On relative K-motives, weights for them, and negative K-groups

Transcription

1 arxiv: v1 [math.ag] 26 May 2016 On reative K-motives, weights for them, and negative K-groups Mikhai V. Bondarko May 20, 2018 Abstract Aexander Yu. Luzgarev In this paper we study certain trianguated categories of K-motives DK( ) over a wide cass of nice enough base schemes, and define certain weights for them (so, we introduce Chow weight structures for K-motives cosey reated to the ones introduced by D. Hébert, the first author, and M. Ivanov for reative Voevodsky motives of various types). We reate the weights of particuar K-motives (of the form f (1 Y ), where 1 Y is the tensor unit of DK(Y)) to (negative) homotopy invariant K-groups (tensored by Z[S 1 ] for S being the set of non-invertibe primes ) K ( ). Our resuts yied a (new) resut on the vanishing of K i (Y) and of certain reative K-groups for i being too negative ; this statement is cosey reated to a question of Ch. Weibe. We aso prove that K i (Y) for i < 0 is supported in codimension i. Moreover, we estabish severa criteria for bounding (beow) the weights of the K-motives f (1 Y ); this automaticay impies the vanishing of the corresponding E 2 -terms of Chow-weight spectra sequences (and of the factors of the corresponding Chow-weight fitrations) for any (co)homoogy of these motives. Our methods of bounding weights use a resoution of singuarities resut of O. Gabber. They can be appied to various motivic trianguated categories; this yieds some new statements on (constructibe) compexes of étae sheaves (as we as simiar bounds on the terms of Chow-weight spectra sequences for Voevodsky motives). We aso reate the weights of K-motives with rationa coefficients to that of Beiinson motives; the Chow-weight spectra sequences converging to their Q -étae (co)homoogy yied Deigne-type weights for The main resuts of the paper were obtained under support of the Russian Science Foundation grant no

2 the atter. Somewhat surprisingy, we are abe to prove in certain ( extreme ) cases that the corresponding weight bounds coming from étae (co)homoogy are precise; we iustrate these statements by some simpe exampes. Contents 1 Some preiminaries: notation, K-motives, and weight structures Some notation, definitions, and auxiiary statements On K-motives over a base Weight structures: reminder On weight compexes and weight spectra sequences On the Chow-weight structure for K-motives; reating the motivic ength with negative K-groups On the Chow weight structure for compact K-motives Motivic weight bounds in the terms of negative K-groups: the finite type version On non-compact Chow weight structures and the K-weight bounds for X-schemes that are not of finite type Studying weights using genera Gabber s resoution of singuarities resuts On dimension functions On the motivic Gabber s emma and its appications An appication to bounding weights and to the vanishing of too negative K-groups Some axiomatic generaizations Studying weights using dh-descent, Voevodsky motives, and their étae (co)homoogy Comparison with the categories DM( ) On detection of weights via étae (co)homoogy over a fied On the reation to Deigne s weights and maximay singuar exampes On detection of weights via étae homoogy over a genera base X Expicit detector schemes over varieties and the reation to the singuarity fitration (on K-theory and singuar cohomoogy) 68 2

3 Introduction In [Wei80] Ch. Weibe has asked whether K n (X) = {0} for any noetherian scheme X of dimension ess than n (where K denotes the Bass Thomason Trobaugh K-theory). Reca here that negative K-groups of reguar schemes vanish; so the question is reated to the singuarities of X. The positive answer to Weibe s question for schemes essentiay of finite type over a fied of characteristic zero was given in [CHSW08]. The next step in this direction was made in [Ke12] where (essentiay) the so-caed homotopy invariant K-theory was considered. 1 In the current paper we propose a categorification of this question, and reate negative K-groups to weights of K-motives (simiar to the weights constructed in [Heb11], [Bon14], and [BoI15] for the corresponding versions DM( ) of the Voevodsky motivic categories foowing the pattern introduced in [Bon10a]). For this purpose we consider the homotopy categories of modues over the symmetric motivic ring spectra KG (instead of modues over the motivic cohomoogy spectra); we ocaize them by inverting a the primes that are not invertibe over our base schemes (for this set of primes S the ocaization Z[S 1 ] wi be denoted by Λ) to obtain our main motivic categories DK( ). The version of K-theory corresponding to categories of this type is the homotopy invariant K-theory tensored by Λ that is denoted by K( ) beow. The Voevodsky s four functor formaism (as deveoped in the treatise [CiD12] that reied on an earier book [Ayo07]) yieds exact functors f and f! : DK(Y) DK(X) for any separated finite type morphism f: Y X (as we as f and f! : DK(X) DK(Y)). The properties of these functors and categories (aong with the vanishing of negative K-theory of fieds and reguar schemes) aow us to define the Chow weight structure for DK(S) for S being a (Λ)-nice scheme (see Remark 1.1.4(1) beow). Somewhat simiary to [BoI15], the non-negative K-motives DK(S) wchow 0 over S are generated by v (1 V ) for v: V S running through finite type separated morphisms with reguar domain, whereas DK(S) wchow 0 is generated by v! (1 V ), where 1 V denotes the tensor unit object of DK(V). 2 1 The reason for this was the usage of cdh-descent provided by [Cis13] for homotopy invariant K-theory; note that it fais for the usua K-theory (of singuar schemes). Aso, the usage of Gabber s resoution of singuarities resut has forced Key and us to invert a the positive residue fied characteristics of X in the coefficient ring. Yet note that our resuts certainy yied certain properties of K-theory when restricted to schemes of fixed positive characteristic; see Remark 3.3.2(4) beow and Theorem 3.7 of [Ke14] for more detai. 2 One may consider ony reguar V/S or a finite type separated S-schemes when generating DK(S) wchow 0. The main distinctions of our weights for DK( ) from the weight 3

