Weight structures and motives; comotives, coniveau and Chow-weight spectral sequences, and mixed complexes of sheaves: a survey

W eigh t structures and motiv es; comotiv es, coniv eau and Cho w-w eigh t sp ectral sequences, and mixed complexes of shea v es: a surv ey M.V. Bondark o, St. P etersburg State Univ ersit y ∗ Octob er 26, 2018 Con ten ts 1 In tro duction 2 2 Categoric notation; definitions of V o ev o dsky 2 3 Main motivic results 4 4 W eight structures: basics 6 5 On functorialit y of weigh t de compos itions; truncations for cohomolo gy 9 6 Dualities of triangulated categorie s ; orthogonal and adjacen t weigh t and t -structures 10 7 W eight sp ectral sequences 12 8 More on w eight structures 13 8.1 ’F unctoriality’ of weight structures: lo calizations and gluing . . . . . . . . . . 13 8.2 The weight complex functor . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 8.3 Certain K 0 -calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 8.4 A ge ner alization: relative w eight structures . . . . . . . . . . . . . . . . . . . 15 9 ’Motivic’ w eigh t structures; comotiv es; glui ng Chow and Gers te n struc- tures from ’bi rational s lices’ 15 9.1 Chow w eight structure(s); rela tio n wit h the motivic t -structure and weigh t filtration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 9.2 Comotives; the Gersten weigh t structure . . . . . . . . . . . . . . . . . . . . . 16 ∗ The author gratefully ackno wledges the supp or t from Deligne 2004 Balzan prize in mathematics. The wo rk is also supported b y RFBR (gran ts no. 08-01-00777a and 10-01-00287). 1 9.3 Comparison o f weigh t structures; ’gluing fro m birational slices’ . . . . . . . . 17 9.4 W eights for r elativ e motives and mixed sheaves . . . . . . . . . . . . . . . . . 18 10 P o ssible applications to finite-dimensionali t y of motiv es 20 1 In tro duction This article is a surv ey of author ’s r esults on V o ev o dsky’s motiv es and w eig h t structures; yet it is supplied with detailed refer e nce s. W e ig h t str uctur es are natural counterparts of t -structures (for triangulated ca tegories) intro duced b y the author in [Bon07] (and also in- dependently b y D. P auks ztello in [P au08]). They allow to to construct weight c omplexes , weight filtr ations , and weight sp e ct ra l se quenc es . Partial c a ses of the latter are: ’classical’ weigh t sp ectral seq uence s (for sing ula r and étale cohomolo gy), coniveau spe c tral s equences, and Atiy ah-Hir z ebruc h sp ectral sequences (w e mention all of these b elow). The details , pro ofs, and several mor e res ults c o uld b e found in [Bon0 7], [B on10a], a nd [Bon0 9] (we also men tion certa in results o f [Bo n10b], [Heb10], and [Bon10p]). W e descr ibe more motiv atio n for the theory o f weigh t structures, and define w eight structure s in §4. Though our ’main’ weigh t structures will b e defined on certai n ’mo tivic’ catego ries, the author tried to make this sur v ey a ccessible to reader s that are rather int erested in ge ner al triangulated categorie s (or possibly , the stable homotopy category in top o logy). Those rea d- ers ma y freely ig nore all definitions and results that are r elated with a lgebraic geometry (a nd motives). On the other hand, the main mo tivic r e s ults (see §3 ) co uld b e understo od without knowing anything ab out weight structures (a fter §3 a ’mo tivic’ rea der may pro ceed dir ectly to §9 to find some more motiv ation to study w eight structures). Alternativ ely , it is quite po ssible for an y r e a der to rea d section §3 only after studying the g eneral theor y of weight structures (§§4–8). The author chose not to pay m uch attention to the differential graded approa c h to motives in this text; y et it is des cribed in detail in [Bon09 ] a nd in §6 of [Bon07] (see also [Be V0 8 ] and §8.2 b e low). This text is ba s ed o n the talks presented by the author at the conferences "Finiteness for motives and motivic cohomo logy" (Regensburg, 9–13 th of F ebr ua ry , 2 009) and "Motivic homotopy theory" (Münster, 27 -31st of July , 2009 ); yet some more recent topics ar e a dded. The a uthor is deeply grateful to prof. Uw e Jannsen, pr o f. Eric F r iedlander, a nd to other organizer s of these conferences for their effor ts. 2 Categoric notati on; definitions of V o ev o dsky F or a catego ry C , A, B ∈ O bj C , we denote by C ( A, B ) the set of C -morphisms fro m A to B . Below B will b e some additiv e category; K b ( B ) ⊂ K ( B ) will denote the homotopy category o f (b ounded) B - complexes. C and D will b e tr ia ngulated categories ; for f ∈ C ( X, Y ) , X, Y ∈ O bj C , we will denote the third vertex of (any) distinguished triangle X f → Y → Z b y Cone( f ) . F or D , E ⊂ Obj C we will write D ⊥ E if C ( X , Y ) = { 0 } for all X ∈ D , Y ∈ E . 2 A will be an ab elian catego ry , D ( A ) is its derived ca tegory; H : C → A will usually b e a cohomolog ical functor (i.e. it is contrav ar ian t, and co nverts distinguished triangles in to long exact sequences in A ). Kar( B ) for any B w ill deno te the Karoubiza tion of B i.e. the catego r y of ’forma l images’ of idemp otent s in B (so B is embedded into an idemp oten t complete catego r y). A full sub category C ⊂ B is called Kar oubi-close d in B if C con tains all B -retr a cts o f its ob jects; Ka r B C will denote the smalles t K aroubi-closed subca tegory of B that contains C (i.e. its ob jects are a ll re tr acts of ob jects of C that b elong to B ). Ab is the ca tegory of ab elian groups. Now we introduce our ’motivic’ definitions; they could b e esp ecially interesting to rea ders that a r e aw a re of ’classical’ mo tives but do not know muc h abo ut V o ev o dsky’s ones. k is our p erfect bas e field. F rom time to time w e will hav e to assume that e ither char k = 0 or that we co nsider (co)motives and co homology with ra tional (or Z [ 1 p ] -) co efficients. S mP rV ar ⊂ S mV ar ⊂ V ar are the sets of (smo oth pro jective) v arieties over k . The definition of V o ev o dsky’s motives starts from smo o th corr espondences (see [V o e00]): Ob j S mC or = S mV ar ; S mC or ( X, Y ) = Z { U } : U ⊂ X × Y is clo sed reduced, finite dominant ov er a co mponent of X . Comp ositions of mor phisms a re g iven b y a natura l a lgebraic ana logue of the comp o sition o f m ulti-v alued functions. R emark 2.1 . 1. So, in co n trast to the ’class ical’ definition, w e c o nsider only those primitive corresp ondences (i.e. closed sub v arieties of X × Y of a certain dimension) that are finite ov er X . Note her e that any ’clas sical’ c orrespo ndence is rationa lly equiv alent to some finite one. The adv antage of finite co r respondences is that the c omposition is well-defined without factorizing mo dulo a n equiv alence rela tion. This is v ery imp ortan t! 2. F or any commutativ e asso ciative ring with a unit R instead of S mC or one can consider a certain category S mC or R ; in o r der to define it one should just replace Z { U } b y R { U } in the definition of S mC or ( X, Y ) . This allows to c o nstruct a reaso nable theory of V o evodsky’s motives with R -co efficients; see [MVW06]. Usually one takes R = Z or R = Q . In [Bon10b] the author a ls o co nsiders certain in termediate co efficient s rings . The case R = Z / ( n ) (for n > 1 ) is also interesting. Cartesian pro duct of v arieties yields tenso r structure for S mC or (as well as for K b ( S mC or ) ). One can define (homolo gical) Chow motives in terms o f S mC or . One s tarts from the cate- gory of rational corre s pondences: Ob j C or r r at = S mP r V ar ; C or r r at ( X, Y ) = S mC or ( X, Y ) /ra tional equiv alence. Now, one has C how ef f = Kar( C or r r at ) (this yields a catego ry that is isomorphic to the ’classical’ effective Chow mo tiv es). F o rmal tensor in version of Z (1)[2] (t he Lefschetz motif i.e. the ’complement’ of a p oint to the pro jective line) yields the who le ca teg ory C how . D M ef f gm is defined a s the Karo ubization of a certain lo calization of K b ( S mC or ) (so it is triangulated). T ensor inv er sion of Z (1)[2] in it y ields D M gm . W e denote by M the comp osition S m V ar → S mC or → K b ( S mC or ) → D M ef f gm ; this defines motives o f smo oth v arieties. If c har k = 0 , in D M ef f gm there also exist motives and certain motives with c omp act supp ort for arbitra ry v arieties. V o ev o dsky constructed the following diagram of functors: 3 C how ef f − − − − → C how   y   y D M ef f gm − − − − → DM gm (1) Here all arrows are full embeddings of a dditi ve ca teg ories. In §3.1 of [V o e00] V o evodsky a ls o defined a certain triang ula ted categ o ry DM ef f − ⊃ D M ef f gm . 