case w e ha v e D ( M g m (Sp ec K l ) , M g m ( P )[ i ]) = 0 (this is a trivial case of Prop osition 3.5 .1 ); hence lo c. cit. yields the result. 2. W e take the same D and X as ab o v e. Let c ha r k = 0 . W e c ho ose P l ∈ S mP r V ar suc h that K l are their function ﬁelds. Since all M g m ( P l ) are co compact in D , we ha v e a natural morphism X → Q M g m ( P l ) . By Prop osition 2 .7.3(I2), it suﬃces to c hec k that Cone( X → Q M g m ( P l )) ∈ D w C how ≥ 0 , H ( X ) ∼ = H ( Q M g m ( P l )) , and E ∗∗ 2 T w C how ( H , Cone( X → Q M g m ( P l ))) = 0 . By Prop osition 4.7 .1(7) w e obtain: it suﬃces a g ain t o v erify these state- men ts in the case when L consists of a single elemen t. No w, w e ha v e Sp ec ( K l ) = lim ← − M g m ( U ) for U ∈ S mV ar, k ( U ) = K l . Therefore (27) yields: it suﬃces to v erify assertions required for Z = M g m ( U → P ) instead, where U ∈ S mV ar , U is o p en in P ∈ S mP r V ar . The Gysin distinguished triangle f or V o ev o dsky’s motiv es (see Prop osi- tion 2.4.5 of [Deg08a]) easily yields b y induction that Z ∈ O bj D M ef f g m (1) . Since C how ef f is − ⊗ Z (1)[2] - stable, w e obtain that there exists a w C how - P ostnik ov tow er for Z suc h that all of its terms are divisible b y Z (1) ; t his 83 yields the v a nis hing of E ∗∗ 2 T w C how ( H , Z ) . Lastly , the f act that Z ∈ D M ef f g m w ′ C how ≥ 0 w as (essen tially) prov ed b y easy induction (using the Gysin triangle) in the pro of of Theorem 6.2.1 of [Bon09]. In the case c har k > 0 , de Jong’s alterations a llow us to replace M g m ( P l ) in the reasoning ab o v e by some Cho w motiv es (with ra t io na l co eﬃcie n ts); see App endix B of [HuK06]; w e will not write down the details here. R emark 4.8.2 . 1. Hence for any H : D → A , X ∈ O bj D s there exists a comparison morphism M : T w C how ( H , X ) → T w ( H ′ , X ) , where H ′ is the restriction o f H to D s . It is canonical starting fr o m E 2 ; see Proposition 2.7.3(I I). 2. Assertion 2 is not ve ry actual for cohomology o f smo oth v arieties since an y Z ∈ S mP r V ar is birationally isomorphic to P ∈ S mP r V ar (a t least fo r c har k = 0 ). Y et the statemen t b ecomes more in teresting when applied for X = M c g m ( Z ) . 4.9 Birational motiv es; constructing the Gersten weigh t structure b y gluing; o ther p ossible w eigh t structures An alternativ e wa y to pro v e Proposition 4.8.1(2) is to consider (follow ing [KaS02]) the category of bir ational c omotives . It satisﬁes the follow ing prop- erties: (i) All birational cohomolo g y theories factorize through it. (ii) Cho w and Gersten weigh t structures induce t he same w eigh t structure on it (see Deﬁnition 2.7.1(4)). (iii) More generally , for any n ≥ 0 Cho w and Gersten w eigh t structures induce w eight structures on the lo calizations D ( n ) / D ( n + 1) ∼ = D bir (w e call these lo calizations slic es ) that diﬀer only b y a shift. Moreo v er, o ne could ’almost reco v er’ orig inal Cho w and G ers ten w eigh t structures starting from this single w eigh t structure. No w w e describ e the constructions and facts men tioned in more detail. W e will be rather sk etc h y here, since we will not use the results of this subsection elsewhere in the pap er. P ossibly , the details will b e written do wn in another pap er. As w e will sho w in §5.4.3 b elo w, the T a te twis t functor could b e extended (as an exact functor) from D M ef f g m to D ; this functor is compatible with (small) pro ducts. 84 Prop osition 4.9.1. I The functor − ⊗ Z (1)[1] is weight-exact with r es p e ct to w o n D s ; − ⊗ Z (1)[2] is weight-exact with r esp e ct to w C how on D (w e wil l say that w is − ⊗ Z (1)[1] -stable, an d w C how is − ⊗ Z (1)[2] -stable). II L et D bir denote the lo c alization of D by D ( 1 ) ; B i s the lo c ali z ation functor. W e denote B ( D s ) by D s,bir . 1. w C how induc es a weight structur e w ′ bir on D bir . Beside s, w ind uc es a weight structur e w bir on D s,bir . 2. W e have D w bir ≤ 0 s,bir ⊂ D w ′ bir ≤ 0 bir , D w bir ≥ 0 s,bir ⊂ D w ′ bir ≥ 0 bir (i.e. the emb e dding ( D s,bir , w bir ) → ( D bir , w ′ bir ) is weight-exact). 3. F or any pr o-scheme U we have B ( M g m ( U )) ∈ D w bir =0 s,bir . Pr o of. I This is easy , since the functors mentione d obvi ously map the corre- sp onding hearts (of w eigh t structures) into themselv es. I I 1. By assertion I, w C how induces a weigh t structure on D (1) (i.e. D (1) is a tria ngulat ed category , O bj D ( 1 ) ∩ D w C how ≤ 0 and O bj D (1) ∩ D w C how ≥ 0 yield a w eigh t structure on it). Hence b y Prop osition 8 .1.1(1) of [Bon10] w e obtain existence (and uniqueness) o f w ′ bir . The same argumen t a lso implies the existence of some w bir on D s,bir . 2. No w w e compare w bir with w ′ bir . Since w is b ounded, w bir also is (see lo c. cit.). Hence it suﬃces to c hec k that H w bir ⊂ H w ′ bir (see Theorem 2.2.1(13)). Moreo v er, it suﬃces to che c k that for X = Q l ∈ L M g m (Sp ec K l ) we ha v e B ( X ) ∈ D w ′ bir =0 bir (since D w ′ bir =0 bir is Karoubi-closed in D bir , here w e also ap- ply Prop osition 4.7.1(2)). As in the pro of of Prop osition 4 . 8 .1(2), w e will consider the case c ha r k = 0 ; t he case c har k = p is t r eated similarly . Then w e ch o ose P l ∈ S mP r V ar suc h that K l are their function ﬁelds; w e hav e a natural morphis m X → Q M g m ( P l ) . It remains to c hec k that Cone( X → Q M g m ( P l )) ∈ D s (1) . No w, since D s (1) and the class of dis- tinguished t ria ngles are closed with resp ect to small pro ducts, it suﬃces to consider the case when L consists of a single elemen t. In this case the state- men t is immediate f ro m Corollar y 3.6.2. 3. Immediate from Corollary 3.6 .2 . R emark 4.9.2 . 