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Motivically Functorial Coniveau Spectral Sequences;

Direct Summands of Cohomology of Function Fields

Mikhail V. Bondarko

Received: September 4, 2009

Revised: June 7, 2010

Abstract.

Thegoalofthispaperistoprovethatoniveauspetral sequenesare motivially funtorialforall ohomologytheoriesthat ouldbefatorizedthroughmotives.Tothisendthemotifofasmooth varietyoveraountable eld

k

is deomposed (in thesense of Post-nikovtowers)intotwisted(o)motivesofitspoints;thisisgeneralized to arbitrary Voevodsky's motives. In order to study the funtorial-ity of this onstrution, we use the formalism of weight strutures (introdued in the previouspaper). Wealso developthis formalism (forgeneraltriangulatedategories)further, andrelateitwithanew notionofanieduality (pairing)of (twodistint) triangulated ate-gories;thispieeofhomologialalgebraouldbeinterestingforitself. We onstrut a ertain Gersten weight struture for a triangulated ategoryofomotivesthatontains

DM

ef f

gm

aswellas(o)motivesof funtioneldsover

k

. Itturnsoutthattheorrespondingweight spe-tralsequenesgeneralizethelassialoniveauones(toohomologyof arbitrarymotives). Whenaohomologialfuntorisrepresentedbya

Y

∈

Obj

DM

−

ef f

, theorrespondingoniveauspetralsequenesan beexpressedin termsofthe(homotopy)

t

-trunations of

Y

; this ex-tendstomotivestheseminaloniveauspetralsequeneomputations ofBlohandOgus.

We also obtain that the omotif of a smooth onneted semi-loal sheme is a diret summand of the omotif of its generi point; o-motivesof funtion elds ontain twisted omotives of their residue elds(forallgeometrivaluations). Henesimilarresultsholdforany ohomologyof(semi-loal)shemesmentioned.

2010 Mathematis Subjet Classiation: 14F42, 14C35, 18G40, 19E15,14F20,14C25,14C35.

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3 Categories of comotives (main properties)

79

3.1 Comotives: an'axiomatidesription' . . . 80

3.2 Pro-shemesandtheiromotives . . . 82

3.3 Primitiveshemes: reminder. . . 84

3.4 Basimotivipropertiesof primitiveshemes . . . 84

3.5 Onmorphismsbetweenomotivesofprimitiveshemes. . . 86

3.6 The Gysin distinguished triangle for pro-shemes; 'Gersten' Postnikovtowersforomotivesofpro-shemes. . . 86

4 Main motivic results

88

4.1 TheGerstenweightstruturefor

D

s

⊃

DM

ef f

gm

. . . 89

4.2 Diretsummand resultsforomotives . . . 91

4.3 Onohomologyofpro-shemes,and itsdiret summands. . . . 92

4.4 Coniveauspetralsequenesforohomologyof(o)motives . . 93

4.5 An extensionofresultsofBlohandOgus . . . 94

4.6 Baseeld hange foroniveauspetralsequenes; funtoriality foranunountable

k

. . . 96

4.7 TheChowweightstruturefor

D

. . . 98

4.8 ComparingChow-weightand oniveauspetralsequenes . . . 100

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5 The construction of

D

and

D

′

; base change and Tate twists104

5.1 DG-ategoriesandmodulesoverthem . . . 104 5.2 Thederivedategoryofadierentialgraded ategory . . . 106 5.3 Theonstrutionof

D

′

and

D;

theproofofProposition3.1.1 . 106 5.4 BasehangeandTatetwistsforomotives. . . 108

5.4.1 Indutionandrestritionfordierentialgradedmodules: reminder . . . 108 5.4.2 Extensionandrestritionofsalarsforomotives . . . 108 5.4.3 Tensor produts and 'o-internal Hom' for omotives;

Tatetwists . . . 109

6 Supplements

110

6.1 Theweightomplexfuntor;relationwithgenerimotives . . . 111 6.2 Therelationoftheheartof

w

with

HI

('Brownrepresentability')112 6.3 Motivesandomotiveswithrationalandtorsionoeients . . 113 6.4 Anotherpossibilityfor

D;

motiveswithompatsupportof

pro-shemes . . . 114 6.5 Whathappensif

k

isunountable. . . 114

Introduction

Let

k

beourperfetbaseeld.