4 Now, some of the properties of the Chow weight structure for DK( ) are quite simiar to its DM-versions (as described in the aforementioned papers); furthermore, some of them are immediate consequences of the genera formaism of weight structures. Certainy, for any Λ-nice S the weight structure w Chow (S) yieds weight fitrations and weight spectra sequences T wchow (H, ) for any (co)homoogica functor H that factors through DK(S) (that converge to H (M) for M ObjDK(S)). In the case where S is (separated) of finite type over fied (whose characteristic is invertibe in Λ; for Λ = Q it suffices to assume that S is of finite type over an exceent noetherian scheme of dimension at most 3) these spectra sequences reate the cohomoogy of M to that of (the DK(S)-versions of) Chow motives over S. The atter are the DK(X)-retract of objects of the form v! (1 V ) for v: V S being a projective morphism with reguar domain (see Remark 2.1.3(3) beow). Moreover (for arbitrary S) the weights of M are detected by E2 (T wchow (H,M)) (that are DK(S)-functoria in M) ; see Coroary beow. Furthermore, in the case Λ = Q the weights of K-motives can be expressed in terms of the weights of the associated Beiinson motives (i.e., of the objects of the Q-inear version of DM(S) obtained from M via the functor M per (S); see Proposition beow). Next, certain conjectures on the so-caed mixed motivic sheaves predict that it suffices to consider H being the corresponding perverse étae homoogy functor to compute the weights of the Beiinson motives obtained (see [Bon15] where this functor was treated in detai and the aforementioned conjectures were reated to standard motivic conjectures over universa domains). We succeed in estabishing (a form of) this weightdetection conjecture in some particuar cases that are rather important for the purposes of the current paper in ; we iustrate the reevance of our weight bounds by certain simpe exampes. Let us now describe the main motivation for treating K-motives in this paper. The orthogonaity axiom of weight structures yieds for anyn 0 that an object M beongs to DK(X) wchow n if and ony if DK(X) wchow n 1 M (i.e., there are ony zero morphisms between objects of weights ess than n into M). Now, for M = f (1 Y ) (where f: Y X is a separated morphism of Λ-nice schemes) these conditions are equivaent to the foowing one: for any separated finite type morphism P X the groups K BM,Y i (P X Y) (see Remark 1.2.2(1) vanish for i < n; moreover, it suffices to consider those P that are affine over X and reguar in this criterion. Here K BM,Y i ( ) are certain Bore Moore K-groups; hence the atter condition is equivaent to the structure construction used in [BoI15] is that we don t have to twist the generators due to the fact that K-motives are 1 = (1)[2]-periodic (see Remark 1.2.2(8) beow); moreover, we treat non-compact motives in much more detai than in ibid. 4

5 vanishing of K i (U X Y) for a smooth U/X and i < n combined with the surjectivity of the natura homomorphism K n (U X Y) K n (V X Y) for any such U and any open (dense) V U. Thus we obtain a cose reation between negative K-groups and the weights of objects of the type f (1 Y ); the aforementioned motivic conjectures (aong with the particuar cases in which we can prove them) yied quite unexpected reations of these matters with the weight spectra sequences converging to étae (co)homoogy (see 4.3). Note here that there certainy exist connecting reguator homomorphisms between K-groups with rationa coefficients and Q -étae cohomoogy; yet these maps are very far from being surjective or injective (in genera). Next, to obtain resuts reated to the Weibe s question we bound beow the weights of 1 X and of f (1 Y ); we aso study the Chow-weight decompositions of these objects. It is no wonder that this requires certain resoution of singuarities resuts (note that f (1 Y ) DK(X) wchow 0 whenever Y is reguar). To make our weight bound resuts as genera as possibe, we consider a form of Gabber s resoution of singuarities resuts that was previousy used in [Ke14]. We use an argument cosey reated to the ones that were used in [ILO14, XIII.3], [CiD16, 6.2], and [CiD12, 4.2] for the proof of the constructibiity of the corresponding anaogues of M = f (1 Y ) for f being of finite type. Now, the reasonings in the aforementioned papers are far from being short and simpe, whereas we are interested in much more precise information on M. So, we use somewhat technica definitions (incuding a modification of Gabber s dimensions functions) and arguments to put objects of the type f (1 Y ) and f! (1 Y ) into certain enveopes (i.e., we describe more or ess expicit sets of objects of DK(X) such that the motives that interest us can be obtained from them by means of extensions and retracts). It does not make sense to formuate these resuts in the introduction; so we wi try to to describe some of their consequences instead (cf. aso Remark 3.4.3(4)). We prove that for any s 0 there exists a cosed scheme Z X such that dim(x) dim(z) s+1 and for U = X\Z we have 1 U DK(U) w s. According to the aforementioned reation of DK(X)-weights of 1 X = id X, (1 X ) to K-groups, it foows that K s is supported in codimension s ; this statement appears to be competey new (see Remark 3.3.2(2)). We aso estabish some more criteria for f (1 Y ) to beong to DK(X) wchow n (for f being of finite type). In particuar (see Theorem 3.3.1(II.2)), it suffices to verify that the groups K BM,Y i (P X Y) vanish for i < n ony in the case where P is reguar and affine and dim(p) d i, where d is the dimension of an X- compactification of Y. 3 Furthermore, it suffices to compute these K-groups 3 Reca that X-compactifications of any finite type separated Y/X exist, and their dimensions are equa. Moreover, d equas dim(y) whenever X is of finite type over a fied 5

6 for a finite number of test schemes that can be (more or ess) expicity described in terms of f; see Remark 3.3.2(3) for more detai. Note aso that the reader wiing to avoid the compicated arguments of 3.2 may repace most of them (incuding the construction of the aforementioned test schemes ) by a much easier Proposition in the case where X is of finite type over a fied. On the other hand, in 3.4 the K-motivic enveope formuations of 3.2 were extended to a wide range of motivic categories; in particuar (appying them to étae motives as studied in [CiD16]) one obtains certain enveope statement that appear to be quite new over genera schemes (yet we do not treat this matter in detai). Let us now describe the contents of the paper. Some more information of this sort can be found at the beginnings of sections. In 1 we introduce the basics on trianguated categories DK( ) of K- motives over (exceent separated Noetherian finite dimensiona) schemes. Since the properties of these categories are quite simiar to the ones of more or ess usua reative Voevodsky motives (as studied in severa papers of Ayoub, Cisinski, and Dégise, we ony sketch their proofs. We aso reca those aspects of the theory of weight structures that wi be needed beow. In 2 we introduce the Chow weight structures on DK( ) (starting from its restriction to the subcategory of compact motives) and study its properties. We aso formuate severa criteria for the motif f (1 Y ) for f: Y X to be of weights n in DK(X); they are formuated in terms of negative homotopy invariant K-groups of certain Y -schemes. A remarkabe distinction of this section from the study of the Chow weight structures in [Heb11], [Bon14], and [BoI15] is that we study (in detai) the weights of non-compact motives (aso); this enabes us not to restrict our criteria to the case where f is of finite type. In 3 we study these criteria aong the weight bounds for f (1 Y ) in more detai. For this purpose we use resoution of singuarities resuts of Gabber for putting the objects f (1 Y ) and f! (1 Y ) (for f : Y X being of finite type) into the enveope of v b, 1 Vb [m b ] for certain finite type morphisms v b : V b X with reguar domains and m b 0; the pairs (V b,m b ) may be (more or ess) expicity described. These resuts can be extended to a wide range of reative motivic categories; so our methods yied (in particuar) a more precise version of the constructibiity of direct images of (tensor) unit h- motives as proved in [CiD16, 6.2] foowing the pattern of [ILO14, XIII.3] (and it yieds a new property of the corresponding constructibe compexes of étae sheaves). We bound beow the weights of f (1 Y ); this yieds vanishing resuts cosey reated to the ( homotopy invariant version of the) question or over SpecZ; see Remark beow for more detai. 6