3 Main m otivic results W e list our main results. Assertions 1– 6 requir e char k = 0 (y et see Remark 3.2(5) below). Theorem 3.1 . 1. In §3 of [Bo n09] D M ef f gm was describ e d ’explicitly’ in terms of twiste d c omplexes over a c ertain differ ent ia l gr ade d c ate gory J (se e §2.4 of ibid .); t he obje cts of J ar e cubic al Suslin c omplexes of smo oth pr oje ctive varieties. 2. This description i s somewhat similar to (yet ’mor e c onvenient’ than) those of Hana- mur a’s motives (se e [Han04]). T his al lowe d to c omp ar e V o evo dsky’s motives with Hana- mur a’s ones: in §4 of [Bon09] it was pr ove d t ha t D M gm Q is anti-isomorphic to Hanamu ra ’s motives. 3. ’Kil ling all arr ows of ne gative de gr e es’ in the ’descriptio n ’ of D M ef f gm yields an exact weight c omplex functor t : D M ef f gm → K b ( C how ef f ) ; it c ould also b e extende d to t gm : D M gm → K b ( C how ) . In §6 of [Bon09] it was also pr ove d that these functors ar e c onservative (i.e. t gm ( X ) = 0 = ⇒ X = 0 ). 4. t gives K 0 ( D M ef f gm ) ∼ = K 0 ( C how ef f ) and K 0 ( D M gm ) ∼ = K 0 ( C how ) (se e §6.4 of [Bon09]; a gener alization and c ertain variations of these r esults ar e describ e d in §§5.3-5.5 of [Bon07]). S e e §8.3 b elow for the definitions of these gr oups and the formulatio n of t he gener alization. 5. Motivic al ly functorial weight sp e ctr al se qu en c es for any c ohomolo gy the ory H : DM ef f gm → A (gener alizing Deligne’s ones for étale and singular c oho molo gy of varieties) wer e c on- structe d (se e §6.6 and Rema rk 2.4.3 of [B on07]; t he y wer e c al le d Chow-wei ght sp e ctr al se- quenc es sinc e they c orr esp ond to t he Chow weight structur e; se e §7 b elow). 6. Al l triangulate d sub c at e gories and lo c alizations o f D M ef f gm wer e ’describ e d’ (s e e §8.1– 8.2 of [Bon09]). In p articular, one obtains ’r e asonable’ descrip tions of T ate motives and of the (triangulate d) c ate gory of bir ational motives (i.e. of the lo c alization of D M ef f gm by D M ef f gm (1) ; se e [KaS02]) this way. 7. A c ertain c ate gory D (of c omotives) that c ontains ’nic e homo topy limits’ of V o evo dsky’s motives was c onstructe d (se e §3.1 and §5 of [ Bon10a], a nd also §9 .2 b elow). In p articular, it c ontains c ertain (c o)motives for al l funct io n fields over k . Some of t he pr op erties of D ar e dual (in a c ert ain sense) to the c orr esp onding pr op erties of the ’usual lar ge motivic’ c ate gories. In p articular, though we have a c ovariant emb e dding D M ef f gm → D , it yiel ds a family o f c o c omp act c o gener ators for D . This is why we c al l the obje cts of D c omotives. Comotives ar e cru cial for the pr o of of the fol lowing r esu lts. 4 8. Ther e exist (c ertain) motivic al ly functorial c onive au sp e ctr al se quenc es for c ohomolo gy of arbitr ary motives (se e §4.2 of [Bon10a]; cf. §7.4 of [Bon07]). Besides, for H r epr esente d by a motivic c omplex (i.e. an obje ct of D M ef f − ) we pr ove that these sp e ctr al se quenc es c an b e describ e d in t erms of the homotopy t -trunc ations of H . This vastly extends seminal r esults of Blo ch and Ogus (se e [BOg94]). 9. L et k b e c ountable. Then the c ohomolo gy of any smo oth semi-lo c al scheme (over k ) is a dir e ct summand of the c ohomolo gy of its generic p oint; the c ohomolo gy of funct io n fields c ontain twiste d c ohomolo gy of their r esidue fields (for al l ge ometric valuations) as dir e ct summands. R emark 3.2 . 1. The term ’w eight complex’ w as proposed b y Gillet and Soulé in [GiS96]. Their functor was essentially the restriction of t to motives with compact s upport of v arie ties (see §6.6 of [B on09 ]). Besis des, in [GNA02] a functor that is essentially t ◦ M was defined. An y of these functors a llo ws to compute E 2 of the corr esponding (Cho w)-weigh t sp ectral sequences (see a ssertion 5). Hence for (rationa l) singular/étale cohomolo gy of v ar ieties (and motives) it computes the factors o f the (’ordinary’) w eight filtration; whence the name. 2. Parts 3–5 o f the Theorem will b e v astly g eneralized b elow (to triangulated categ ories endow ed with weigh t structures). They follow from the existence of a certa in Chow weight s t ructur e for D M ef f gm ; whereas assertions 8 –9 follow from the existence of a certain Gersten wei ght structur e for some tri- angulated D s such that D M ef f gm ⊂ D s ⊂ D . 3. Recen tly (indep enden tly in [Heb10] a nd in [Bon10p]) it was also prov ed that the Chow weigh t structure c o uld b e defined for th e catego ry of V oevo dsky’s motives with ra tional co efficien ts o ver a n y ’reasonable’ ba se scheme S (in [CiD09] where the ba sic prop erties of S -motives were es tablished, they were ca lled Beilinson’s mo tiv es; one could either cons ider the ’large’ ca tegory DM ( S ) of S -motives or its subc ategory DM c ( S ) o f c onstructible i.e. ’geometric’ ob jects here). The heart of this w eig h t structur e is ’generated’ b y (cer tain) motives of r egular schemes that a r e pro jective ov er S (tensor ed b y Q ( n )[2 n ] for all n ∈ Z ; see §9.4 b elow) . So, we obtain certain a nalogues of pa rts 3 –5 of the Theor em for S -mo tiv es also. In §3 of [B o n10p ] the weigh ts for S -motives were also related with the ’classical’ weights of mixed complexes of sheaves. T o this end the notion of a r elative weight structu r e was in tro duced; s e e Definition 8.1 b elo w. Besides, in [Lev09] for S b eing a v ariety ov er a characteristic zero field an explicit differen- tial gr aded description o f a certain sub category of DM c ( S ) was given; this is a gener alization of assertion 1 of the Theorem. Possibly , this r esult could b e extended to the whole D M c ( S ) (at lea st, with rational co efficient s). 4. No explicit comparison functor in the ’description’ of part 1 is known (the t wo tri- angulated c ategories in questio n are co mpared by means of a third triangula ted catego r y). Note also that the category of twisted complexe s cons ider ed is a ’t wisted’ a nalogue of K b ( B ) i.e. o ne considers morphisms and ob jects up to (a certain) homotopy equiv alence. Hence in order to work with V o evodsky’s motives one needs constructions that do not depend on the choice o f r epresen tatives in these homotopy equiv alenc e cla sses. W eight structures really help her e ! 5. All the asser tions of the theo rem remain v a lid if we r eplace motives with integral co efficien ts b y those with ra tional (or Z /n Z -) ones; see Rema rk 2.1(2). 5 Besides, the requirement char k = 0 is only needed to apply the resolution of singularities (that is required to prove s ome of the sta temen ts in [V o e00], which are necessary to deduce our results). F or motives with r ational co efficient s (we denote them b y C how ef f Q ⊂ DM ef f gm Q ⊂ D M gm Q ) for most of purp oses it suffices to apply de Jong’s alter a tions. In pa rticular, this allows to prove the ’rationa l’ analogues of as sertions 3– 5 also for any per fect k o f characteristic p . Moreov er, a recent resolution of singularities r esult of Gabb er (see Theorem 1.3 of [Ill0 8]) allows also to prov e the analogues of ass e r tions 3 –5 with Z [ 1 p ] -co efficien ts (over k ). Note here: Gabber’s theorem co uld be called ’ Z ( l ) -resolution of singularities’ (for all l ∈ P \ { p } ); yet weigh t structure metho ds a llo w to deduce motivic results with Z [ 1 p ] -co efficien ts (that is a prio r i mor e difficult); see [Bon1 0b ]. 6. In §6.3 of [Bon09] a certain length of motives w a s defined (this is the ’length’ of t ( X ) ). This is a mo tivic analogue o f the length of the weigh t filtration for mixed Ho dge structures (co ming from cohomolog y of v arieties). In particular, the length of a mo tif of a smo oth v ariety is is not grea ter than its dimension and not less than the leng th of the weigh t filtration for its co homology . 7. One ca n prove more than conserv ativity for t . In par ticular, X ∈ O bj D M gm is mixed T ate whenever t gm ( X ) is (see Corollar y 8.2.3 of [Bon09]). 