1. Assertion I I easily implies Prop osition 4.8.1(2). Indeed, any extended birational H (as in lo c. cit.) can b e f actorized as G ◦ B for a cohomological G : D bir → A . Since B is w eigh t-exact with r esp ect to w C how (and its restriction to D s is w eigh t-exact with resp ect to w ), (t he 85 trivial case of ) Prop osition 2.7.3(I2 ) implies t ha t for an y X ∈ O bj D (a n y c hoice) of T w ′ bir ( G, B ( X )) is natura lly isomorphi c starting from E 2 to an y c hoice of T w C how ( H , X ) ; for an y X ∈ O bj D s (an y ch oice) of T w bir ( G, B ( X )) is naturally isomorphic starting from E 2 to an y choic e of T w ( H , X ) . It is also easily seen that the isomorphism T w C how ( H , X ) → T w ( H , X ) is compatible with t he comparison morphism M (see lo c. cit.). 2. The pro of of existence o f w bir and of assertion 3 w orks with in tegral co eﬃcie nts ev en if c har k > 0 . Hence w e obtain that that the category imag e B ( M g m ( U )) , U ∈ S mV ar , is negativ e. W e can apply this statemen t in C b e- ing the idemp oten t completion of B ( D M ef f g m ) i.e. in the category o f biratio nal comotiv es. Hence Theorem 2.2.1(18 ) yields: there exists a w eigh t structure for C whose heart is the category of birational Cho w motiv es (deﬁn ed as in §5 o f [K a S02 ]). Note also tha t one can pass to the inductiv e limit with respect to base c hange in this statemen t (cf. §4.6); hence one do es not need to require k to b e coun table. No w we explain that w and w C how could b e ’almost recov ered’ from ( D bir , w ′ bir ) . Exactly the same reasoning as ab o v e show s that for an y n > 0 the lo calization of D b y D ( n ) could b e endo w ed with a w eigh t structure w ′ n compatible with w C how , whereas the lo calization of D s b y D s ( n ) could b e endo w ed with a w eigh t structure w n compatible with w . Next, w e hav e a short exact sequence of triangulated categories D / D ( n ) i ∗ → D / D ( n + 1) j ∗ → D bir . Here the notation for functors comes from the ’classical’ gluing data setting (cf. §8.2 o f [Bo n1 0 ]); i ∗ could b e giv en b y − ⊗ Z (1)[ s ] for an y s ∈ Z , j ∗ is just the lo calization. Now, if we c ho ose s = 2 then i ∗ is w eight-exact with respect to w ′ n and w ′ n +1 ; if w e c ho ose s = 1 then the restriction of i ∗ to D s / D s ( n ) is w eigh t-exact with respect to w n and w n +1 . Next, an argumen t similar to the one used in §8.2 of [Bon10] sho ws: for an y short exact sequence D i ∗ → C j ∗ → E of triangulated categories, if D and E are endow ed with wei ght structures, then there exist at most one w eigh t structure on C suc h that b oth i ∗ and j ∗ are w eigh t-exact (see also Lemma 4.6 of [Bei98] for t he pro of of a similar statemen t for t -structures). Hence one can reco v er w n and w ′ n from (copies of ) w ′ bir ; the main diﬀerence b et wee n them is that the ﬁrst one is − ⊗ Z (1)[1] -stable, whereas the second one is − ⊗ Z (1)[2] -stable. It is quite amazing that weigh t structures corresp onding to sp ec tral sequence s of quite distinct geometric orig in diﬀer just b y [1] here! If one calls the ﬁltration of D b y D ( n ) the slic e ﬁl tr ation (this term w as already used b y A. Hub er, B. Kahn, M. Levine, V. V o ev o dsky , and other 86 authors for ot her ’motivic’ categories), then one may sa y that w n and w ′ n could b e recov ered f r o m slices; the diﬀerence b et w een them is ’ho w w e shift the slices’. Moreo v er, Theorem 8.2 .3 o f [Bon10] sh ows : if b oth adjoints to i ∗ and j ∗ exist, then one can use this gluing data in o rder to glue (a ny pair) of w eigh t structures fo r D and E in to a w eight structure for C . So, supp ose that w e ha v e a w eigh t structure w n,s for D / D ( n ) that is − ⊗ (1)[ s ] -stable and compatible with w ′ bir on all slices (in the sense described ab ov e; so w ′ n = w n, 2 , w n is the restriction of w n, 1 to D s / D s ( n ) , and all w 1 ,s coincide with w ′ bir ). General homological algebra (see Prop osition 3.3 of [Kra05]) yields that all the adjoints required do exist in our case. Hence one can construct w n +1 ,s for D / D ( n + 1) that satisﬁes similar prop erties. So, w n,s exist for all n > 0 and all s ∈ Z . Hence Gersten and Cho w w eight structures (for D s / D s ( n ) ⊂ D / D ( n ) ) are mem b ers of a rather natural f amily of we ight structures indexed b y a single in tegral par a me ter. It could b e in teresting to study other mem b ers of it (for example, the one that is − ⊗ Z (1) -stable), though p ossibly w ′ n is the only mem b er of this family whose heart is co compactly cogenerated. This approac h could allow constructing w in the case of a not neces- sarily countable k . Note here that the system of D s / D s ( n ) yields a ﬁne appro ximation of D s . Indeed, if X ∈ S mP r V ar , n ≥ dim X , then P oincare dualit y yields: f o r any Y ∈ O bj D M ef f g m w e ha v e D M ef f g m ( Y ( n ) , M g m ( X )) ∼ = D M ef f g m ( Y ⊗ X ( n − dim X )[ − 2 dim X ] , Z ) ; this is zero if n > dim X since Z is a birational motif. Hence (b y Y oneda’s lemma) for an y n > 0 the f ull sub catego r y of D M ef f g m generated by motiv es of v arieties of dimension less than n fully em b eds in to D M ef f g m /D M ef f g m ( n ) ⊂ D / D ( n ) . It follows tha t the restrictions of w n,s to a certain series of (suﬃcien tly small) subcategories of D / D ( n ) are induce d by a single − ⊗ (1) [ s ] -stable w eigh t structure w s for the corresp onding sub category of D . Here for the correspo ndin g sub category of D / D ( n ) (or D ) one can tak e the union of the sub catego r ies of D / D ( n ) (resp. D ) cog enerated (in an a ppropriate sense) b y the comotiv es of (smo oth) v arieties of dimension ≤ r (with r running through all natural n um b ers). Note that this subcatego r y of D contains D M ef f g m . W e also relate brieﬂy our results with the (conjectural) picture fo r t - structures describ ed in [Bei98]. The re another (geometric) ﬁltra tio n f o r mo- tiv es w as considered; this ﬁltrat io n (roughly) diﬀers from the ﬁltration con- sidered ab o v e by (a certain v ersion of ) Poinc are duality . No w, conjecturally the g r n of t he category of birational motive s with rational co eﬃcien ts (cf. §4.2 of ibid.) should b e (the homotopy category of complexes ov er) an ab elian 87 semisimp le category . Hence it supp orts a t -structure whic h is sim ultaneously a we ight structure. This structure should b e the building blo c k of all rele- v an t w eigh t and t -structures for (co)motiv es. Certainly , this picture is quite conjectural at the presen t momen t. R emark 4.9.3 . The author also hop es to carry o ver ( some of ) the results of the curren t pap er to relativ e motive s (i.e. motiv es ov er a base sc heme that is not a ﬁeld), relativ e comotiv es, and their cohomology . One of the p ossible metho ds for this is the usage of gluing of w eigh t structures (see §8.2 of [Bon10], esp eci ally Remark 8 .2.4(3) of lo c. cit.). P o ss ibly for this situation the ’v ersion of D ’ that uses motiv es with compact suppo rt (see §6.4 b elo w) could b e more appropriate. 5 The construction of D and D ′ ; base c hange and T ate t wists No w w e construct our categories D ′ and D using the diﬀeren tial graded categories formalism. In §5.1 w e recall the deﬁnitions of diﬀeren tial graded categories, mo dules o v er them, shifts and cones (of morphisms). In §5.2 we recall main prop erties of the deriv ed category of (mo dules o v er) a diﬀeren tial graded category . In §5.3 w e deﬁne D ′ and D as the categories opp osite to the corresp onding categories of mo dules; then we pro v e that they satisfy the prop erties required. In §5.4 w e use the diﬀeren tial graded mo dules formalism to deﬁne base c hange for motiv es (extension and restriction of scalars). This yields: our results on direct summands of the comotiv es (and cohomolo g y) of function ﬁelds (prov ed ab o v e) could b e carried o ve r to pro-sc hemes obtained from them via base c hange. W e also deﬁne tensoring of comotiv es by motiv es, as w ell as a certain ’co-in ternal Hom’ (i.e. the corresp onding left adjoin t functor to X ⊗ − for X ∈ O bj D M ef f g m ). These results do not require k to b e countable. 5.1 DG-categories and m o dules o v er them W e recall some basic deﬁnitions; cf. [Kel06] and [Dri04]. 88 An a dditiv e category A is called g r a ded if for any P , Q ∈ O bj A there is a canonical decomp osition A ( P , Q ) ∼ = L i A i ( P , Q ) deﬁned; this decom- p osition satisﬁes A i ( ∗ , ∗ ) ◦ A j ( ∗ , ∗ ) ⊂ A i + j ( ∗ , ∗ ) . A diﬀeren tial graded cat- egory (cf. [Dri04]) is a graded category endo w ed with an additiv e o per- ator δ : A i ( P , Q ) → A i +1 ( P , Q ) fo r all i ∈ Z , P , Q ∈ O bj A . δ should satisfy the equalities δ 2 = 0 (so A ( P , Q ) is a complex of ab elian groups); δ ( f ◦ g ) = δ f ◦ g + ( − 1) i f ◦ δ g for a n y P , Q, R ∈ O bj A , f ∈ A i ( Q, R ) , g ∈ A ( P , Q ) . In particular, δ ( id P ) = 0 . W e denote δ restricted to morphisms of degree i by δ i . No w we giv e a simple example of a diﬀeren tia l graded category . F or an additiv e category B w e consider the category B ( B ) whose ob- jects are the same as for C ( B ) whereas for P = ( P i ) , Q = ( Q i ) w e deﬁne B ( B ) i ( P , Q ) = Q j ∈ Z B ( P j , Q i + j ) . Ob viously B ( B ) is a graded category . W e will also consider a full sub category B b ( B ) ⊂ B ( B ) whose ob jects are b ounded complexes . W e set δ f = d Q ◦ f − ( − 1) i f ◦ d P , where f ∈ B i ( P , Q ) , d P and d Q are the diﬀeren tia ls in P and Q . Note that the ke rnel of δ 0 ( P , Q ) coincides with C ( A )( P , Q ) (the morphisms of complexes); the image o f δ − 1 are the morphisms homotopic to 0 . Note also that the oppo site category to a diﬀere ntial graded category b ecomes diﬀeren tia l graded also (with the same gradings and diﬀeren tials) if w e deﬁne f op ◦ g op = ( − 1) pq ( g ◦ f ) op for g , f b eing comp osable homogeneous morphisms of degrees p and q , resp ectiv ely . F or any diﬀeren tial graded A w e deﬁne an additiv e category H ( A ) ( some authors denote it b y H 0 ( A ) ); its ob jects are the same as for A ; its morphisms are deﬁned as H ( A )( P , Q ) = Ker δ 0 A ( P , Q ) / Im δ − 1 A ( P , Q ) . In the case when H ( A ) is t r ia ng ulated (as a full subcategory of the category K ( A ) describ ed b elo w) w e will sa y that A is a (diﬀerential gra ded ) enhan c e- ment for H ( A ) . W e will a lso need Z ( A ) : O bj Z ( A ) = O bj A ; Z ( A )( P , Q ) = K er δ 0 A ( P , Q ) . W e hav e an obvious functor Z ( A ) → H ( A ) . Note that Z ( B ( B )) = C ( B ) ; H ( B ( B )) = K ( B ) . No w we deﬁne (left diﬀeren tial graded) mo dules o v er a small diﬀeren- tial graded category A (cf. §3.1 of [Kel06] o r §14 of [Dri04]): the ob jects DG-Mo d( A ) are those additiv e functors of the underlying additiv e categories 89 A → B ( Ab ) that preserv e gradings and diﬀerentials for morphisms. W e de- ﬁne DG-Mo d( A ) i ( F , G ) as the set of transformations o f additive functors of degree i ; for h ∈ DG -Mo d( A ) i ( F , G ) w e deﬁne δ i ( h ) = d G ◦ f − ( − 1) i f ◦ d F . W e hav e a natura l Y oneda em b edding Y : A op → D G-Mo d ( A ) (one should apply Y oneda’s lemma for the underlyin g additiv e categories); it is easily seen to b e a full em b edding o f diﬀeren tial graded categories. No w w e deﬁne shifts and cones in D G-Mo d ( A ) comp onen tw isely . F or F ∈ O bj D G -Mo d( A ) w e set F [1]( X ) = F ( X )[1] . F or h ∈ Ker δ 0 DG-Mo d( A ) ( F , G ) w e deﬁne the ob j ect Cone( h ) : Cone ( h )( X ) = Cone ( F ( X ) → G ( X )) for all X ∈ O bj A . Note that for A = B ( B ) b oth of these