7 of Ch. Weibe on the vanishing of the K-theory of Y in degrees ess than dim(y). One may say that our weight bounds are certain categorifications of this vanishing question. In 4 we prove that the weights of K-motives can be expressed in terms of the ones of the corresponding Beiinson motives. Then we proceed to study the weights of the atter (and of some simiar motivic categories); we mosty study the case where the base scheme X is the spectrum of a fied or a variety. Under this restriction the weight zero X-motives come from reguar projective X-schemes; so the corresponding weight spectra sequences T wchow (X)(H,M) reate the (co)homoogy H of X-motives with that of X- schemes of this sort. Note here that (according to the genera theory of weight structures) the non-vanishing of E pq 2 T wchow (X)(H,M) 0 for some q Z and p > n impies that M contains weights ess than n. When X = Speck this yieds a cose connection of the weights of X-motives and schemes with Deigne s weights of their étae (co)homoogy; for X being a variety one can use the weights from [BBD82] and [Hub97] here. We have an inequaity between these two versions of weight bounds; it is conjecturay an equaity, and we prove this conjecture for certain (amost) maximay singuar X-schemes. We aso iustrate our notion of maximay singuar schemes (this is a certain motivic characterization of the singuarities of X) by some easy exampes. Lasty; we describe some simpe substitute of the motivic resoution of singuarities arguments of 3 in the case where X is a variety; this suggests a cose reation of our weight fitrations and spectra sequences to the singuarity ones considered in [PaP09] and [CiG14] in the characteristic 0 case. The authors woud ike to apoogize for possibe typos and inaccuracies in the first version of this text. Moreover, some of the arguments beow aong with the reation of our work to that of other authors wi possiby be described in more detai in the next versions of the paper. The authors are deepy gratefu to prof. F. Dégise and prof. S. Key for their iuminating remarks. 1 Some preiminaries: notation, K-motives, and weight structures The resuts of these section are (more or ess) easy consequences of the formaism of (reative) motivic categories and weight structures (as deveoped in [CiD12] and [Bon10a], respectivey). In 1.1 we introduce some notation and conventions for (mosty, triangu- 7

8 ated) categories. We aso reca some basics on ocaizing coefficients in trianguated categories and reca the main resut of [BoS15] (that describes the enveope of a set of object of C in terms of cohomoogica functors from C). In 1.2 we introduce our version of the categories DK( ) of (reative) K-motives. Our resuts are essentiay the KG-modue versions of the corresponding resuts of [CiD12] and [CiD16]; so we just skip most of the proofs. In 1.3 we reca some basics properties on weight structures and prove a few new statement. In 1.4 we introduce the (more compicated) notions of weight compexes and weight spectra sequences; a reader ony interested in 2 3 may skip this section. 1.1 Some notation, definitions, and auxiiary statements For a category C the symbo C op wi denote its opposite category. For a category C, X,Y ObjC, C(X,Y) is the set of C-morphisms from X to Y. We wi say that X is a retract of Y if id X can be factored through Y. Note that if C is trianguated or abeian then X is a retract of Y if and ony if X is its direct summand. For categories C,D we write D C if D is a fu subcategory of C. For any D C the subcategory D is caed Karoubi-cosed in C if it contains a retracts of its objects in C. We wi ca the smaest Karoubi-cosed subcategory of C containing D the Karoubi-cosure of D in C; sometimes we wi use the same term for the cass of objects of the Karoubi-cosure of a fu subcategory of C (corresponding to some subcass of ObjC). The Karoubi enveope Kar(D) (no ower index) of an additive category D is the category of forma images of idempotents in D. In this paper a compexes wi be cohomoogica, i.e., the degree of a differentias is +1; respectivey, we wi use cohomoogica notation for their terms. K(B) wi denote the homotopy category of compexes over an additive category B; for n Z the notation K(B) n is used to denote the cass of compexes isomorphic (i.e., homotopy equivaent to) compexes concentrated in degrees n. For a C-morphism f: X Y the symbo Cone(f) denotes the third vertex of the corresponding triange X Y Cone(f) (so, Cone(f) is we-defined up to a non-canonica isomorphism). C and D wi usuay denote some trianguated categories. We wi use the term exact functor for a functor of trianguated categories (i.e., for a functor that preserves the structures of trianguated categories). For any distinguished triange A B C in C we wi ca B a (C-) extension of C by A. A cass D ObjC wi be caed extension-cosed if 8