4 W eigh t s tructures: basics Now we define weigh t structures. They a re related with stupid truncations o f complexes (i.e. of ob jects of K ( B ) ) in a wa y s imila r to the relation of t - structures w ith canonical truncations (see [BBD82] for the foundations of th e theory of t - structures); certainly , the distinctions here ar e a ls o very s ignifican t! Stupid truncations are not very p opular since they are not canonical (whereas canonica l truncations a re canonica l and functorial). Y et we will expla in (starting fro m §5 below) how they do yield plent y of functorial (’cohomolog ical’) information; these results ar e new even for C = K ( B ) . There are a lot of ex a mples when non-ca nonical constructions yield im- po rtan t functorial information: pro jective and injective resolution of ob jects a nd complexes ov er ab elian catego ries allow to define derived functor s ; nice co mpactifications and smo oth h yp er-resolutions of v arieties a llo w to define weigh t sp ectral seq uences fo r étale a nd singular cohomolog y; skeletal filtration for top ological spec tr a allow to co ns truct Atiy ah-Hir z ebruc h sp e ctral sequences for their cohomology . All o f these o bserv ations hav e very natural ’expla- nations’ inside the theo ry of weigh t s tructures! W eight structures ha ve (a t least) t wo distinct incarna tions imp ortant f or V o evodsky’s motives (related to weight and coniveau sp e c tr al sequences), and also one that is relev ant for the stable homotopy category (in top ology). Y et first we illustrate some ba sics of the theory on a (more) s imple (though quite interesting) example. F or C = K ( B ) we denote by C w ≤ 0 the cla ss of complexes, ho motop y equiv a lent to those concentrated in non-p ositiv e degrees; we denote b y C w ≥ 0 the class complexes , equiv alent to those concentrated in deg r ees ≥ 0 . Then the classe s of co mplexes descr ibed satisfy the following prop erties (we write them down in the for m that reminds the axioms of t -structures ; this is quite conv enient). 6 Definition 4.1 (Axioms o f weigh t structures) . (i) C w ≥ 0 , C w ≤ 0 are additive and Ka roubi- closed in C . (ii) ’Sem i-in v ariance’ with respect to tr anslations. C w ≥ 0 ⊂ C w ≥ 0 [1] , C w ≤ 0 [1] ⊂ C w ≤ 0 . (iii) Orthogonali ty . C w ≥ 0 ⊥ C w ≤ 0 [1] . (iv) W eig h t de compos itions . F or any X ∈ O bj C there exists a disting uis hed triangle B [ − 1] → X a → A f → B (2) such that A ∈ C w ≤ 0 , B ∈ C w ≥ 0 . F or an y triangulated ca tegory C we will say that the classes ( C w ≤ 0 , C w ≥ 0 ) y ield a weigh t structure if they s a tisfy the prop erties listed. R emark 4.2 . 1. F o r C = K ( B ) we can ta ke weigh t decomp ositions coming ’stupid trunca- tions’ of complexes; s ee the illustration: X = . . . − − − − → X − 2 − − − − → X − 1 − − − − → X 0 − − − − → X 1 − − − − → X 2 − − − − → . . .   y a A = . . . − − − − → X − 2 − − − − → X − 1 − − − − → X 0 − − − − → 0 − − − − → 0 − − − − → . . .   y f B = . . . − − − − → 0 − − − − → 0 − − − − → X 1 − − − − → X 2 − − − − → X 3 − − − − → . . . 2. In this partial ca se ( C = K ( B ) ) we a ls o ha ve an o pposite orthogona lit y prop ert y ( C w ≤ 0 ⊥ C w ≥ 1 ); yet this additional orthog o nalit y is not imp ortant, and do es not genera lize to other (more interesting) exa mples. 3. F o r t -structures the orthogo nalit y axiom is opp osite; also, the arrows in t -decomp ositions ’go in the conv er se dir e c tio n’. These distinctions r esult in a dras tic difference b et ween the prop erties o f these tw o t y pes of structures. Note that dualization do es not change anything here (since the axioma tics of t -s tructures is se lf-dual, a s w ell as the one of weigh t structures). 4. W e demand (in (i)) C w ≥ 0 and C w ≤ 0 to b e K aroubi-closed; this is a technical condition that is not really imp ortant . The corr esponding condition for t -structure s is also true (though in co n trast to the weigh t structure s itua tio n, it follows from the remaining a xioms). W e also define the he art H w of w (similarly to hear ts of t -str uctur es): O bj H w = C w =0 = C w ≥ 0 ∩ C w ≤ 0 , H w ( X, Y ) = C ( X , Y ) for X , Y ∈ C w =0 . Now we list s o me very basic pr operties of weigh t structures (and their hear ts). Theorem 4.3. 1. C w ≤ 0 , C w ≥ 0 , and C w =0 ar e extension-stable i.e. for a distinguishe d triangle A → B → C if A, C b elong to C w ≤ 0 (r esp. to C w ≥ 0 , r esp. to C w =0 ) then B b elongs to the c orr esp onding class also. 2. If A → B → C → A [1 ] is a distinguishe d triangle and A, C ∈ C w =0 , then B ∼ = A ⊕ C . 3. H w is ne gative i.e. H w ⊥ ∪ i> 0 H w [ i ] . 7 4. Conversely, for a triangulate d C let an additive D ⊂ O bj C b e ne gative; supp ose that the sm al lest triangulate d sub c ate gory of C c ontaining D is C itself. Then t her e exist s a unique weight structu re w f or C such that D ⊂ C w =0 ; for it we have H w = Kar C D (se e The or em 4.3.2 of [Bon07]). One c an c onst ru ct al l b ounde d weight structur es (i.e. those ones that satisfy ∩ i ∈ Z C w ≤ 0 [ i ] = ∩ i ∈ Z C w ≥ 0 [ i ] = { 0 } ) t his way. R emark 4.4 . 1. Example s Assertion 4 allows to co nstruct the ’stupid’ weight structure for K b ( B ) mentioned above (note: as for t -structur e s , a single C may supp ort more than o ne dis tinct weigh t structures). Besides, in the stable homo top y categ ory S H there a re no morphisms of p ositive deg rees betw een c o products of the spher e sp ectrum S 0 . Hence asser tion 4 a llo ws to construct a certain weigh t structure for the sub category S H f in ⊂ S H o f finite sp ectra. In §4 of [Bon07] several other exis tence of weigh t s tr uctures r e sults (for u n b ounded weigh t structures) were prov ed. In pa rticular, they allow to construct a cer tain w S 0 for the whole S H (see §4 .6 of ibid.). The corr esponding weigh t decomp ositions corresp ond to cellular filtration of sp ectra; one can obta in Atiy a h-Hirzebruc h sp ectral s equences this way (as weight sp e ct ra l se quenc es ; see b elo w)! Lastly , C how ef f is n egative inside DM ef f gm ⊂ D M ef f − ; C how is negativ e inside D M gm (see (1)). This allows to constr uct certain Chow weight structures for all o f these ca tegories. W e denote a ll of them b y w C how , since they a re compatible; see §§6 .5-6.6 of [Bon07], a nd also Remark 3.2 a bov e. 2. Assertio n 4 demonstrates that in the bo unded case a weigh t structure could b e com- pletely des cribed in terms o f its heart; so instead of weigh t str uctures in this case one can consider o nly negative Karoubi-clos e d genera ting subcatego r ies of C . Y et w eig h t deco m- po sitions are very impor tan t (so it do es no t seem wise to avoid men tioning them in the axioms)! 3. The ob vious a nalogue of assertio n 4 for t -structures (i.e. w e w ant to construct a t -structure such that a given p ositive D ⊂ C lies in its heart) is very far from b eing true. So, negative s ubcategor ies of triangulated categor ies are muc h more v aluable than p ositiv e ones! Besides, weight structures ’are more likely to exist for small tria ngulated categories ’ (than t -s tr uctures); see Remark 4 .3.4 of [Bon07]. 4. Y et another distinction of w eight structures fro m t - structures is demo nstrated by assertion 3: distinguished tria ngles in C do not yield non-trivial extensions in H w . In fact, one may say that the notion of the heart of a weigh t structure is a ’triangulated analogue’ of the categor y of pro jectiv e (or injectiv e ) ob jects of an ab elian categor y A . Note here: w e ha ve D ( A )( P, Q [ i ]) = { 0 } if i 6 = 0 and P , Q are b oth pro jective (or injectiv e ) ob jects of A ; this allows to construct resolutions of ob jects of A (and hyperreso lutions o f complexes) that are functoria l up to ho motop y equiv ale nce . The theory of weight structures demonstrates that one mostly needs D ( A )( P, Q [ i ]) = { 0 } if i > 0 ; the absence o f ’po sitiv e extensions’ is sufficient to prov e certain functoria lit y of the cor responding ’reso lutions’ (i.e. P ostniko v tow ers); s e e b elow. So, weigh t s tructures yield a v ast genera lization of pro jectiv e and injective hyperreso lutions! 8 5 On functorial it y of w eigh t decomp ositions; truncations for cohom o logy Now w e discuss to what extent weigh t decompo sitions are functorial, and how this allows to define nice canonical ’tr unca tions’ a nd filtration for co ho mology . W eight decomp ositions (as in (2)) ar e (almost) nev er unique. Still w e w ill denote any pair o f ( A, B ) as in (2) by X w ≤ 0 and X w ≥ 1 . X w ≤ l (resp. X w ≥ l ) will denote ( X [ l ]) w ≤ 0 (resp. ( X [ l − 1]) w ≥ 1 ). w ≤ i X (resp. w ≥ i X ) will denote X w ≤ i [ − i ] (res p. X w ≥ i [ − i ] ). Now we observe that w eight decomp ositions a r e ’weakly functorial’. Prop osition 5.1. 1. Any g ∈ C ( X , Y ) c ould b e c omplete d (non-uniquely) to a morp hism weight de c omp ositions. 2. Mor e over, for any i ∈ Z , j > 0 , g ext ends to a diagr am w ≥ i +1 X − − − − → X − − − − → w ≤ i X   y   y g   y w ≥ i + j +1 Y − − − − → Y − − − − → w ≤ i + j Y (3) in a u nique way if we fix the c orr esp onding weight de c omp ositions. R emark 5.2 . 