deﬁnitions are compatible with the correspo ndin g notions for complexes (with respect to the Y oneda em b edding). W e hav e a na tura l triangle of morphisms in δ 0 DG-Mo d( A ) : P f → P ′ → Cone( f ) → P [1] . (30) 5.2 The deriv ed category of a diﬀeren tial graded cate- gory W e deﬁne K ( A ) = H (DG-Mo d( A )) . It w as sho wn in §2.2 of [Kel06] that K ( A ) is a triangulated category with resp ect to shifts and cones of morphisms that w ere deﬁned ab ov e (i.e. a triangle is distinguished if it is isomorphic to those of the form (30)). W e will say that f ∈ Ker δ 0 DG-Mo d( A ) ( F , G ) is a quasi-isomorp h ism if for an y X ∈ O bj A it yields an isomorphism F ( X ) → F ( Y ) . W e deﬁne D ( A ) as t he localization of K ( A ) with resp ect to quasi-isomorphisms; so it is a triangulated category . Note that quasi-isomorphisms yield a lo calizing class of morphisms in K ( A ) . Moreov er, the functor X → H 0 ( F ( X )) : K ( A ) → Ab is corepresen ted b y D G-Mo d ( A )( X , − ) ∈ O bj K ( A ) ; hence for any X ∈ O bj A , F ∈ O bj K ( A ) w e hav e D ( A )( Y ( X ) , F ) ∼ = K ( A )( Y ( X ) , F ) . (31) Hence we ha v e an embedding H ( A ) op → D ( A ) . W e deﬁne C ( A ) as Z (DG-Mo d( A )) . It is easily seen that C ( A ) is closed with resp ec t to (small ﬁltere d) direct limits, and lim − → F l is giv en by X → lim − → F l ( X ) . 90 No w w e recall (brieﬂy) that diﬀeren tial graded mo dules admit certain ’resolutions’ (i.e. an y ob ject is quasi-isomorphic to a sem i -fr e e one in the terms of §14 of [Dri04]). Prop osition 5.2.1. Ther e exists a ful l triangulate d K ′ ⊂ K ( A ) such that the pr oje ction K ( A ) → D ( A ) induc es an e q uivalenc e K ′ ≈ D ( A ) . K ′ is close d with r esp e ct to al l (smal l) c opr o ducts. Pr o of. See §14.8 of [Dri04] R emark 5.2 .2 . In fact, there exists a (Quillen) mo del structure for C ( A ) suc h that D ( A ) its homotopy category; see Theorem 3.2 of [Kel06]. Moreo v er (for the ﬁrst mo del structures men tioned in loc. cit.) all ob jects of C ( A ) are ﬁbran t, all ob jects coming from A are coﬁbrant. F or this mo del structure t w o morphisms are homotopic whenev er they b ecome equal in K ( A ) . So, o ne could take K ′ whose ob jects are the coﬁbran t ob jects of C ( A ) . Using these facts, one could ve rify most of Prop osition 3.1.1 (f or D ′ and D describ ed b elo w). 5.3 The construction of D ′ and D ; the pro of of Prop o- sition 3.1. 1 It w as pro v ed in §2.3 of [ Be V08 ] (cf. also [Lev98] that D M ef f g m could b e described as H ( A ) , where A is a certain (small) diﬀeren tial graded category . Moreo v er, t he functor K b ( S mC o r ) → D M ef f g m could b e presen ted as H ( f ) , where f : B b ( S mC o r ) → A is a naturally deﬁned diﬀeren tial gr a ded functor (note still: it seems diﬃcult t o construct suc h a functor fo r t he ’mo del’ of D M ef f g m constructed in [Bon09]). W e will not describe the details for (any of ) these constructions since we will not need them. W e deﬁne D ′ = C ( A ) op , D = D ( A ) op , p is the natural pro j ec tion. W e v erify that these categories satisfy Prop osition 3.1.1. Ass ertion 10 follo ws from the fact t hat an y lo calization of a triangulated category that p ossess es an enhancemen t is enhanceable also (see §§3.4–3 .5 of [Dri04 ]). The em b edding H ( A ) op → D ( A ) yields D M ef f g m ⊂ D ′ . Since a ll ob j ects coming from A are co compact in K ( A ) op , Propo sition 5.2.1 yields that the same is true in D . W e o bta in assertion 1. D ′ is closed with resp ect to in v erse limits since C ( A ) is closed with resp ect to direct ones. Since the pro jection C ( A ) → K ( A ) resp ects copro ducts (as 91 w ell a s a ll other (ﬁltered) colimits), Prop osition 5.2.1 yields that p resp ects pro ducts also. W e obtain assertion 2. The descriptions of C ( A ) and D ( A ) yields all the prop erties of shifts and cones required. This yields assertions 3, 4, a nd 6. Since D ( A ) is a lo calization of K ( A ) , w e a lso obtain assertion 5. Next, since D ( A ) is a lo calization of K ( A ) with resp ect to quasi-isomorphism s, w e obtain assertion 8 . Recall that ﬁltered direct limits of exact sequences of ab elian groups a re exact. Hence for an y X ∈ O bj A ⊂ O bj D ′ , Y : L → DG -Mod( A ) we hav e K ( A )(DG -Mo d ( A )( X , − ) , lim − → l Y l ) = H 0 ((lim − → Y l )( A )) = H 0 (lim − → ( Y l ( A ))) = lim − → H 0 ( Y l ( A )) = lim − → l K ( A )(DG -Mo d ( A )( X , − ) , Y l ) . Applying (31) w e obtain assertion 7. It remains to v erify assertion 9 o f lo c. cit. Since the inv erse limit with respect to a pro jectiv e system is isomorphic to the inv erse limit with resp ec t to an y its un b ounded subsystem, and the same is true for lim ← − 1 in the coun t- able case, w e can assume that I is the categor y of natural n um b ers, i.e. w e ha v e a sequence of F i connected by morphisms. In this case we ha v e functorial morphisms lim ← − F i f → Q F l g → Q F i as in (27). Hence it suﬃces t o che c k that these morphisms yield a distinguished triangle in D . Note that g ◦ f = 0 ; hence g can b e factorized through a morphism h : Cone f → Q F i in D ′ . Since fo r a n y X ∈ O bj A the mor- phism h ∗ : Q D ′ F i ( X ) → Cone F ( X ) is a quasi-isomorphism, h b ecomes an isomorphism in D . This ﬁnishes the pro of. R emark 5.3.1 . 1. Note that the only part of our arg ument when w e needed k to b e coun table in the pro of of assertion 9 of lo c. cit. 2. The constructions of A (i.e. of the ’enhancemen t’ for D M ef f g m men- tioned ab o v e) tha t w ere described in [Be V08] and [Lev98] are easily seen to b e functorial with respect to base ﬁeld change (see b elo w). Still, the con- structions men tioned are distinct and far from b eing the only ones p ossible ; the author do es not kno w whether a ll p ossib le D are isomorphic. Still, this mak es no diﬀerence for cohomology (of pro-sc hemes); see Remark 4 .3 .2. Moreo v er, note that assertion 10 of Prop osition 3.1 .1 w as not v ery im- p ortan t for us (without if we w ould only hav e to consider a certain we akly exact w eigh t complex functor in §6.1 b elo w; see §3 of [Bon10]). The a uthor doubts that this condition follow s from the other part s of Prop osition 3.1 .1 . 