9 D contains 0 as we as a extensions of its eements by its eements. In particuar, an extension-cosed D is strict (i.e., contains a objects of C isomorphic to its eements). The smaest extension-cosed D containing a given D ObjC wi be caed the extension-cosure of D. The smaest extension-cosed Karoubi-cosed subcass of Obj C containing D (resp. containing i 0 D[i], resp. containing i 0 D[i]) wi be caed the enveope (resp. the eft enveope, resp. the right enveope) of D. Beow A wi aways denote some abeian category. We wi ca a covariant (resp. contravariant) additive functor H: C A homoogica (resp. cohomoogica) if it converts distinguished trianges into ong exact sequences. For X,Y ObjC we wi write X Y if C(X,Y) = {0}. For D,E ObjC we write D E if X Y for a X D, Y E. For D C the symbo D denotes the cass {Y ObjC X Y X D}. Sometimes we wi use the notation D to denote the corresponding fu subcategory of C. Duay, D is the cass {Y ObjC Y X X D}. We wi say that some C i ObjC, i I, weaky generate C if for X ObjC we have: C(C i [j],x) = {0} i I, j Z = X = 0 (i.e., if {C i [j] j Z} contains ony zero objects). M ObjC wi be caed compact if the functor C(M, ) commutes with a sma coproducts (we wi ony consider compact objects in those categories that are cosed with respect to arbitrary sma coproducts). We wi say that a trianguated category C (cosed with respect to arbitrary sma coproducts) is compacty generated if the (trianguated) subcategory of compact objects in it is essentiay sma and its objects weaky generate C. For a set of objects C i ObjC, i I, we wi use the notation C i to denote the smaest stricty fu trianguated subcategory containing a C i ; for D C we wi write D instead of ObjD. We wi ca the Karoubicosure of C i in C the trianguated category generated by C i (reca that it is trianguated indeed). For a C cosed with respect to a sma coproducts and D C (D coud be equa to C) we wi say that C i generate D as a ocaizing subcategory if D is the smaest fu strict trianguated subcategory of C that contains C i and is cosed with respect to a sma coproducts (this actuay means that C i weaky generate D). We ist the main properties of the ocaization of coefficients functors for compacty generated trianguated categories. 9

10 Proposition Let C be a trianguated category that is compacty generated by its (fu) trianguated subcategory C ; et S Z be a set of prime numbers. Denote Z[S 1 ] by Λ; denote by C S tors the ocaizing subcategory of C generated by cones of c s c for c ObjC, s S. Then the foowing statements are vaid. 1. C S tors is (aso) weaky generated by c s c for c ObjC, s S. 2. The Verdier quotient category C[S 1 ] = C/C S tors exists (i.e., the morphism groups of the target are sets); the ocaization functor : C C[S 1 ] respects a coproducts, converts compact objects into compact ones. Moreover, C[S 1 ] is generated by (ObjC) as a ocaizing subcategory. 3. For anyc ObjC, c ObjC, we havec[s 1 ]((c ),(c)) = C(c,c ) Z Λ. 4. C[S 1 ] is an Λ-inear category. 5. possesses a right adjointgthat is a fu embedding functor. G(ObjC[S 1 ]) consists of those M Obj C essentiay (i.e., up to C-isomorphisms) such that s: M M is an automorphism for any s S (i.e, G(C) is essentiay the maxima fu Λ-inear subcategory of C). Proof. See Proposition A.2.8 and Coroary A.2.13 of [Ke12] (cf. aso Appendix B of [Lev13]). Remark Sometimes we wi have to increase S. For an S S we certainy have obvious exact (ocaization) comparison functor C[S 1 ] C[S 1 ] that respects compact objects and coproducts. Note that one can obtain it by setting the new starting category being equa to D = C[S 1 ]. 2. In the case S = P\{} (for P\S) the corresponding comparison functor wi be denoted by c D. Certainy, Proposition 1.1.1(3) yieds that the restriction of c D onto the subcategory D of compact objects is the naive Z () -inearization functor, i.e., it is the exact functor that tensors the morphism groups by Z (). 3. Actuay, we wi not need assertion 5 of this proposition beow. Yet note that composing (exact, homoogica, or cohomoogica) functors with G yieds an interesting way of Λ-inearizing them. 10

11 The foowing obvious modification of the main resut of [BoS15] appears to be quite usefu for the contro of enveopes. Proposition Let C be a sma Λ-inear trianguated category, D {M} ObjC. Then M beongs to the enveope of D if and ony if for any homoogica functor F: C Ab we have F(M) = {0} whenever the restriction of F to D is zero. Moreover, if Λ Q then it suffices to verify this condition under the assumption that the target of F is Z () -modues for some P\S. Proof. The first part of the assertion is immediate from Theorem 0.1 of [BoS15] (appied to the category C op ). To obtain the moreover part one shoud note (simiary to ibid.) that a the functors F( ) z Z () for S are cohomoogica, and we have F(M) = {0} whenever F(M) z Z () = {0} for a S. Throughout the paper we wi ony consider schemes that are exceent separated of finite Kru dimension; we wi ca schemes that satisfy a of these conditions nice ones (for the sake of brevity). A the morphisms we wi consider wi be separated; they wi aso usuay be of finite type. Remark Moreover, we wi usuay fix a set of primes S (and set Λ = Z[S 1 ] as in Proposition above). We wi ca a (nice) scheme X Λ-nice if a the primes in P \ S are invertibe on it (so, the characteristics of a the residue fieds of X beong to S {0}; in particuar, a nice schemes are Q-nice with S = P). By defaut, a the schemes we consider wi be Λ-nice aso (so, we wi assume that a scheme is Λ-nice if it wi not be said expicity that is just nice). 2.. Our reason for concentrating on Λ-nice schemes is that this restriction is necessary for the appication of Gabber s resoution of singuarities resut (see Remark beow) for the contro of the compactness of Λ-inear motives (see Theorem 1.2.1(II.4)). A morphisms of schemes we consider wi be separated. We wi say that a morphism is smooth ony if it is aso of finite type. The foowing versions ofk-theory wi be one of the main subjects of this paper: for a scheme X we wi use the notation KH (X) for the (Weibe s) homotopy invariantk-theory ofx (cf. [Cis13], [Ke14]); K(X) = KH(X) Z Λ. We wi say that a projective system S i, i I, of schemes is essentiay affine if the transition morphisms g ji : S j S i are affine whenever i i 0 (for some i 0 I). 11