1. A nice illustration for a s sertion 1 is: for C = DM ef f gm , w = w C how , it implies (in particular) that any morphism o f smoo th v arieties (coming fro m S mV ar , S mC or , o r D M ef f gm ) could be completed in DM ef f gm to a mo rphism of (any c hoices of ) their smo oth com- pactifications. Note: though o ne can prov e this statement eas ily without weight s tructures, yet it is s o mewhat ’co un terintuitiv e ’. 2. F or C = K ( B ) ass e r tion 2 mea ns: if we fix the choice o f w eight decomp ositions, then the diag ram . . . − − − − → X − 2 − − − − → X − 1 − − − − → X 0 − − − − → X 1 − − − − → X 2 − − − − → . . .   y g − 2   y g − 1   y g 0   y g 1   y g 2 . . . − − − − → Y − 2 − − − − → Y − 1 − − − − → Y 0 − − − − → Y 1 − − − − → Y 2 − − − − → . . . is compatible with a unique c hoice of the following dia gram ( . . . − − − − → X − 2 − − − − → X − 1 − − − − → X 0 ) f − − − − → ( X 1 − − − − → X 2 − − − − → . . . )   y g − 2   y g − 1   y g 1   y g 2 ( . . . − − − − → Y − 2 − − − − → Y − 1 ) f ′ − − − − → ( Y 0 − − − − → Y 1 − − − − → Y 2 − − − − → . . . ) in C (i.e. if we cons ider all mo r phisms up to ho mo top y equiv alence). Prop osition 5.1 immediately a llo ws to co nstruct some functorial filtration and ’trun- cations’ for coho mo logy (i.e. for some contrav ar ia n t H : C → A , tha t will usually b e cohomolog ical). 9 Prop osition 5.3. 1. F or any c ontr avariant H : C op → A , j > 0 , Pr op osition 5.1(1) yields that the weight filtr ation W i H ( X ) = Im( H ( w ≤ i X ) → H ( X )) of H ( X ) is C -fun ctori al in X . 2. Applying b oth p arts of the pr op osition we obtain that H i 1 : X 7→ Im( H ( w ≤ i X ) → H ( w ≤ i + j X )) also defines a functor. 3. If H is c ohomolo gic al, j = 1 , H i 1 is c ohomolo gic al also. 4. H i 2 = Im( H ( w ≥ i X ) → H ( w ≥ i +1 X )) is also functorial and c ohomolo gic al (if H is); ther e is a l ong exact se quenc e of funct ors (i. e. it b e c omes a long exact se qu enc e in A when applie d t o any obje ct of C ) · · · → H i 2 ◦ [1] → H i 1 → H → H i 2 → H i 1 ◦ [ − 1] → . . . W e ca ll H i 1 and H i 2 virtual t-tru n c ations of H . The reaso n for this is that they ’b eha ve as’ if H is ’represe nted’ by an o b ject of some triangula ted category D , and the truncations are ’represented’ by its a ctual t -tr unca tions with resp ect to some t -structure of D . W e will observe that it is often the c a se in the next section; yet note that virtual t -truncations can be defined (and hav e nice proper ties) without sp ecifying any D and an y t -structure for it (in fact, it is fa r from being ob v ious that such D and t ex ist alwa y s; even if they do, D is definitely not determined by C in a functorial wa y)! Virtual t -truncations are studied (in detail) in §2.5 of [Bon0 7 ] (there this concept w as also developed for co v ariant functors; certainly , the difference is quite formal) a nd in §§2.3–2.5 of [Bon10a]. Also, ˜ W t H B M n ( − , − ) in Definition 5.8 of [F rH04] are essen tially (restrictions to motives of v arieties of ) virtual t -tr uncations of Borel-Mo ore ho mology with resp ect to w C how . 6 Dualities of triangul a ted categories ; orthogonal and ad- jacen t w eig h t and t -structures Let D also b e a triangulated category . Definition 6.1. 1. W e will call a (cov ar ian t) bi-functor Φ : C op × D → A a duality if it is bi-additive, homological with resp ect to b oth ar gumen ts; and is equipp e d with a (bi)natural transformation Φ( X , Y ) ∼ = Φ( X [1] , Y [1]) . 2. Supp ose now that C is endow ed with a weigh t str uctur e w , D is endowed with a t - structure t . Then we will say that w is (left) ortho gonal to t with r e s pect to Φ if the following ortho gonality 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 . (4) R emark 6.2 . 1. If t is orthog onal to w , then: for any X ∈ C w =0 the functor Y 7→ Φ( X , Y ) is exact when res tricted to H t . Virtual t -truncations of Φ( − , Y ) are ’represented’ b y t -truncations of Y : for example, Φ( X , Y t ≥ i [ j ]) ∼ = Im(Φ([ X w ≥ − j , Y [ i ]) → Φ( X w ≥ − 1 − j , Y [ i − 1 ])) . 2. Adj acen t structures A v ery important example of a dualit y is: D = C , Φ( X , Y ) = C ( X , Y ) . T his duality is also nic e (see Definition 2.5.1 of [Bon10a]); nice nes s is a tec hnical condition needed for sp e ctral sequences ca lculations (see b elo w). 10 In this situation, we call or thogonal w and t adjac ent struct ur es ; w is (left) adjac e nt to t whenever C w ≤ 0 = C t ≤ 0 ; see §4.4 of [Bon07]. 3. W eig h t-exact functors; relation with adjoint functors . Recall now: if a n exact functor C → C ′ is t -exact with resp ect to some t -structures o n these categ ories, its (left or right) adjoint is usua lly not t -exact (it is only left o r rig h t t -exa c t, resp ectiv ely). This problem could be fixed if there exist adja c en t weight structures for these t -structures (see Prop osition 4.4.5 o f ibid.). W e a s sume that C is endow ed with a weigh t str uctur e w a nd its a djacen t t -s tr ucture t ; C ′ is endow ed w ith a weigh t structure w and its adjacent t -structure t ; F : C → C ′ is exac t, G : C ′ → C is its left adjoint . W e will say that G is left (resp. righ t) weigh t-exa ct if G ( C ′ w ≤ 0 ) ⊂ C w ≤ 0 (resp. G ( C ′ w ≥ 0 ) ⊂ C w ≥ 0 ). Then: G is left (resp. r igh t) weigh t-e x act whenever F is right (res p. left) t -exact (in the well-kno wn and similarly defined sense). 4. Examples . A s imple exa mple of a djacen t structures is: if Pro j A ⊂ A denotes the full sub category of pr o jectiv e ob jects, D ? ( A ) (i.e. some version of D ( A ) ) is iso morphic to the c orresp o nding K ? (Pro j A ) , then for C = D ? ( A ) the canonic t -structure for C is adjacent to the ’stupid’ weigh t structure for C ∼ = K ? (Pro j A ) (mentioned above). Note that this exa mple allows to compute extension functors for A (and a ls o hyper extension ones i.e. mor phisms in D ? ( A ) )! Besides, the spherical weigh t structure ( w S 0 for S H mentioned above) is adjacen t to the P ostniko v t -str uctur e t P ost (for S H ). Moreov er, a proce s s similar to the construction of Eilenberg- Maclane sp ectra allows to construct a Chow t -structu r e fo r D M ef f − such that H t C how ∼ = AddF un( C how ef f , Ab ) (see §7.1 o f [Bon07]). t C how is a djacen t to the Chow weigh t s tructure for D M ef f − ; it is related with unramified coho mology (see §7.6 of ibid.). Other r elated calculations of hea r ts o f orthogonal structures were made in §§4.4 –4.6 of ibid. and in §6.2 of [Bo n 10a ]. Lastly , there also exis ts a nice dua lit y D op × D M ef f − → Ab (see §4.5 of [Bon1 0a ]). If (the base field) k is co un table, there a lso exists a triangulated catego ry D s (such that D M ef f gm ⊂ D s ⊂ D ) endow ed with a Gersten weigh t structure (see §4.1 of ibid.), that is o rthogonal to the homotopy t - structure for D M ef f − (defined in §3 of [V o e00]). So, the ob jects of its heart induce exact cov ariant functors from H t (i.e. the category o f homotopy inv a rian t shea ves with transfers) to Ab . It is no surprise that this hear t is ’g enerated’ by co motiv es of (sp ectra of ) function fields (over k ). Note that in this case C 6 = D . 5. The recen tly proved Beilinson-Lich tenbaum conjecture i mplies that the homotop y t -truncations of complexes of sheav es that repres en t Z / n Z -étale cohomology yield Z /n Z - motivic c o homology . There fore one can express torsion motivic cohomology (of smoo th v arieties, motives, and comotives) in ter ms of vir tual t -truncations o f torsion étale cohomol- ogy with resp ect to the Gersten weigh t structure. This allows to obtain s o me new formulae for motivic cohomology ; cf. §§ 7 .4–7.5 o f [Bon07] a nd Remar k 4 .5 .2 of [Bon10a]. 