92 5.4 Base c hange and T ate t wists for comotiv es Our diﬀeren tial graded formalism yields ce rtain functorialit y of comotiv es with resp ect to em b eddings of base ﬁelds. W e construct b oth extensi on and restriction of scalars (the latter one for the case of a ﬁnite extension of ﬁelds only). The construction of base change functors uses induction f or diﬀeren tial graded mo dules. This metho d also allo ws deﬁning certain tensor pro ducts and C o − H om for comotiv es. In particular, we obtain a T ate tw ist functor whic h is compatible with (29) (a nd a left adjoin t to it). W e no t e that the results of this subsection (probably) could not b e de- duced from the ’axioms’ of D listed in Prop osition 3 .1.1; y et they are quite natural. 5.4.1 Induction and restriction for diﬀeren tial graded mo dules: reminder W e recall certain results of §14 o f [Dri04] on functoriality of diﬀeren tial graded mo dules. These extend the corresp onding (more or less standard) statemen t s for mo dules ov er diﬀeren tial graded algebras (cf. §14.2 of ibid.). If f : A → B is a functor of diﬀeren tial graded categories, w e ha v e an ob vious r estriction functor f ∗ : C ( B ) → C ( A ) . It is easily seen that f ∗ also induces functors K ( B ) → K ( A ) and D ( B ) → D ( A ) . Certainly , the la t ter functor resp ects homotopy colimits (i.e. the direct limits from C ( B ) ). No w, it is not diﬃcult to construct an induction functor f ∗ : DG- Mod( A ) → DG-Mo d( B ) whic h is left adjo in t to f ∗ ; see §14.9 of ibid. By Example 14.10 of ibid, for any X ∈ O bj A this functor sends X ∗ = A ( X , − ) to f ( X ) ∗ . f ∗ also induces functors C ( A ) → C ( B ) and K ( A ) → K ( B ) . Restricting the latter one to the category of semi-free mo dules K ′ (see Prop osition 5.2.1) one obtains a functor Lf ∗ : D ( A ) → D ( B ) whic h is also left adjoin t to the correspo ndin g f ∗ ; see §14.12 of [Dri04 ]. Since all functors of the type X ∗ are semi-free b y deﬁnition, w e ha v e Lf ∗ ( X ∗ ) = A ( X , − ) = Lf ( X ) ∗ . It can also b e sho wn tha t Lf ∗ respects direct limits of ob j ects of A op (considered as A -mo dules via the Y o ne da em b edding). In the case of coun table limits this follo ws easily f rom the deﬁnition of semi-free mo dules and the express ion of the homotop y colimit in D ( A ) a s lim − → X i = Cone( ` X i → ` X i ) (this is just the dual to (27)). F or uncoun ta ble limits, one could pro v e the fact using a ’resolution’ of the direct limit similar to those describ ed in §A3 of [Nee01]. 93 5.4.2 Extension and restriction of scalars for comotives No w let l /k b e an extension of p erfect ﬁelds. Recall t hat D ′ and D were describ ed (in §5.3) in terms of mo dules ov er a certain diﬀeren tia l graded category A . It was sho wn in [Lev98] that the correspo ndin g v ersion of A is a tensor (diﬀeren tial graded) category; w e also ha v e a n extension of scalars functor A k → A l . It is most probable that b oth of these prop erties hold for the v ersion of A describ ed in [Be V08] (note that they o b viously hold f o r B b ( S mC o r ) ). Moreov er, if l /k is ﬁnite, then w e ha v e the functor o f restriction of scalars in in v erse direction. So, the induction f or the correspo ndin g diﬀeren tial graded mo dules yields an exact functor of extension of scalars Ext l/k : D k → D l . The reasoning ab o v e sho ws that Ext l/k is compatible with the ’usual’ extension of scalars for smo oth v arieties (and complexes of smo oth correspondences). Moreo ver, since Ext l/k respects homotop y limits, this compatibilit y extends to the co- motiv es o f pro-sc hemes and their pro ducts. It can also b e easily sho wn that Ext l/k respects T ate t wists. W e immediately obtain t he followi ng result. Prop osition 5.4.1. L e t k b e c ountable (a nd p erfe ct), let l ⊃ k b e a p erfe ct ﬁeld. 1. L et S b e a c onn e cte d primitive scheme ove r k , let S 0 b e its generic p oint. Then M g m ( S l ) is a r etr act of M g m ( S 0 l ) in D l . 2. L et K b e a function ﬁeld over k . L et K ′ b e the r esidue ﬁeld for a ge ometric v a luation v of K of r ank r . Then M g m ( K ′ l ( r )[ r ]) is a r etr a ct of M g m ( K l ) in D l . As in 4 .3 , this result immediately implies similar statemen ts for any cohomology of pro-sc hemes men tioned ( constructed from a cohomological H : D M ef f g m l → A via Prop osition 1.2.1). Next, if l /k is ﬁnite , induction for diﬀeren tial graded mo dules applied to the restriction of scalars for A ’s a lso yields a restriction of scalars functor Res l/k : D l → D k . Similarly to Ext l/k , this functor is compatible with restric- tion of scalars for smo oth v arieties, pro-sc hemes, and complexes of smo oth correspo nde nces; it a ls o resp ects T ate t wists. It follow s: l /k is ﬁnite, then Ext l/k maps D s k to D s l ; Res l/k maps D s l to D s k . Besides, if w e also assume l to b e coun table, then b oth of these functors respect w eigh t structures (i.e. they map D s w ≤ 0 k to D s w ≤ 0 l , D s w ≥ 0 k to D s w ≥ 0 l , and vice v ersa). 94 R emark 5.4.2 . It seems that one can also deﬁne restriction of scalars via restriction of diﬀeren tial graded mo dules (applied to the extension of scalars for A ’s). T o this end one needs to che c k t he corresp onding a djunc tion for D M ef f g m ; the corresp onding (and related) statemen t for the motivic homotop y categories w as prov ed by J. A y oub. This w ould allow deﬁning Res l/k also in the case when l /k is inﬁnite; this seems to b e ra ther in teresting if l is a function ﬁeld o v er k . Note that R es l/k (in this case) w ould (probably) a lso map D s w ≤ 0 l to D s w ≤ 0 k and D s w ≥ 0 l to D s w ≥ 0 k (if l is countable ). 