12 A presentation of a scheme X as X α, where Xα, 1 n (it wi be convenient for us to use this numbering convention throughout the paper), are pairwise disjoint ocay cosed subschemes of X and each X α is open in i Xi α, wi be caed a stratification of X.4 The corresponding embeddings X α X wi be denoted by j α. We we say that this stratification (α) is reguar whenever a X α are reguar. The symbo X red wi denote the reduced scheme associated to a scheme X. Note that X and X red are equivaent from the motivic point of view (see Theorem 1.2.1(II.5) beow); so one may assume that a the schemes we consider are aso reduced (and consider a the Cartesian diagrams of this paper in the category of reduced schemes). 1.2 On K-motives over a base For any nice scheme S consider the symmetric motivic ring spectrum KG S ObjSH(S) defined in of [CiD12] (that reies on the main resut of [RSO10]); this spectrum is a minor modification of Voevodsky s KG S. Foowing of ibid., we take the category DK Z (S) of modues over KG S ; here we consider KG S-modues in SH(S) endowed with the ( projective ) structure of a Quien mode category (see of ibid.), and definedk Z (S) as the homotopy category of the mode category obtained. We wi not give detaied proofs of (most of) the statements beow since very simiar statements were aready proved in ibid. (and in other papers of Cisinski and Degise). Theorem Let X,Y be any (nice) schemes, f: Y X be a (separated) scheme morphism, i Z. I. Then the foowing statements are vaid. 1. The category DK Z (X) is a tensor trianguated category; its unit object wi be denoted by Z X. DK Z (X) is cosed with respect to a (sma) coproducts, and the tensor product respects them. 2. We have exact functors f : DK Z (X) DK Z (Y) and f : DK Z (Y) DK Z (X); f is eft adjoint to f. Any of them (when f varies) yieds a 2-functor from the category of (nice) schemes with separated morphisms to the 2-category of trianguated categories. 4 This somewhat weak notion of a stratification was used in some previous papers of the first author. 12

13 3. If f is of finite type, then we aso have adjoint functors f! : DK Z (Y) DK Z (X) :f!. Simiary, these two types of functors yied 2-functors from the category of (nice) schemes with separated finite type morphisms to the 2-category of trianguated categories. 4. For a Cartesian square of (separated) morphisms Y f g X g (1.1) Y f X such that g and g are of finite type we have g! f = f g! and g! f = f g!. 5. f is symmetric monoida; f (Z X ) = Z Y. 6. f = f! if f is proper; f! = f if f is an open immersion. 7. If i: Z X is a cosed immersion, U = X \Z, j: U X is the compementary open immersion, then the motivic image functors yied a guing datum for DK( ) in the sense of of [BBD82]; cf. Proposition 1.1.2(10) of [Bon14]). In particuar, for any M ObjDK Z (X) the pairs of morphisms and j! j (M) M i i (M) (1.2) i i! M M j j M (1.3) can be (uniquey and) functoriay competed to a distinguished triange. Moreover, i j! = 0, i! j = 0, and the adjunctions transformations i i 1 DKZ (Z) i! i! and j j 1 DKZ (U) j! j! are isomorphisms. 8. f! = f if f is smooth (of finite type). 9. If i: S S is a cosed immersion of reguar (nice) schemes then Z S = i! (Z S ). 10. If X,Y are reguar, and O Y is a free O X -modue of finite rank d, then for anym ObjDK Z (X) there exists a morphismv DK Z (X)(f f (M),M) such that its composition with the unit morphism M f f (M) equas did M. 13

14 11. The fu subcategory DK c Z(X) DK(X) of compact objects is trianguated. DK c Z (X) is generated by g!(z X ) for g: X X running through a smooth (finite type) morphisms. 12. f preserves the compactness of objects; this is aso true for f! if f is of finite type. 13. f and f commute with arbitrary sma coproducts; the same is true for f! and f! if f is of finite type. 14. Let a scheme S be the imit of an essentiay affine (see 1.1) fitering projective system of schemes S i for i I (certainy, we assume that a of these schemes are nice). Denote the corresponding transition morphisms S j S i (resp. S S i ) by g ji (resp. by h i ). Then DK c Z(S) is isomorphic to the 2-coimit of the categories DK c Z (S i); in this isomorphism the corresponding connecting functors are given by g ji and by h i, respectivey. Furthermore, for anyi 0 I, M ObjDK c Z(S i0 ), and N ObjDK Z (S i0 ), the natura map im DK Z (S i )(g i i 0 ii 0 (M),gii 0 (N)) DK Z (S)(h i 0 (M),h i 0 (N)) is an isomorphism. 15. We have DK Z (Z X [i],z X ) = KH i (X); these isomorphisms (when X varies) are compatibe with the motivic functors of the type f. II. Assume in addition that X and Y are Λ-nice. For any S that is aso Λ-nice we define the category DK(S) as DK Z (S)[S 1 ] (see Proposition 1.1.1); the ocaization functor DK Z (S) DK(S) wi be denoted by S. 1. DK(X) is a tensor trianguated category; the object 1 X = X (Z X ) is a unit one for this tensor product. Furthermore, DK(X) is cosed with respect to a (sma) coproducts. 2. We have DK(1 X [i],1 X ) = K i (X); these isomorphisms (when X varies) are compatibe with the motivic functors of the type f. In particuar, if X is reguar and i < 0 then 1 X [i] 1 X. 3. The natura anaogues of assertions I.2 14 for the categories DK( ) aong with their subcategories DK c ( ) of compact objects (and for Λ- nice schemes) are aso vaid. 14

15 4. If f is of finite type, then f and f! respect the compactness of objects (aso; cf. assertion I.12). 5. If f is a finite universa homeomorphism, then f, f, f!, and f! are equivaences of categories. Moreover, f! (1 Y ) = f (1 Y ) = 1 X and f (1 X ) = f! (1 X ) = 1 Y. 6. If X is of finite type over a fied, then DK c (X) (as a trianguated category) is generated by {p (1 P )}, where p: P X runs through a projective morphisms such that P is reguar. In particuar, if X is the spectrum of a perfect fied itsef, then we consider (a) smooth projective p here. Moreover, if Λ = Q then it suffices (in the first of these statements) to assume that X is of finite type over an exceent noetherian scheme of dimension at most 3. Proof. I. Coroary of [CiD12] states that DK Z ( ) is a motivic trianguated category (see Definition of ibid.). Moreover, it is aso oriented (see Remark and Exampe (3) of ibid.). Hence Theorem of ibid. (aong with Definitions and of ibid.) impies our assertions 1 7. Moreover, we aso obtain that assertion 8 is fufied up to Tate twists (cf. Remark 1.2.2(8) beow). Since DK Z ( ) is periodic (see (K4) in of [CiD12]), the Tate twists are automorphisms of DK Z ( ); this finishes the proof of this assertion. Assertion 9 foows easiy from Theorem of ibid. Assertion 10 can be easiy estabished simiary to Theorem of [CiD12], using Proposition of ibid. To prove assertion 11 we note thatg! is eft adjoint tog whenever g: Z X is a smooth morphism. Hence the definition ofdk Z (X) (see the adjunction in [CiD12, ] and the definition of SH(X)) yieds that a objects of the form g! (Z Z ) (for a finite type g) are compact, and they weaky generate DK Z (X) (cf. Remark 4.4 of [CiD15]). Hence the trianguated subcategory of DK Z (X) that is generated by a g! (Z Z ) consists of compact objects. Lasty, it contains a compact objects of DK Z (X) by Lemma of [Nee01] (cf. aso Lemma A.2.10 of [Ke14]). I.12. f respects the compactness of objects according to assertions I.4 and I.11. The same is true for f! (if f is of finite type) by Remark 1.2.2(3) beow. By Proposition of [CiD12], these fact impy that f and f! (for a finite type f) respect coproducts. To concude the proof of assertion 13 it 15