11 7 W eigh t s p ectral sequences Applying H to (shifted) weight deco mpositions of X one o btains an exact couple C w ( H, X ) with: D pq 1 = H ( X w ≤ − p [ − q ]) , E pq 1 = H ( X − p [ − q ]) . Here X i ∈ C w =0 are the terms of the weight c omplex of X ; the la tt er coincides with X for C = K ( B ) , was mentioned in Theorem 3.1(3) for C = D M ef f gm or = D M gm , and will be co nsidered in §8.2 in the gener al ca se. W e will call the sp ectral sequence corre sponding to C w ( H, X ) a w eight sp e ctr al se quenc e and denote it b y T w ( H, X ) (w e will often omit w in this notatio n). Under certain (quite weak) b oundedness co nditi ons this sp ectral sequence conv erg es to E p + q ∞ T ( H , X ) = H ( X [ − p − q ]) . Note that is na tural to denote H ( X [ − i ]) by H i ( X ) ; s ee a lso §2.3–2.4 of [B on07] for mo r e details. C w ( H, X ) (and s o also T w ( H, X ) ) is functor ia l in H (in the obvious w ay). Y et C w ( H, X ) (as well as E 1 ( T w ( H, X )) ) is not c a nonically determined b y X and H (though an y g ∈ C ( X, X ′ ) co uld be extended to a mor phism C w ( H, X ′ ) → C w ( H, X ) for any choices of those). Still, such a n extension is (almost) never unique. Y et this problem v a nishes completely if one pas ses to the derived exact co uple! It is easily seen that D 2 -terms are virtual t - tr uncations of H (defined in §5 ab o ve); E 2 are certain ’truncations from b oth sides’; s o b oth are given by cohomolog ical functors C → A (see lo c.cit. and §2 .4 of [Bon10a]). Hence T ( H, X ) is (also ) C -functorial (in X ) starting from E 2 . Besides, the rela tio n b et ween virtual t -truncations and truncations with r espect to an orthogo na l t -s tructure (descr ibed ab o ve) yields: for a nice dualit y Φ , H = Φ( − , Y ) , Y ∈ Ob j D , one has a functorial description of T ( H , − ) (star ting from E 2 ) in ter ms of t - truncations of Y ; see Theo rem 2.6.1 of [Bo n10a ]. This is a pow erful to ol for co mparing sp ectral sequences (in this situation); it does no t req uir e constructing any complexes (and filtra tio ns for them) in co n trast to the metho d of [P ar96] (pro bably , originating fro m Deligne). R emark 7.1 (Examples; change of weigh t structures) . 1 . W eight s p ectral seq uences gener alize Deligne’s weigh t sp ectral sequences, co niv eau, and Atiy ah- Hir zebruc h sp ectral sequences. W eight sp ectral sequences corresp o nding to w C how (w e c all them Chow-weigh t sp e ctr al se quenc es since they relate cohomo logy of V o evodsky’s motives with those of Chow motiv e s ) essentially gener alize Deligne’s weigh t spectr a l sequences; se e Remark 2.4 .3 and §6 of [Bon0 7]. F or H being étale or singular cohomology (of motives) this yields motivic functoria lity o f T w C how ( H, − ) for integral (or torsio n) co efficien ts. Note that the ’classical’ w ay of proving uniqueness of these spectra l sequences uses Deligne’s weigh ts for shea ves, and so requir es rational co efficients (one also uses hea v ily the fact that in this particular cas e w eight s pectral sequences deg enerate at E 2 ). One co uld also take the motivic cohomolo gy theory fo r H . This yields co mpletely new sp e ctral se quences (yet s ee Remar k 2.4.3(2) of ibid.). This T w C how ( H, − ) do es not degener ate at any fix e d level (even with rational co efficien ts, in genera l), a nd so its functoriality definitely cannot b e proved by ’clas sical’ methods . 2. Let F : C → C ′ be an exact functor that is right weight -exact with resp ect to w for C and w ′ for C ′ (see Remark 6 .2( 3)); let H : C ′ → A be cohomological. Then in §2.7 of [B on10a] it w as proved: for an y X ∈ O bj C there exists some comparison morphism of weigh t spectr al sequences M : T w ( H ◦ F, X ) → T w ′ ( H, F ( X )) . Moreover, t his morphism is unique and additively functoria l star ting from E 2 . The proo f uses a natural (and easy) generaliza tio n of (3). 12 In particular, this yields co mparison functors fr om Chow-weigh t spectra l sequences t o coniveau ones (cf. §9.3 below for more details ). If F is left weigh t-exact, there exists a comparison transformation N in the inv er s e direction. W e call both M and N ’change of weight structures ’ tra ns fo rmations. 3. Using the Gersten weigh t structure (for D s , see ab ov e) one can extend ’classical’ coniveau sp ectral sequences from (motives o f ) smo oth v arieties to D s ⊃ DM ef f gm in a natural wa y (for an arbitrary co homology theory H that factorizes through D M ef f gm , such that A satisfies AB5). This also yields motivic functoriality of coniveau sp ectral sequences (which is far from b eing obvious fro m their definition; see Remark 4.4.2 of [Bon10 a ]). Note a lso that we obtain this functoria lit y for a not necessarily countable k , since o ne ca n alwa ys define the coniveau sp ectral sequence for ( H, X ) over k as the limit of the rela ted coniveau sp ectral sequences over countable per fect fields of definition of X (see §4.6 of ibid.). Here we use the ’change of weigh t s tr ucture’ tra nsformations (that we deno ted by N above). The orthogona lit y o f the Gersten w eight structure with the homotop y t -structure (for D M ef f − ; se e the previous sec tion) yields that the coniveau sp ectral s equence for H repre- sented by s o me Y ∈ Obj D M ef f − could b e des cribed in terms of the ho motop y t -truncations of H . This ex tends v a stly the coniveau sp ectral sequence calcula tions of B lo ch & O gus (in [BOg94]; se e §4.5 of [Bon10a]). 4. Since t P ost and w S 0 are adjacent, w e obtain the well-kno wn fact: t he Atiy ah-Hirz e br uc h sp e ctral sequence c o n verging to [ X , Y ] for X , Y ∈ Obj S H could b e expressed either in terms of the t P ost -truncations of Y or in terms of w S 0 -truncations of X (i.e., in terms of cellular filtration of X ). 8 More on w ei gh t structures 8.1 ’F unctorialit y’ of w eigh t structures: lo calizations and gluing W eight structures could be car ried over to lo calizations and also ’glued’ similar ly to t - structures. If w (for C ) induces a w eig ht structure also on some triang ula ted D ⊂ C , then it als o induces a co mpatible weigh t structure on the V erdier quotient C /D ; its heart could b e easily describ ed (in terms o f the hearts of C and D in a wa y that is quite distinct from those for t -structures; see §8.1 of [Bon07]). Moreov er, one can glue w eight structures (i.e. reco ver a weigh t structure for C from those for D and C / D when c e r tain adjoint functors exis t) in a way that is just slightly different from those fo r t -s tr uctures (see §8.2 of ibid.). W e discuss an interesting exa mple of such a gluing in §9.3 b elo w. This statemen t was also used in §2.3 of [Bo n1 0p] in (one of the metho ds of ) the co nstruc- tion o f the Chow weight structure for motives ov er S . 8.2 The w eigh t complex functor There are tw o ways to co nstruct the weight co mplex functor for a general ( C , w ) (that generalizes the exact cons erv a tiv e functor t : DM ef f gm → K b ( C how ef f ) mentioned in Theorem 3.1). 13 First we describ e the ’rigid’ method. Suppose that C has a ’descr iption’ in ter ms of twiste d c omplexes ov er a negative differential gr aded catego r y (i.e. a differ ential gr ade d enhanc ement ; see §2 of [Bo n0 9 ] or §6 of [Bon07]). Supp ose als o that w is compatible with this enhancement (i.e. that w co incides with the weigh t s tr ucture g iv en by Propo sition 6 .2.1 of ibid.). Then ther e exists an exa ct weight complex fu nctor t : C → K ( H w ) ; see §6.3 o f ibid. (actually , in lo c.cit. o nly b ounded twisted co mplexe s a r e considere d, so the targ et of t is K b ( H w ) ). The main disadv antage of this metho d is that it requires so me extra information on C . A different ial graded enhance men t do es not hav e to exist at all (for a g e ner al C ; for exa mple, S H has no differential graded enhancements); a n exac t functor do es not have to extend to enhancements (and if such an ex tension exis ts, it is not nece ssarily unique). Luckily , in [Bo n0 7] another metho d was dev elo ped; it always works and do es not dep end on any extra structures . There is a co nstruction that a ssocia tes a cer tain complex to ea c h X ∈ O bj C for any C and depends o nly on w . It is closely related with the definition of a weight Postnikov tower for X (see Definitions 1.1.5 a nd 2.1.2 of [Bo n10a]) . The terms of the (w eight) co mplex t ( X ) ar e X i = Cone( w ≤ i − 1 X → w ≤ i X )[ i ] ∼ = Cone( w ≥ i X → w ≥ i +1 X )[ i − 1 ] (see Remark 2.1 .3 o f lo c.cit.); the cor responding triangles yield some b oundary morphisms X i → X i +1 (see §2.2 of [Bon10 a ]). It is easily seen that an y g ∈ C ( X, X ′ ) is compa tible with some t ( g ) : t ( X ) → t ( X ′ ) . This metho d has the following serio us disadv a n tage: in general, t ( g ) is only well-defined up to morphisms of the form d f + g d (i.e. mo dulo an equiv alence relation that is more coarse than homo to p y equiv alence of morphisms of complexes). Still this equiv alence r elation has ce r tain nice prop erties: equiv alent morphisms yield the sa me ma p on the cohomolog y of complexes ; the homoto py equiv alence class of t ( X ) do es not dep end on the choices ment ioned. So, we obtain a ce r tain we akly exact functor C → K w ( H w ) (see Definition 3.1 .5 of loc .cit.). F or any H one has E pq 1 T ( H , X ) = H ( X − p [ − q ]) ; hence E ∗∗ 2 T ( H , X ) can b e describ ed in terms of t ( X ) (in a functoria l wa y); see Remark 3.1.7 of lo c.cit. In the case C = S H w e hav e K w ( H w ) = K ( H w ) ; so t is actually an exact functor (see Remark 3 .3.4 of ibid.). Moreov er, this (’weak’) w eight complex functor is compatible with the ’stro ng’ one given b y the differential graded appr oac h; see §6.3 of ibid. It is conser v ative if w is b oun de d (i.e. if ∩ i ∈ Z C w ≤ 0 [ i ] = ∩ i ∈ Z C w ≥ 0 [ i ] = { 0 } ); see Theor em 3.3.1 o f ibid. for the pro of of this fact and of s e v era l other nice prop erties o f t . 