5.4.3 T ensor pro ducts and ’co-in ternal Hom’ for comotiv es; T ate t wists No w, for X ∈ O bj A w e apply restriction and induction of diﬀeren tial g raded mo dules fo r the functor X ⊗ − : A → A . Induction yields a certain functor X ⊗ − : D → D , whereas restriction yields its left adjoin t whic h we will denote b y C o − H om ( X , − ) : D → D . Both of them resp ect homotop y limits. Also, X ⊗ − is compatible with tensoring by X on D M ef f g m . Bes ides, the isomorphisms classes of these functors only dep end o n the quasi-isomorphism class o f X in DG -Mo d( A ) . Indeed, it is easily seen that b oth X ⊗ Y and C o − H om ( X , Y ) are exact with resp ec t to X if we ﬁx Y ; since they are ob viously zero f or X = 0 , it remains to note that quasi-isomorphic ob jects could b e connected b y a c hain of quasi-isomorphisms . No w supp ose that X is a T ate motif i.e. X = Z ( m )[ n ] , m > 0 , n ∈ Z . Then w e obtain that the formal T ate twists deﬁned b y (29) are the true T ate twists i.e. they are give n by tensoring b y X on D . Then recall the Cancellation Theorem for motive s: (see Theorem 4.3.1 of [V o e00a], and [V o e10])): X ⊗ − is a full em b edding of D M ef f g m in to itself. Then one can deduce that X ⊗ − is fully faithful on D also (since all ob jects of D come from semi-free mo dules ov er A ). Moreo v er, C o − H om ( X , − ) ◦ ( X ⊗ − ) is easily seen t o b e isomorphic to the iden tit y on D (for suc h an X ). 6 Supplemen ts W e describ e some more prop erties of comotiv es, as well as certain p ossible v ariations of our metho ds and results. W e will b e somewhat sk etc h y some- times. In §6.1 w e deﬁne an a dditiv e category D g en of generic motiv es (a v ariation 95 of those studied in [Deg08a]). W e also pro v e that the ex act conserv ativ e weight c omplex f unctor (tha t exists b y the general theory of w eigh t structures) could b e mo diﬁed to an exact conserv ativ e W C : D s → K b ( D g en ) . Besides, w e pro v e assertions on retracts of the pro-motif of a function ﬁeld K/k , that are similar to (and follo w from) those for its comotif. In §6.2 we prov e that H I has a nice description in terms of H w . This is a sort of Brow n represen tabilit y: a cofunctor H w → Ab is represen table b y a (homotop y in v arian t) sheaf with transfers whenev er it con v erts all small pro ducts in to direct sums. This result is similar to the corresp onding results of §4 of [Bon10] (o n the connection b et w een the hearts of adjacent structures). In §6.3 we note that o ur metho ds could b e used for motives (and como- tiv es) with co eﬃcien ts in an arbitrary commu tative unital ring R ; the most imp ortan t cases ar e rational (co)motiv es a nd ’torsion’ (co)motiv es. In §6.4 w e note that there exis t natural motive s of pro-sc hemes with compact suppo rt in D M ef f − . It seems that one could construct alternativ e D and D ′ using this observ ation (ye t this probably w ould not aﬀect our main results signiﬁcan tly). W e conclude the section b y studying whic h of our arg uments could b e extended to the case of an uncoun table k . 6.1 The w eight complex functor; relation wi th generic motiv es W e recall that the general formalism of we ight structures yields a conserv a- tiv e exact w eigh t complex functor t : D s → K b ( H w ) ; it is compatible with Deﬁnition 2.1.2(9). Next we pro v e that one can comp ose it with a certain ’pro jection’ functor without losing the conserv ativit y . Lemma 6.1.1. Ther e exists a n exact c o nservative functor t : D s → K b ( H w ) that se nds X ∈ O bj D s to a choic e of its weight c omplex ( c oming fr om a ny choic e of a weight Postniko v tower for it). Pr o of. Immediate from Remark 6.2.2(2) and Theorem 3.3.1 ( V) of [Bon10 ] (note that D s has a diﬀeren t ia l graded enhancemen t by Prop osition 3.1.1(10)). No w, since all ob jects of H w a re r etracts of those t hat come via p from in v erse limits of ob jects of j ( C b ( S mC o r )) , w e hav e a natura l additiv e functor H w → D naive (see §1.5). Its categor ical image will b e denoted b y D g en ; this is 96 a sligh t mo diﬁcation of Deglise’s category of generic motive s. W e will denote the ’pro jection’ H w → D g en and K b ( H w ) → K b ( D g en ) b y pr . Theorem 6.1.2. 1. T he functor W C = pr ◦ t : D s → K b ( D g en ) is exact and c onservative . 2. L et S b e a c onne cte d primitive scheme, le t S 0 b e i ts generic p oint. Then pr ( M g m ( S )) is a r etr a c t of pr ( M g m ( S 0 )) in D g en . 3. L et K b e a func tion ﬁeld over k . L et K ′ b e the r esidue ﬁeld for some ge ometric valuation v of K of r ank r . Then pr ( M g m (Sp ec K ′ )( r )[ r ]) is a r etr act of pr ( M g m (Sp ec K )) in D g en . Pr o of. 1. The exactness of W C is ob vious (fro m Lemma 6.1.1). Now w e c hec k that W C is conserv ativ e. By Prop osition 3.1.1(8), it suﬃces to c hec k: if W C ( X ) is a cyclic for some X ∈ O bj D s , then D ( X , Y ) = 0 for all Y ∈ O bj D M ef f g m . W e denote the t erms of t ( X ) b y X i . W e consider the coniv eau sp ec tral seque nce T ( H , X ) for the functor H = D ( − , Y ) (see Remark 4.4.2). Since W C ( X ) is acyclic, we obtain that the complexes D ( X − i , Y [ j ]) are acycl ic for all j ∈ Z . Indeed, note that the restriction of a functor D ( X − i , − ) to D M ef f g m could b e expressed in terms of pr ( X − i ) ; see Remark 3.2.1. Hence E 2 ( T ) v anishes. Since T con v erges (see Prop osition 4.4.1(2)) w e obtain the claim. 