16 remains to note that f and f! (if f is of finite type) respect coproducts since they possess right adjoints. Next, assertion 14 can be estabished simiary to Proposition 4.3 of [CiD15] (see aso Proposition 2.7 of ibid.). 15. The adjunction used in of [CiD12] yieds thatdk Z (Z X [i],z X ) = SH(X)(Σ X (X +),KG X). It remains to appy Theorem 2.20 of [Cis13]. II. Assertions 1 3 easiy foow from the corresponding statements in part I of our theorem if we combine assertions I.11, 13 with Proposition Assertion 4 can be proved simiary to Coroary of [CiD16]; see Remark beow for more detai. The first part of assertion 6 can be proved simiary to Proposition 7.2 of [CiD15], whereas in the Λ = Q-case one shoud combine Coroary of [CiD12] with Theorem of [Tem15]; cf. 2.4 of [BoD15]. Remark DK( ) is the main motivic category of this paper. We wi need the foowing observations reated to it beow. 1. For a finite type (separated) f: Y X we set MK BM X (Y) = f!(1 Y ) (this is a certain Bore Moore motif of Y ; cf. [BoD15] and I.IV.2.4 of [Lev98]). For g: X X being a morphism of (Λ-nice) schemes (that is separated but not necessariy of finite type) one easiy sees that g (MK BM X (Y)) = MK BM X (Y X X ). Next, foru Y being an open subscheme, Z = Y\U, the distinguished triange (1.2) easiy yieds the natura distinguished triange MK BM X (U) MKBM X (Y) MKBM X (Z). (1.4) (cf (BM3) of [BoD15]). Certainy, this triange yieds the corresponding ong exact sequence for any cohomoogy theory H defined on DK(X). We define Hi BM (Y) as H(MK BM X (Y)[ i]) (so, we ift the index i and change the sign of the degree in the usua way ; the reason for doing this is that we want this notation to be compatibe with the usua one for K-theory). 2. More generay, et Y α be the components of some stratification α of Y red. Then combining obvious induction with (1.2) and Theorem 1.2.1(II.5) we obtain that anym ObjDK(Y)beongs to the extensioncosure of {j,! j (M)}, where j : Y α Y are the corresponding morphisms. It easiy foows that MK BM X (Y) beongs to the extensioncosure of {MK BM X (Y α )}. This observation is especiay usefu for us if α is a reguar stratification. 16

17 3. Certainy, we aso have distinguished trianges simiar to (1.4) indk Z (X) (for any nice X). Hence DK c Z (X) contains g!(z Z ) for g: Z X being an arbitrary finite type morphism. This aows to concude the proof of Theorem 1.2.1(I.12). 4. Now we appy part 1 of this remark to (our version of) K-theory; this corresponds to H: M DK(X)(M,1 X ). We fix a quasi-projective Z/X and choose a certain smooth Y/X containing it as a cosed subscheme, U = Y \Z. Then we obtain a ong exact sequence K BM,X i (Z) K i (Y) K i (U)... (1.5) here we use the fact that g! is eft adjoint to g if g is smooth, and consider the cohomoogy theory represented by 1 X. Now et g: X X be a morphism of schemes (that is separated and not necessariy of finite type). Then, considering the cohomoogy theory represented by g (1 X ) and appying the adjunction g g, we get a ong exact sequence... K i+1 (Y X X ) K i+1 (U X X ) K BM,X i (Z X X ) K i (Y X X ) K i (U X X ) = DK(X)(MK BM X (Z),g (1 X )[ i]) (1.6) It certainy foows that K BM,X i (Z X X ) vanishes for a i < n, where n is a fixed integer, if and ony if K i (Y X X ) = K i (U X X ) for i < n and K n (Y X X ) surjects onto K n (U X X ). 5. We suspect that K BM,X i (Z) is naturay isomorphic to the K-theory of Y with the support on Z; possiby we wi treat this question in a subsequent paper (perhaps, using the resuts of [PPR12] or Theorem 1.18 of [Nav16]). At east, part 4 of this remark yieds the foowing: K BM,X i (Z X X ) vanishes for a i n if and ony if the same property is fufied for the K-theory of Y X X with the support on Z X X. 17

18 6. We wi aso need a certain (Verdier) dua to part 2 of this remark. Let Y α be the components of some stratification α of Y red. Then (1.3) combined with Theorem 1.2.1(II.5) yieds that any M ObjDK(Y) beongs to the extension-cosure of {j, j! (M)} (for the morphisms j : Y α Y ). Now assume that Y and a Y α are reguar. Then parts (I.9,II.3) of the theorem yied that j! (1 Y ) = 1 Y α for any. Hence f (1 Y ) (for any separated f: Y X) beongs to the extension-cosure of {f, (1 Y α )}, where f = f j. It certainy foows that f 1, (1 Y α 1 ) beongs to the extension-cosure of {f! (1 Y )} {(f, (1 Y α ) : 2}[1]. Lasty, assume that f is of finite type. Then for any N ObjDK(Y) the object f! (N) beongs to the extension-cosure of j, f! (N). Now take N = 1 X and assume that a Y α are reguar and quasi-projective over X. Then combining parts I.8, I.9, and II.3 of Theorem we obtain that f!(n) = f! (1 X ) = 1 Y α. Thus f! (1 X ) beongs to extensioncosure of {j, (1 Y α )}. 7. Beow we wi need ony those properties of DK( ) that are isted in our Theorem. Thus one may consider it as a ist of axioms for a system of trianguated categories. In particuar, the authors do not caim that a possibe constructions of the categories DK( ) possessing these properties are isomorphic. Moreover, one can probaby consider the foowing generaization of our setting: for R being an arbitrary torsion-free coefficient ring one may define DK(S) as the homotopy category of the category of modues over KG S R (in SH(S)). Yet such a generaization wi not affect our main resuts significanty. 8. We wi not need use the tensor structure much in this paper. Yet we note that for Beiinson motives (i.e., for Voevodsky motives with rationa coefficients that were the centra subject of [CiD12], [Heb11], and [Bon14], and wi aso be considered in 4 beow) there is a certain particuar case of tensor products that is very important (this is aso the case for cdh-motives considered in [CiD15] and [BoI15]). InK-motives we havef! (1 P 1 (X)) = 1 X 1X for any (R)-niceX. Yet for Beiinson motives (as we as for R-inear cdh-motives over characteristic p nice schemes, where R is unita ring such that p is invertibe in R whenever it is positive) we have f! (1 P 1 (X)) = 1 X 1X 1 for a certain -invertibe object 1 X 1 (that is often denoted by 1 X ( 1)[ 2]; it is 18