8.3 Certain K 0 -calculations Suppos e that w is b o unded, H w is idempo ten t complete. Then C is idemp oten t co mplete also; see Lemma 5.2 .1 of ibid. In pa rticular, this yields that D M ef f gm is gener ated by C how ef f (i.e. the only strict full triangulated sub c ategory of D M ef f gm containing C how ef f is D M ef f gm itself ); it seems that §3.5 of [V o e00] do es not contain a complete pr oof o f this statement. Besides, we hav e K 0 ( C ) ∼ = K 0 ( H w ) . Recall that the generator s o f K 0 ( C ) (res p. K 0 ( H w ) ) are [ X ] , X ∈ O bj C ( X ∈ O bj H w ), and the relations are: [ B ] = [ A ] + [ C ] if A → B → C is a distinguished triangle (resp. B ∼ = A L C ). In particular , we obtain Theorem 3 .1 (4) this w ay . 14 8.4 A generaliza tion: relativ e w eigh t structures Now we des cribe a formalism that generalizes those of weigh t s tructures. It is actual since in the (derived) ca tegory of mixed complexes of sheaves ov er a v a riet y X 0 defined over a finite field F q the s ub categories of ob jects of non-p ositive and non-neg ativ e weigh ts do not quite satisfy the orthog o nalit y axiom (iii) of Definition 4.1. So , we adjust this axiom in order mak e it co mpatible with Pro position 5 .1.15 of [BBD82]. Note here: in our notation the roles of C w ≤ 0 and C w ≥ 0 are p erm uted with r espect to the notation of (§5.1.8 o f ) ibid. Definition 8.1. Let F : C → D b e an exa ct functor (of triang ulated ca tegories). A pair of extension- stable (see Theorem 4 .3( 1)) Karoubi-closed sub classes C w ≤ 0 , C w ≥ 0 ⊂ Ob j C for a triangulated categ ory C will b e said to define a rela tiv e weigh t structure w for C with resp ect to F (or just and F -weight structure) if they satisfy conditions (ii) and (iv) of Definition 4.1, as well as the following or thogonalit y assumptions: C w ≥ 0 ⊥ C w ≤ 0 [2] ; F kills a ll mor phisms b et w een C w ≥ 0 and C w ≤ 0 [1] . Relative weight structures satisfy several prop erties similar to thos e of ’abso lute’ w eight structures (note: an ’abso lute’ weight structure is the s ame thing as an id C -weigh t s tr ucture); see b elo w. 9 ’Motivic’ w eigh t s tructures; comoti v es; gluing Cho w and Gersten structures from ’ bi rational slices’ W e briefly summarize ho w w eight structures help in th e proo f o f Theorem 3 .1 (this infor- mation could be found ab o ve, y et it is somewhat s cattered). W e also mak e sev e ral other remarks. As explained ab o ve, w eight structures y ield a migh ty instr umen t for constructing and studying cer ta in functor ial sp e c tr al sequences for cohomo logy functors (defined on a trian- gulated categ ory C ); so they a lso yield certain functoria l (’w eight’) filtration. They also describ e how ob jects of C could be ’constructed fro m’ ob jects o f a ’more simple’ additiv e H w ⊂ C . W e have tw o main ’motivic’ weigh t s tr uctures. They corr espond to (Chow)-w eight and coniveau sp ectral seq uence s, r espectively . Note that b oth of these s pectral sequences were ’classically’ defined only for cohomolo gy of v arieties; still our approach allows to define them for ar bitrary V oevo dsky’s motives, a nd also y ie lds their motivic functoriality (which is very far from being obvious). 9.1 Cho w w eigh t struct ure(s); relation with the motivic t -struct ure and w eight filtration Our firs t (’motivic’) weigh t structure (being more pr e c is e, we hav e a system of compati- ble weight structures o n v arious ’motivic’ categories) is w C how ; it is defined on DM ef f gm ⊂ D M gm , its heart is C how ef f ⊂ C how ; w C how can also be extended to DM ef f − and D . So, it clos e ly relates D M ef f gm with C how ef f (in par ticular, the w eight co mplex functor D M ef f gm → K b ( C how ef f ) is conserv ative; note that DM ef f gm is very far from being isomorphic 15 to K b ( C how ef f ) !). So, the cohomolog y of V oevo dsky’s motives can be ’functorially r elated’ with the cohomolog y o f Chow ones; one obtains a v ast generalization of Deligne’s w eig h t sp e ctral sequences. Besides, there exists a Chow t -structur e for D M ef f − such that H t C how ∼ = AddF un( C how ef f , Ab ) ; t C how is adjacent to the Chow weight structure for D M ef f − . Now we rela te w C how with the ’us ua l exp ectations for w eights o f mo tiv es’; see §8.6 of [Bon07] for more details. Conjecturally , DM ef f gm Q (and D M gm Q ) should support a certain (’mixed’) motivic t - structure ( t M M , who se heart is the ab elian catego ry M M ef f ⊂ M M of mixed motives) and a weight fi ltr ation (by certain triang ulated subca teg ories); the latter one comes from cer ta in weigh t filtration functors M M → M M (co mpatible via cohomology with the weigh t filtration of mixed Ho dge structur e s and o f mixed Galois mo dules; these functor s are idempotent). So, there should b e three imp ortant filtrations for D M ef f gm Q ⊂ D M gm Q altogether. Now, one ca n easily verify that the (widely b elieved to b e true, y et conjectural) pr operties of the t wo c o njectural filtrations ment ioned yield: for a subca tegory of ob jects that are ’pure of some fixed w eight i ’ with resp ect to a n y one o f the three filtrations mentioned, the filtra- tions induced by tw o remaining structures differ only by a shift o f indices (that dep ends o n i ). In pa rticular, t M M ’should split’ Chow motives in to comp onen ts that are ’pur e with res pect to the weigh t filtratio n’; w C how -weigh t decomp ositions induce the (conjectural!) w eight filtra- tion for mixed mo tiv es. Note here: though w eig ht deco mpositions (of ob jects of triangulated categories ) a re (usually) highly non- unique, for a n y i ∈ Z , X ∈ Ob j M M ⊂ O bj D M ef f gm Q , there ’should ex ist’ a unique w eig h t decompo sition of X [ i ] such that w ≤ i X , w ≥ i +1 X ∈ M M ; this choice of w ≥ i +1 X is what one exp ects to b e the corresp onding level of the weight filtra- tion o f X in M M . In [Wil09 at ] this (conjectura l) picture was justified in the case when k is a num b er field for the triang ulated ca teg ory D AT ⊂ D M ef f gm Q (of so- called Ar tin-T a te motives; this is the triangulated subcateg o ry of D M ef f gm Q generated by T ate twists o f motives of sp ectra of finite extensions o f k ). It was also shown that the restriction o f w C how to D AT ca n be completely characterized in terms of weigh ts of singular homolo gy . A ctually , this corresp onds to the fact that the triangula ted catego ry D H S of mixed Ho dge complexes has a weigh t filtration (by triangulated sub categories) and could be endow ed with a w eight structure; these filtrations and the ’canonical’ t -s tructure for D H S are connected by the s a me relations as those that ’should connect’ the co rrespo nding filtratio ns of D M ef f gm Q ⊂ D M gm Q . Besides, it c o uld b e easily se e n that sing ular (co)homolo gy is w eight-exact. 9.2 Comotiv es; the Gersten w eigh t structure Our seco nd ’motivic’ weigh t structure is the Gers ten weight s tructure w defined on the category D s ⊃ D M ef f gm (for a countable k ). Here D s is a full triangulated sub category of a certain ca teg ory D of c omotives (already men tioned in Theorem 3 .1). The idea is that w should b e or thogonal to the homotopy t -structure on D M ef f − (recall that the la tt er is the r estriction of the canonica l t -structur e of the derived category of Nis- nevich sheav es with trans fers). So, H w is ’g enerated’ by comotives of function fields ov e r k (note that these ar e Nisnevich p oin ts). It follows th at w cannot be defined on D M ef f gm (or on D M ef f − ). The problem with 16 D M ef f − ⊃ DM ef f gm is that there are no ’nice’ ho motop y limits in it. In order to have them one needs ’nice’ (small) pro ducts; one also nee ds the ob jects of D M ef f gm to b e co compact (in this ’category of homotopy limits’). D M ef f − definitely do es not s atisfy these conditions. Instead in §5 o f [Bo n10a] a catego ry D ′ that is opp osite to a certa in c a tegory o f differ ent ia l gr ade d mo dules (i.e. cov aria n t differen tial gr a ded functors from the differ ential gr ade d enhanc ement of D M ef f gm to complexes of abelia n groups ) w as considered; D is its