2. Immediate from Corollary 4.2 .2 ( 1 ). 3. Immediate from Corollary 4.2 .2 ( 2 ). R emark 6.1.3 . F or X = M g m ( Z ) , Z ∈ S mV ar , it easily seen that W C ( X ) could b e describ ed as a ’naive’ limit of complexes of motiv es; cf. §1.5. No w, the terms of t ( X ) are just the factors of (some po ss ible) w eigh t P ostnik ov to w er for X ; so o ne can calculate them (at least, up to an isomor- phism) f or X = M g m ( Z ) . Unfortunately , it seems diﬃcult to desc rib e the b oundary for t ( X ) completely since H w is ﬁner than D g en . 6.2 The relatio n of the heart of w wi th H I (’Bro wn rep- resen ta bi lit y’) In Theorem 4.4.2(4) of [Bon10], f or a pair of adjacen t structures ( w , t ) for C (see Remark 2.5.7 ) it w as pro v ed that H t is a full sub category of H w ∗ (= 97 AddF un( H w op , Ab )) . This result cannot b e extended to arbitrary orthog- onal struc tures since o ur deﬁnition of a duality did not include an y non- degeneratenes s conditions (in pa rt icular, Φ could b e 0 ). Y et for our main example of orthogonal structures the statemen t is true; moreo ve r, H I has a natural description in terms o f H w . This statemen t is ve ry similar to a certain Bro wn represen tabilit y-t yp e result (for adjacen t structures) prov ed in Theorem 4.5.2( I I.2) of ibid. Note that H w is closed with resp ect to arbitrary small pro ducts; see Prop osition 4.1.1(2). Prop osition 6.2.1. H I is natur al ly isomorp h ic to a ful l ab elian sub c ate gory H w ′ ∗ of H w ∗ that c onsists of functors that c o n vert al l pr o ducts in H w into dir e c t sums (o f the c orr esp onding ab e lian gr oups). Pr o of. First, no t e that for an y G ∈ O bj D M ef f − the functor D → Ab that sends X ∈ O bj D to Φ( X , G ) ( Φ is t he duality constructe d in Prop osition 4.5.1) is cohomological. Moreov er, it con v erts homotopy limits in to injectiv e limits (of the correspo ndin g ab elian g roups); hence its restriction to H w b elongs to H w ′ ∗ . W e obtain an additiv e functor D M ef f g m → H w ′ ∗ . In f a ct, it factorizes thro ugh H I (b y (25) ) . F or G ∈ O bj H I w e denote the functor H w → Ab obtained b y G ′ . Next, fo r an y (additiv e) F : H w op → Ab w e deﬁne F ′ : D s → Ab b y: F ′ ( X ) = (Ker( F ( X 0 ) → F ( X − 1 )) / Im( F ( X 1 ) → F ( X 0 )); (32) here X i is a w eigh t complex for X . It easily seen from Lemma 6.1.1 that F ′ is a w ell-deﬁned cohomological functor. Moreov er, Theorem 2 .2 .1 (19) yields that F ′ v anishes on D w ≤− 1 s and on D w ≥ 1 s (since it v a nis hes on D w = i s for all i 6 = 0 ). Hence F ′ deﬁnes an additiv e functor F ′′ = F ′ ◦ M g m : S mC or op → Ab i.e. a presheaf with transfers. Since M g m ( Z ) ∼ = M g m ( Z × A 1 ) for any Z ∈ S mV ar , F ′′ is homotopy in v ariant. W e should c hec k tha t F ′′ is actually a (Nisnevic h) sheaf. By Prop osition 5.5 of [V o e00b], it suﬃces t o chec k tha t F ′′ is a Zariski sheaf. Now, the Ma y er-Vietoris triangle for motiv es (§2 of [V o e00a]) yields: to an y Zariski co v ering U ` V → U ∪ V there corresponds a long exact sequenc e · · · → F ′ ( M g m ( U ∩ V )[1]) → F ′′ ( U ∪ V ) → F ′′ ( U ) M F ′′ ( V ) → F ′′ ( U ∩ V ) → . . . Since M g m ( U ∩ V ) ∈ D w ≤ 0 s b y part 5 of Prop osition 4.1.1, we hav e F ′ ( M g m ( U ∩ V )[1 ]) = { 0 } ; hence F ′′ is a sheaf indeed. 98 So, F 7→ F ′′ yields an additiv e functor H w ∗ → H I . No w we c hec k that the f unctor G 7→ G ′ (described ab o v e) and the restric- tions o f F 7→ F ′′ to H w ′ ∗ ⊂ H w ∗ yield m utually inv erse equiv a lences of the categories in question. (24) immediately yields that the functor H I → H I that sends G ∈ O bj H I to ( G ′ ) ′′ is isomorphic to id H I . No w for F ∈ O bj H w ′ ∗ w e should c hec k: for a n y P ∈ D w =0 s w e ha v e a natu- ral isomorphism ( F ′′ ) ′ ( P ) ∼ = F ( P ) . Since H w is t he idemp oten t completion of H , it suﬃces to consider P b eing of the form Q l ∈ L M g m (Sp ec K l )( n l )[ n l ] (here K l are function ﬁelds o v er k , n l ≥ 0 ; n l and the transcendence degrees of K l /k are bo unde d); see part 2 of Prop osition 4.1.1. Moreo v er, since F con v erts pro ducts into direct sums, it suﬃces to consider P = M g m (Sp ec K ′ )( n )[ n ] ( K ′ /k is a f unction ﬁeld, n ≥ 0 ). Lastly , part 2 of Corollary 4.2.2 reduces the situation to the case P = M g m (Sp ec K ) ( K /k is a function ﬁeld). No w, b y the deﬁnition of the functor G 7→ G ′ , w e hav e ( F ′′ ) ′ ( M g m (Sp ec K )) = lim − → l ∈ L F ′′ ( M g m ( U l )) , where K = lim ← − l ∈ L U l , U l ∈ S mV ar . W e hav e F ′′ ( U l ) = Ker F ( M g m (Sp ec K )) → F ( Q z ∈ U 1 l M g m ( z )(1)[1]) ; here U 1 l is the set o f p oin ts of U l of co dimension 1 . Since F ( Q z ∈ U 1 l M g m ( z )(1)[1]) = ⊕ z ∈ U 1 l F ( M g m ( z )(1)[1]) ; w e hav e lim − → l ∈ L F ( Q z ∈ U 1 l M g m ( z )(1)[1]) = { 0 } ; this yields the result. 6.3 Motiv es and como tiv es with rational and torsion co eﬃcien ts Ab o ve we considered (co)motives with in tegral co eﬃcien ts. Y et, as w as show n in [MVW06], one could do the theory of motive s with co eﬃci en ts in an ar bi- trary comm utativ e asso ciativ e ring with a unit R . One should start with the naturally deﬁned category of R -correspondences: O bj ( S mC or R ) = S mV ar ; for X , Y in S mV ar w e set S mC or R ( X , Y ) = L U R for all integral closed U ⊂ X × Y that are ﬁnite ov er X and dominan t ov er a connected comp onen t of X . Then one obtains a theory of motives that w ould satisfy all prop erties that are required in or der to deduce the main results of this pap er. So, w e can deﬁne R -comotiv es and extend our r esults to t hem. A w ell-kno wn case o f motiv es with co eﬃcien ts are the motives with ra- tional co eﬃcien ts (note that Q is a ﬂat Z -algebra). Y et, one could a lso take R = Z /n Z for any n prime to c