19 not isomorphic to 1 X ) instead (so, Beiinson motives are not periodic ). Tensor products by 1 X 1 and by its tensor powers (incuding negative ones; these product functors are caed Tate twists) commute with a the motivic image functors, whereas the corresponding axioms of Beiinson motives differ from their K-anaogues in part I.8 and I.9 of Theorem by certain (ocay constant) Tate twists; see the beginning of 3.4 beow for more detai. 1.3 Weight structures: reminder Definition I. For a trianguated category C, a pair of casses C w 0, C w 0 ObjC wi be said to define a weight structure w for C if they satisfy the foowing conditions: (i) C w 0 andc w 0 are Karoubi-cosed inc (i.e., contain ac-retracts of their objects). (ii) Semi-invariance with respect to transations: C w 0 C w 0 [1], C w 0 [1] C w 0. (iii) Orthogonaity: C w 0 C w 0 [1]. (iv) Weight decompositions: for any M ObjC there exists a distinguished triange such that A C w 0 [1], B C w 0. B M A f B[1] (1.7) II. The fu category Hw C whose object cass is C w=0 = C w 0 C w 0 wi be caed the heart of w. III. C w i (resp. C w i, resp. C w=i ) wi denote C w 0 [i] (resp. C w 0 [i], resp. C w=0 [i]). IV. We wi ca C b = ( i Z C w i ) ( i Z C w i ) the cass of bounded objects of C. We wi say that w is bounded if C b = ObjC. Besides, we wi say that eements of i Z C w i are bounded beow. 19

20 V. Let C andc be trianguated categories endowed with weight structures w and w, respectivey; et F : C C be an exact functor. The functor F wi be caed eft weight-exact (with respect to w,w ) if it maps C w 0 into C w 0 ; it wi be caed right weight-exact if it maps C w 0 to C w 0. F is caed weight-exact if it is both eft and right weight-exact. Remark A weight decomposition (of any M ObjC) is (amost) never canonica; sti (any choice of) a pair (B,A) coming from (1.7) wi be often denoted by (w 0 M,w 1 M). More generay, for any m Z shifting a weight decomposition of M[ m] by [m] we obtain a distinguished triange w m M M w m+1 M with some w m+1 M C w m+1, w m M C w m ; we wi ca it an m-weight decomposition of M. 2. A simpe (and yet usefu) exampe of a weight structure comes from the stupid fitration on K(B) (for an arbitrary additive category B; see 1.1). We take K(B) w 0 = K(B) 0 (in the notation of ibid.), and take K(B) w 0 being the simiary defined K(B) 0 ; we ca this weight structure the stupid one. 3. In the current paper we use the homoogica convention for weight structures; it was previousy used in [Heb11], [Bon14], [Bon15], [Bon13], and [BoI15], whereas in [Bon10a] and in [Bon10b] the cohomoogica convention was used. In the atter convention the roes of C w 0 and C w 0 are interchanged, i.e., one considers C w 0 = C w 0 and C w 0 = C w 0. For exampe, a compex M ObjK(B) whose ony non-zero term is the fifth one has weight 5 (with respect to the stupid weight structure) in the homoogica convention, and has weight 5 in the cohomoogica convention. Thus the conventions differ by signs of weights. Now we reca some basic properties of weight structures. Proposition Let C be a trianguated category endowed with a weight structure w, M ObjC, i,j Z. Then the foowing statements are vaid. 1. The axiomatics of weight structures is sef-dua, i.e., for D = C op (so ObjC = ObjD) there exists the (opposite) weight structure w for which D w 0 = C w 0 and D w 0 = C w C w i, C w i, and C w=i are Karoubi-cosed and extension-cosed in C (and so, additive). 3. Let j < i. Then for any choices of weight decompositions corresponding 20

21 to the rows of the square w j M M c ji w i M M id M there exists a unique morphism c ji making it commutative. Moreover, Cone(c ji ) C [j+1,i]. 4. C w i = (C w i 1 ). 5. If A B C A[1] is a C-distinguished triange and A,C C w=0, then B = A C. 6. If A C B is a C-distinguished triange then for any fixedw i 1 A,w i 1 B there exists a weight decomposition of C such that w i 1 C is an extension of w i 1 B by w i 1 A. 7. Let C and D be trianguated categories endowed with weight structures w and v, respectivey. Let F : C D : G be adjoint functors. Then F is eft weight-exact if and ony if G is right weight-exact. 8. Assume that M beongs to the enveope (see 1.1) of some cass of M j ObjC (for j J); we fix a choice of w i 1 M j (for these objects). Then M C w i if and ony if w i 1 M j M. In particuar, if J = J 1 J 2 such that M j C w i for any j J 1 and M j C w i 1 for any j J 2, then it suffices to check whether M j M for a j J 2. Proof. Assertions 1 5are contained in Theorem of [Bon10b] (whereas their proofs reied on [Bon10a]) and assertion 6 is a part of Lemma of [Bon10a] (pay attention to Remark 1.3.2(3)!). Assertion 7 is just Proposition 1.2.3(9) of [Bon14]. It remains to verify assertion 8. Obviousy, the in particuar part of the assertion is just a particuar case of the genera statement preceding it; so we prove the atter. Furthermore, the orthogonaity axiom of weight structures yieds that N M if N C w i 1 and M C w i ; so the ony if part of the statement is cear. We prove the converse impication. We start with the foowing easy observation: for any choice of ani 1-weight decomposition triangew i 1 M M w i M we have M C w i if and ony if w i 1 M M. Indeed, the 21