homotopy catego ry (with resp ect to a certain closed mo del structure; so it is oppo site to the corres ponding derived category of different ial graded modules). So , w e have a contrav ar ia n t Y oneda em bedding of D M ef f gm to the ca tegory opp osite to D whose ima g e consists of co mpact ob jects; in t his category ’nice’ homoto p y colimits exist. Th us, inv er ting arrows we obtain a ’nice’ categor y of comotiv es. Inside D w e define D s as its smallest K aroubi-closed tria ngulated catego ry that co n tains (countable) pro ducts of comotives o f functions fields. No te: w e need k to b e countable since without this the author does not know ho w to prove that (our candidate for) H w is neg ativ e; still comotives ca n b e defined ov er a n y p erfect k . The gener al theory o f weight sp ectral sequences yields thos e for co homological functors D s → A . A (minor) problem here is that D s is ’large’; yet an y H : D M ef f gm → A has a ’nice’ extension to D s (and also to D ⊃ D s ) if A satisfies AB5 (see Prop osition 4.3.1 of [Bon10a]). So, we c a n co nsider w eig h t sp ectral sequences T = T w ( H, X ) for any suc h H and any X ∈ Ob j DM ef f gm (or X ∈ Ob j D s ). It turns out that for X b eing the motif of a smo oth v ariety , T is isomorphic to the coniveau sp ectral sequence (corresp onding to H ) starting from E 2 ; see Prop osition 4 .4 .1 of ibid. So, w e call T a coniveau s pectral sequence for an y X . As in the case of ’classic a l’ co niv eau sp ectral sequences, if H is repr e sen ted by a n ob ject of D M ef f − , T w ( H, X ) ca n b e descr ib ed in terms of co homology of X with the co efficien ts in the homo top y t -truncations of H (see Co rollary 4.5.3 of ibid.); this fact extends the related results of Blo ch-Ogus and P ara njape (see [BO g 94 ] and [Par96]). Our latter result follows from the existence of a nice duality D op × D M ef f − → Ab . R emark 9.1 . w can b e restric ted to the category D AT ⊂ D M ef f gm of Artin-T a te motives (men tioned ab ov e; o ne may take in tegral co efficien ts here; k is any perfect field). Indeed, we don’t need comotives here, since (co)motives of (spe ctra of ) finite e xtensions of k b elong to Ob j DM ef f gm . W e expla in this in more detail. D AT is generated by M ( F )( j )[ j ] , where F runs thro ug h all (spectra of ) finite field extensions of k , j ≥ 0 . D = {⊕ i M ( F i )( j i )[ j i ] } is a negative (additiv e) sub category of D AT , so Theo r em 4.3(4) implies: there exists a weigh t structure w DAT with D ⊂ H w DAT . Since H w DAT ⊂ H w ( ⊂ D ) , we obtain that w DAT is compatible with w (at lea st, for a count able k ). In pa rticular, this implies that coniveau sp ectral seq uences for co homology of a n y X ∈ Ob j DAT ha ve quite ’ec onomical’ descriptions (starting from E 2 ). 9.3 Comparison of w eight structures; ’gluing from birational slices’ First we describ e the relation betw een T ′ = T w C how ( H, X ) and T = T w ( H, X ) (for X ∈ Ob j DM ef f gm ⊂ O bj D s ). The ’c hange of w eight structure tr ansformation’ (see Remar k 7.1) yields some morphism M : T → T ′ (functorially starting from E 2 ; see §4.8 of [Bon10a]). M is an isomorphism if H is bir ational i.e. kills DM ef f gm (1) ; here − ⊗ Z (1) is t he T ate t wist isomorphism of D M ef f gm in to itself. 17 Now, − ⊗ Z (1) can b e ex tended from D M ef f gm to D (see §5 .4.3 of ibid.); this is a lso true for w C how (see §4.7 of ibid.). It is easily seen that w and w C how induce the same weigh t structure w bir on the categor y of bir ational c omotives D bir = D / D (1) (the V erdier quotient); the hear t of this lo calization co n tains images of all (co)motives of all s mooth v ar ieties. One obtains that (roug hly!) w and w C how ’coincide on slices ’ and o nly differ by the v alue of a single integral parameter: w is − ⊗ Z (1)[1] - stable a nd w C how is − ⊗ Z (1)[2] -stable! W e try to make this mo r e prec ise; see §4.9 of ibid. for more details. W e consider the lo calizations D / D ( n ) for all n > 0 . Though none of them is isomorphic to D , they ’approximate it pr ett y well’. Also, for any n we hav e a short exa ct sequence o f triang ula ted categories D / D ( n ) i ∗ → D / D ( n + 1) j ∗ → D bir . Here the nota tion for functors comes from the ’classical’ gluing da ta setting (cf. §8.2 of [Bon07]); i ∗ can b e g iv en by − ⊗ Z (1)[ s ] for any s ∈ Z , j ∗ is just the lo calization. Now, if we choose s = 2 then b oth i ∗ and j ∗ are weigh t- exact with resp ect to weight structures induced by w C how on the corres ponding ca tegories; if we choose s = 1 these functors are weigh t-exac t with resp ect to the w eig h t s tr uctures coming fro m w . So, t he Chow and Gersten weigh t structures induce weight structures on the lo calizations D ( n ) / D ( n + 1) ∼ = D bir (w e call these lo calizations ’slices’) that differ only b y a shift. One can show that for any short exact sequence D i ∗ → C j ∗ → E of triangulated catego ries, if D and E a re endow ed with weigh t s tr uctures, then there exist at most o ne weigh t structure on C such that b oth i ∗ and j ∗ are weigh t-exact. So, if one calls the filtration of D by D ( n ) the slic e filtr ation (this term was already used by A. Huber , B. Kahn, M. L evine, V. V o ev o dsky , and other authors for other ’motivic categories ’), then one may say that the weigh t structures induced by w and w C how on all D / D ( n ) ’can b e recovered from slices’; the only difference b et ween them is ’how we shift the slices’ ! Moreov er, Theo r em 8 .2 .3 of [Bo n07 ] shows that if b oth adjoints to b oth i ∗ and j ∗ exist, then one ca n use this gluing data in order to ’glue’ (any pair) of weigh t structure s for D and E in to a weight structure for C . So , suppose that we have a weigh t s tructure w n,s for D / D ( n ) that is − ⊗ (1)[ s ] -stable and ’compatible with w bir on all slices’. Then we can also cons tr uct w n +1 ,s satisfying similar pr operties, since genera l homologic a l algebra yields that a ll adjoints needed exist in our situation. So, w n,s exist f or a ll n > 0 and all s ∈ Z . Moreov er, there exis ts a ’large’ subcateg ory of D (con taining DM ef f gm ) that f or any s can be endow ed with a weight structure w s compatible with all w n,s . Hence Gers ten and Chow weigh t structures (f or D s / D s ( n ) ⊂ D / D ( n ) ) are mem ber s of a rather natur a l family of weigh t str uctures indexe d b y a single integral para meter! It could b e interesting to study other members of this family (for example, the one that is − ⊗ Z (1) -stable). 9.4 W eights for relativ e motiv es and mixed shea v es Let S b e a scheme of finite t yp e ov er so me ex c ellen t no etherian scheme S 0 of dimension ≤ 2 . As w e hav e already said, on the category D M c ( S ) (of c onst ructible i.e. ’geometric’ motives with ra tional co efficien ts ov er S ) there exists a weigh t structure w C how whose hea rt C how ( S ) is the idempo tent completion o f { p ! Q P ( n )[2 n ] } , for p : P → S running through pro jective (or pro per) morphisms such that P is re g ular, n ∈ Z (see § 3 of [Heb10] and §2.1 of [Bo n10p ]). The corr esponding Chow-w eig h t spectra l sequences yield: for any cohomological H : 18 D M c ( S ) → A , X ∈ Obj D M c ( S ) , there exists a filtration o n H ∗ ( X ) (that is DM c ( S ) - functorial in X ) whose factors are subfactors of coho mo logy of s o me r egular pro jective S - schemes’; see Remark 3.3.2(3) of [Bon10p]. Besides, th e ( Chow)-w eig ht filtration o f coho- mology yields a na tur a l wa y of description of the ’integral pa r t’ of the motivic cohomolo gy of a v ariety ov e r a num b er field (a s co nstructed in [Sch00]; see Remark 3.3.2(4) of [Bon10p]). W e also obtain that K 0 ( D M c ( S )) ∼ = K 0 ( C how ( S )) (cf. §8.3), and define a certain ’motivic Euler characteristic’ for S -schemes (in §3.2 of [Bon1 0p ]). The author ho pes that these r esults co uld b e useful for motivic in tegration. Now deno te by H the étale rea lization functor D M c ( S ) → D S H , where D S H = DS H ( S ) is the category D b m ( S, Q l ) of mixed complexes of Q l -étale sheav es as co ns ider ed in [Hub97] and in [BBD82]. Then H sends C how motives ov er S to pure complexes of sheaves (see Definition 3.3 of [Hub97] and §3.4 of [Bon10p]). W e deduce cer tain co ns equences from this fact. Suppos e that S is a finite type Spec Z - s c heme. W e tak e H per being the p erv erse é ta le cohomolog y theory i.e. H i per ( M ) (for M ∈ O bj D M c ( S ) , i ∈ Z ) is the i - th coho mology of H ( M ) with resp ect to the perverse t -structure o f DS H (see Propo sition 3.2 of [Hub97]). Then T w C how ( H per , M ) for any M ∈ Obj DM c ( S ) yields: all H i per ( M ) have weight filtrations (defined using Definition 3.3 of lo c.cit., for a ll i ∈ Z ). Note that this is no t at a ll automatic (for p erverse shea ves over S ); see Remark 6.8.4(i) of [Jan90]. Certainly , o ne can replace per v ers e sheaves ov er S here by Q l -adic representations o f the absolute Galois gr oup o f the function field of S ; cf. §6.8 o f lo c.cit. Now let S = X 0 be a v ariety ov er a finite field F q ; let X denote X 0 × Sp ec F q Spec F , where F is the algebr aic closure of F q . The re sults of §5 of [BBD82] (along with so me of the results of [Bon10p]) yield that the catego ry D S H (= D b m ( X 0 , Q l )) can b e endow ed with an F - weigh t s tructure w DS H whose hea rt is the categor y of pure complexes of sheaves, for F b eing the extensio n of scalars functor DS H → D b ( X, Q l ) ; see Prop osition 3.6.1 of [Bon10p]. In particular, we obtain that any ob ject M of D S H pos s esses a ’filtra tion’ (a weight Postnikov tower ) whose ’factors’ b elong to H w DS H . Next, our H is a weight-exact fun ctor (i.e. it sends D M c ( S ) w C how ≤ 0 to D S H w DS H ≤ 0 and sends D M c ( S ) w C how ≥ 0 to D S H w DS H ≥ 0 ). Hence this is no wonder that the weigh t-exac tness prop erties o f motivic bas e change functor s (for D M c ( − ) ; s ee Prop osition 3.8 of [Heb10], and Theorem 2.2.1 and Prop osition 2.3.4 o f [Bon10p]) are parallel to the ’stabilities’ 5 .1.14 of [BBD82]. Lastly , let G : D b ( X, Q l ) → A be any cohomo logical functor, H = G ◦ F , M ∈ Ob j DM c ( S ) . Then the w eig ht-exactness of H yields that the ( Chow)-w eight filtra tion for ( H ◦ H ) ∗ ( M ) is exactly the w DS H -weigh t filtration for H ∗ ( H ( M )) ; cf. Prop osition 3.5.5(I I2) of [Bo n10p ]. V ery proba bly , some analogues of these results are v alid for H replaced by a ’Hodg e mo dule’ r ealization o f mo tives (for S b eing a complex v ar iet y); the problem is that (to the knowledge of the a uthor) no such re alization is constructed at the moment . 19 10 P ossi bl e applications to finite-dimensi o nalit y of mo- tiv es Recall that D M ef f gm ⊂ D M gm , as w ell a s their ’rational versions’ DM ef f gm Q ⊂ D M gm Q (see Remark 2 .1( 2)) a re tensor t riangulated categ ories. This allows to define external and symmetric p o wers of ob jects in tw o la tter ca tegories, since those are dire c t summands o f tensor p ow ers (for Q -linear motivic catego ries). M ∈ D M gm Q is called Kimur a-fin ite (or finite-dimensional) if M = M 1 L M 2 , where some external power of M 1 and some symmetric power of M 2 is 0 . In this cas e M 1 is called evenly finite-dimensional . Now, t Q : D M ef f gm Q → K b ( C how ef f Q ) (the ra tional version of the weight complex) is a conserv ative tensor functor; so X ∈ Obj DM ef f gm (or D M gm , or Ob j DM gm Q ) is Kimura-finite whenever t gm Q ( X ) is. Now we describ e a series of motives that ’should b e’ finite-dimensional (v ery similar ob jects were considered b y A. Be ilinson and M. Nori though in somewhat distinct contexts). Let X/k b e smo oth affine of dimension n , Y be its generic hyperplane section (with r e - sp e ct to so me pro jective embedding). Then for M = ( Y → X ) the only non-zero cohomo lo gy is H n et ( M k alg ) . Hence some external p ow er of M ⊗ Q [ − n ] ’should’ v a nish (since a certain external p o w er of its cohomo logy v anishes). So M [ − n ] ’should b e’ evenly finite-dimensional. W e can also pas s to K b ( C how ef f Q ) here (i.e. co nsider t ( M ) instead of M ) since the rational version of the weigh t complex functor is a tensor functor. R emark 10.1 . 1. If all such M ar e K imura-finite at leas t n umerically (i.e. we cons ider their images in K b (Mot num ) obtained via t ), then one ca n pr o ve that Mot num is a tannakia n category . 2. Widely-b eliev ed conser v ativity of étale cohomolog y (as a functor o n D M ef f gm Q ) imme- diately implies that all such M are Kimu ra-finite indeed (as ment ioned a bov e). Alternatively , it is p ossible to deduce Kim ura- finiteness o f M from a ce rtain weak Lefschetz for motivic cohomolog y . The latter ’sho uld be true’ since it easily follows from the (widely believed, yet conjectural!) existence of a ’reaso nable’ motivic t - structure for D M ef f gm Q . Unfortunately , the author has no idea how to pr o ve a n ything here unconditionally . References [BBD82] Beilinson A., Ber nstein J., Deligne P ., F aisceaux per v ers, Asterisque 100, 1982 , 5–171 . [Be V08] Beilinson A., V ologo dsky V. A guide to V oevo dsky motives// Geom. F unct. Analy- sis, vol. 17, no. 6, 2 008, 1 709–178 7, see also ht tp://arxi v.org/abs/math.AG/0604004 [BOg94 ] Bloch S., Ogus A. Gersten’s conjecture and the homolog y of schemes// Ann. Sci. Éc. Norm. Sup. v.7 (1994), 181–2 0 2. [Bon09] Bondarko M.V., Differen tial gra ded mo tiv es: weigh t c o mplex, weight filtra tions and sp e ctral sequences for r e a lizations; V o evodsky vs. Hanamura// J . of the Inst. o f Math. of J ussieu, v.8 (2009 ), no. 1, 39– 97, see a ls o htt p://arxiv .org/abs/math.AG/0601713 20 [Bon07] Bondarko M., W eigh t structures vs. t -structures; weigh t filtrations, sp ectral se- quences, a nd complexes (for motives and in genera l), to app ear in J. of K- theory , http: //arxiv.o rg/abs/0704.4003 [Bon10a] Bondarko M.V., Motivically functorial coniveau spectral sequences; direct s um- mands of coho mo logy of function fields/ / Doc. Ma th., extra volume: Andrei Suslin’s Sixtieth Birthday (2010 ), 33–1 17. [Bon10b] Bondarko M. Z [ 1 p ] -motivic resolution of singularities, preprint, http: //arxiv.o rg/abs/1002.2651 [Bon10p] Bondarko M., W eig h ts for V o ev o dsky’s motives ov er a base; relation with mixed complexes o f sheaves, preprint, h ttp://arx iv.org/abs/1007.4543 [CiD09] Cisinski D., Deglise F., T r ia ngulated ca tegories of mixed motives, prepr in t, http: //arxiv.o rg/abs/0912.2110 [F rH04] F riedlander E, Haesemey er C., W alker M., T echn iques, computations and co njec- tures for semi-top ological K -theory/ / Math. Ann. 330, 20 04. [GiS96] Gillet H., Soulé C. Descent, motives and K -theory// J. reine und angew. Math. 478, 1996, 1 27–176. [GNA02] Guillen F., Na v arr o Azna r V. Un critere d’extension des foncteurs definis sur les schemas lis s es// Publ. Math. Ins t. Hautes Etudes Sci. No . 95 , 2 002, 1 –91. [Han04] Hanamura M. Mixed motives and a lgebraic cy c le s I I// Inv. Math 158 , 2 004, 105– 179. [Heb10] Hébert D., Str uctur e s de poids à la Bondar k o sur les motifs de Beilinso n, prepr in t, http: //arxiv4. library.cornell.edu/abs/1007.0219 [Hub97] Huber A., Mixed p erv e rse sheav es for schemes over n umber fields// Comp ositio Mathematica, 10 8, 10 7 –121. [Ill08] Illusie L., On Gabb er’s r efine d uniformization , T a lks at the Univ. T o ky o, Jan. 17, 22, 31, F eb. 7 , 2008, http: //www.mat h.u- psud.fr/~illusie/refined_uniformizat ion3.pdf [Jan90] Jannsen U., Mixed Motives and Algebraic K -theory , Lecture Notes in Math. 1400 , Springer, New Y ork, 1990. [KaS02] Kahn B., Sujatha R., Birational motives, I, preprint, http: //www.mat h.uiuc.edu/K- theory/0596/ [Kim05] Kim ur a S.-I., Cho w motives are finite-dimensional, in some se ns e // Ma th. Ann. 331 (2005), 17 3 –201. [Lev09] Levine M. Smo oth motives, in R. de Jeu, J.D. Lewis (eds.), Motives and Algebraic Cycles. A Celebration in Honour of Sp encer J. Blo c h, Fields Institute Communications 56, America n Math. So c. (2009), 17 5 –231. 21 [MVW06] Mazza C., V o ev o dsky V., W eibel Ch. Lecture notes on mo- tivic cohomolog y , Clay Mathematics Mo nographs, vol. 2, see also http://www .math.rutg ers.edu/~weibel/MVWnotes/prova-h yperlink.pdf [P a r96] Paranjap e K., Some Sp ectral Sequences for Filtered Co mplexes and Applications / / Journal of Algebra, v. 186 , i. 3 , 1 996, 7 93–806. [P a u08] Pauksztello D., Compact cochain ob jects in triangulated categor ie s and co -t- structures// Central Europea n Journal of Mathematics, vol. 6, n. 1, 2008 , 25–42, see also h ttp://arx iv.org/abs/0705.0102 [Sch 00] Sc holl A., Integral elemen ts in K - theory a nd pro ducts of mo dular curves, The arith- metic and geometry of a lgebraic cycle s (Banff, AB, 1998), NA TO Scie nce Series C, volume 548 (Kluw er , 2 0 00), 46 7–489. [V o e00] V oevo dsky V. T riang ulated categ o ry of mo tives, in: V o evodsky V., Suslin A., a nd F riedlander E. Cy cles, transfers and motivic homo logy theories, Annals of Mathematical studies, vol. 14 3 , Princeton University Press, 2000, 18 8–238. [Wil09at] Wildeshaus J., Notes on Artin-T ate motives, preprint, http: //www.mat h.uiuc.edu/K- theory/0918/ 22