har k . So, the results of this pa p er are also v alid for rational (co)motiv es and 99 ’torsion’ (co)motive s. Still, note that there could b e idemp oten ts for R -motives that do not come from in tegral ones. In particular, for the naturally deﬁned rational motivic categories w e hav e D M ef f g m Q 6 = D M ef f g m ⊗ Q ; also C how ef f Q 6 = C how ef f ⊗ Q (here C how ef f Q ⊂ D M ef f g m Q denote the corresp onding R -hulls). Certainly , this do es not matter at all in the curren t pap er. 6.4 Another p o s sibilit y for D ; motiv es with compact supp ort of pro-sc hemes In the case c har k = 0 , V o ev o dsky dev elop ed a nice theory of motiv es with compact supp ort that is compatible with P oincare duality; see Theorem 4.3.7 of [V o e00a]. Moreo v er, the explicit constructions of [V o e00a] yield that the functor of motif with compact supp ort M c g m : S mV ar op → D M ef f g m is com- patible with a certain j c : S mV ar op f l → C − ( S hv ( S mC or )) (whic h sends X to the Suslin complex of L c ( X ) , see §4.2 lo c. cit.); this observ a tion w as kindly comm unicated to the a uthor by Bruno Kahn). This allows us to deﬁne j c ( V ) for a pro-sc heme V as the corresp onding direct limit (in C ( S hv ( S mC or )) ). Starting fro m this observ ation, one could try to dev elop an analogue of our theory using the functor M c g m . One could consider D = D M ef f − op ; then it w ould con tain D M ef f g m op as the full category of co compact ob jects. It seems that o ur argumen ts could b e carried ov er to this con text. One can construct some D ′ for this D using certain diﬀeren tial graded categories. Though motive s with compact supp ort are P oincare dual to ordinary motiv es o f smo oth v ar ieties (up to a certain T ate t wist), w e do not hav e a cov ariant em b eddin g D M ef f g m → D (for this ’alternativ e’ D ), since ( the whole) D M ef f g m is not sel f-dual. Still, D M ef f g m has a nice embedding in to (V o ev o dsky’s) self-dual category D M g m ; it contains an exhausting system of self-dual sub categories. Hence this alternativ e D w ould yield a theory that is compatible with (t ho ugh not ’isomorphic’ to) the theory dev elop ed ab ov e. Since the alternativ e v ersion of D is closel y r elated with DM ef f − op , it seems reasonable to call its ob jects comotiv es (as w e did fo r the ob jects of ’our’ D ). These observ ations sho w that one can dualize all the direct summands results of §4 to obtain their natural analogues fo r motiv es o f pro-sc hemes with compact suppo rt. Indeed, to prov e them we ma y apply the duals of o ur argumen ts in §4 without an y problem; see part 2 of Remark 3.1.2. Not e that 100 w e obtain certain direct summand statemen ts for ob jects of D M ef f − this w ay . This is an adv antage of o ur ’a xi omatic’ approac h in §3.1. One could also tak e D op = ∪ n ∈ Z D M ef f g m ( − n ) (more precisely , this is the direct limit o f copies o f D M ef f g m with connecting morphisms b eing − ⊗ Z (1) ). Then w e ha v e a co v arian t em b edding D M ef f g m → D M g m → D . Note that b oth of these alternativ e v ersions of D are not closed with respect to all (coun table) pro ducts , and so not closed with resp ect to all (ﬁltered countable ) homotop y limits; yet they contain a ll pro ducts and ho- motop y limits that are required for our main argumen ts. 6.5 What happ ens i f k i s uncoun table W e describe whic h of the a rgumen ts ab ov e could b e applied in the case o f an uncoun ta ble k (and for which of them the author has no idea how to ac hiev e this). The author w arns that he didn’t chec k the details thoroughly here. As w e ha v e already noted ab o v e, it is no problem to deﬁne D , D ′ , or ev en D s for any k . The main problem here that (if k is uncoun table) the comotiv es of generic p oin ts of v arieties (and of other pro-sc hemes) can usually b e presen ted only as uncountable homotopy limits of motive s of v a rietie s. The general f o rmalis m of in v erse limits (applied to the categories of mo dules o v er a diﬀeren tial graded category) allo ws us to extend to this case all parts of Prop osition 3.1.1 except part 9. This actually means that instead of the short exact sequence (28) one o bta ins a sp ectral sequence whose E 1 -terms are certain lim ← − j ; here lim ← − j is the j ’s deriv ed functor of lim ← − I ; cf. App endi x A of [Nee01]. This do es not seem to b e catastrophic; yet the a uthor has absolutely no idea ho w to control higher pro jectiv e limits in the pro of of Prop o si tion 3.5.1; note that part 2 o f lo c. cit. is especially imp ortan t for the construction of the Gersten w eight structure. Besides, the author do es not kno w how to pa ss to a n uncoun table homo- top y limit in the Gysin distinguished triangle. It seems that to this end one either needs to lift the functoriality of the (usual) motivic Gysin triangle to D ′ , or to ﬁnd a w a y to describ e the isomorphism class of an uncoun table ho- motop y limit in D in terms of D -only (i.e. without ﬁxing any lifts to D ′ ; this seems to b e imp ossible in general). So, one could deﬁne the ’G ersten’ w eight to w er for the comotif of a pro-sc heme as the homotop y limit of ’geometric to w ers’ (as in the pro of of Corollary 3.6.2); y et it seems to b e r a ther diﬃcult to calculate factors of suc h a to w er. It seems that the problems men tioned do not b ecome simpler for the alternativ e v ersions of D describ ed in §6.4. 101 So, curren tly the author do es not kno w ho w to pro v e the direct summand results of §4.2 if k is uncoun table (they ev en could b e wrong). The problem here that the splittings of §4.2 are not canonical (see Remark 4 .2 .3), so one cannot apply a limit argumen t (as in §4.6) here. 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