22 atter condition yieds that M is a retract of w i M (whereas the converse impication is immediate). Hence it suffices to verify that any object M in the enveope of {M j } possesses a shifted weight decomposition (as above) such that w i 1 M beongs to the enveope of {w i 1 M j }. To this end it certainy suffices to combine assertion 6 with the foowing statement: for A, some fixed w i 1 A and C being a retract of A there exists a weight decomposition of C such that w i 1 C is a retract of w i 1 A. The atter resut can easiy be estabished using the corresponding arguments from the proof of Lemma of ibid. (at east, in the case where C is Karoubian; cf. the remark beow). Remark Whereas the case of a Karoubian C is certainy sufficient for the purposes of the current paper, we note that there exists an aternative (and more eegant) proof of assertion 8. It reies on the properties of the so-caed virtua t-truncations of cohomoogica functors (see 2.3 of [Bon10b] and Appendix A.3 of [Bon14]). We consider the functor F = τ 1 i (C(,M)): C op Ab; note that it is cohomoogica (aso). Ifw i 1 M j M (for a j) then F(M j ) = 0. Hence in this case we have F(M) = {0}. It remains to note that the atter impies that M C w i. In 2.3 and 4 beow we wi need certain (more or ess, new) properties of weight structures extended from subcategories of compact objects. Proposition Let C C be trianguated categories such that C contains a sma coproducts of its objects, C is is essentiay sma, and the objects of C are compact in C. Let w be a weight structure for C. Then the foowing statements are vaid. 1. The sets C w 0 = C w 1 C and C w 0 = C (C w 1 ) yied a weight structure for C. 2. Hw is (naturay) equivaent to the Karoubi enveope of the category of a coproducts of objects of Hw. 3. Assume that C w 0 is the enveope of some set {C i} of its objects. Then C w 0 = {C i [ 1]} C. 4. The embedding C C is weight-exact (with respect to w and w ). 5. Let (D,D,v,v) satisfy simiar conditions (i.e., D D are trianguated categories, D contains a sma coproducts of its objects; D is is essentiay sma and its objects are compact in D, v is a weight structure for D and v is its extension to D). Let F : C D :G be 22

23 adjoint exact functors, and assume that F maps C into D. Then the foowing statements are equivaent: (a) F(C w 0 ) D v 0. (b) G is right weight-exact. (c) F is eft weight-exact. 6. For (C,w) and (D,v) as above and any exactg: D C that commutes with (sma) coproducts we have the foowing: G is eft weight-exact whenever G(D v 0 ) C w For any M i ObjC (for i running through some index set) we have Mi C w 0 (resp. Mi C w 0 ) if and ony if M i C w 0 (resp. M i C w 0 ) for a i. Proof. 1,2. Immediate from Theorem of [Bon10a] (cf. aso Remark 4.5.3(2) of ibid. and Proposition 1.3.3(4)). 3. Immediate from the foowing obvious fact: C w 1 C = {C i [ 1]} C. 4. The orthogonaity axiom of weight structures immediatey impies that C w 0 C w 0 (i.e., that our embedding is eft weight-exact). Appying the axiom once more, we aso obtain the eft weight-exactness in question. 5. Let N D v 0 and assume F(C w 0 ) D v 0. Then for any M C w 1 we have F(M) N. Next the adjunction yieds M G(N), and so the condition 5b is fufied. Condition 5b impies condition 5c by Proposition 1.3.3(7). Lasty, condition 5c impies condition 5a by the previous assertion. 6. This is an easy consequence of the descriptions of D v 0 and C w 0 that can be obtained by duaizing Theorem of [Bon13]. 7. Obvious. Remark Theorem 5 of [Pau12] states that one can define a weight structure for C starting from the right enveope of any set of objects of C (instead of C w 1 in assertion 1). Yet we do not need this more genera fact in the current paper. 2. One can obtain severa more properties of weight structures obtained this way by duaizing Theorem of [Bon13]. 3. For arbitrary C C and w for C one may turn assertion 1 of the proposition into a definition as foows: a weight structure w is caed the (right) extension of w (to C) if C w 0 = C w 1 C and C w 0 = C (C w 1 ). In particuar, w is the right extension of itsef to C. Certainy, the natura anaogues of assertions 3 5 are vaid for this more genera setting. 23

24 Duay, one may consider eft extensions instead of the right ones; they correspond to the setting considered in ibid. (and to cocompacty cogenerated categories). 4. Some more properties of extended weight structures can be easiy obtained by axiomatizing the setting of Proposition beow. 5. Note that for C and D as in assertion 5 a right adjoint to an exact F : C D exists if and ony if F respects coproducts; see Theorem of [Nee01]. So one can appy the equivaence of conditions 5a and 5c of the assertion for any F satisfying this additiona condition (without specifying G). 1.4 On weight compexes and weight spectra sequences Now we reca some of the properties of weight compexes and weight spectra sequences. The ony pace preceding 4 where they wi be needed is Remark 2.3.3; so that the reader mosty interested in sections 2 3 may ignore the current section. Proposition Let C be a trianguated category endowed with a weight structure w, M ObjC, n Z. Then the foowing statements are vaid. 1. For a i Z fix some choices of w i M and denote Cone(c i 1,i )[ i] (see Proposition 1.3.3(3) by M i. Define (a choice of) the weight compex t(m) for M as the compex whose terms are M j (for j Z) and the connecting morphisms are given as the corresponding compositions M j (w j 1 M)[j +1] M j+1. Then t(m) is a compex indeed (i.e., the square of the boundary is zero); a M i beong to C w=0 (so, we are abe to consider t(m) as an object of K(Hw)). 2. M determines t(m) up to a homotopy equivaence. 3. If M is bounded beow, thenm C w n if and ony ift(m) K(Hw) n. 4. If M beongs to the enveope of certain M i ObjC then t(m) beongs to the K(Hw)-enveope of t(m i ). 5. Let d i : M i 1 M i for i Z be a chain of C-morphisms such that Cone(d i ) C w=i for a i, M i = 0 for i 0, and M i = M (with d i+1 being isomorphisms) for i 0. Then M i give certain choices of w i M and d i yied the corresponding c i 1,i. 24