We reall two very important statements onerning oniveau spetral se-quenes. The rst one is the alulation of

E

2

of the oniveau spetral se-quene for ohomologialtheories that satisfy ertain onditions; see [5℄ and [8℄. ItwasprovedbyVoevodskythat theseonditionsarefullled byany the-ory

H

representedbyamotiviomplex

C

(i.e. anobjetof

DM

ef f

−

;see[25℄); thenthe

E

2

-termsofthespetralsequeneouldbealulatedintermsofthe (homotopy

t

-struture)ohomologyof

C

. This resultimpliestheseond one:

H

-ohomologyof asmooth onnetedsemi-loal sheme (in thesense of 4.4 of [26℄) injets into the ohomology of its generipoint; thelatter statement wasextendedto all(smoothonneted)primitiveshemesbyM.Walker. The main goal of the present paper is to onstrut (motivially) funtorial oniveau spetral sequenes onverging to ohomology of arbitrary motives; there shouldexistadesriptionof thesespetralsequenes(startingfrom

E

2

) thatissimilartothedesriptionfortheaseofohomologyofsmoothvarieties (mentionedabove).

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Ourmainhomologialalgebratoolisthetheoryofweightstrutures(in trian-gulated ategories; weusually denote aweightstruture by

w

)introdued in the previous paper [6℄. Inthis artilewe develop it further; this part of the paperould be interesting also to readers notaquainted with motives (and ould be read independently from the rest of the paper). In partiular, we studyniedualities(ertainpairings)of(twodistint)triangulatedategories; itseemsthatthissubjetwasnotpreviouslyonsideredintheliteratureatall. Thisallowsustogeneralizetheoneptofadjaentweightand

t

-strutures(

t

) in atriangulatedategory(developed in 4.4of [6℄): weintrodue thenotion oforthogonal struturesin(twopossiblydistint)triangulatedategories. If

Φ

is anieduality oftriangulated

C, D

,

X

∈

ObjC, Y

∈

ObjD

,

t

is orthogonal to

w

, then the spetral sequene

S

onverging to

Φ(X, Y

)

that omes from the

t

-trunationsof

Y

isnaturallyisomorphi(startingfrom

E

2

)totheweight spetralsequene

T

forthefuntor

Φ(−, Y

)

.

T

omesfromweighttrunationsof

X

(notethatthosegeneralizestupidtrunationsforomplexes). Ourapproah yieldsan abstratalternativeto themethodof omparingsimilarspetral se-quenes using ltered omplexes (developed by Deligne and Paranjape, and used in [22℄, [11℄, and [6℄). Note also that werelate

t

-trunations in

D

with virtual

t

-trunationsofohomologialfuntorson

C

. Virtual

t

-trunationsfor ohomologialfuntors aredened for any

(C, w)

(wedonot needany trian-gulated 'ategoriesoffuntors' or

t

-struturesforthem here);this notionwas introduedin2.5of[6℄andisstudiedfurther intheurrentpaper.

Now,weexplainwhywereallyneedaertainnewategoryofomotives (on-taining Voevodsky's

DM

ef f

gm

), and so the theory of adjaent strutures (i.e. orthogonalstruturesinthease

C

=

D

,

Φ =

C(−,

−)

)isnotsuientforour purposes. Itwasalreadyprovedin[6℄thatweightstruturesprovidea power-fultoolforonstrutingspetralsequenes;theyalsorelatetheohomologyof objetsoftriangulatedategorieswith

t

-struturesadjaenttothem. Unfortu-nately,aweightstrutureorrespondingtooniveauspetralsequenesannot existon

DM

ef f

−

⊃

DM

gm

ef f

sinetheseategoriesdonotontain(any)motives forfuntioneldsover

k

(aswellasmotivesofothershemesnotofnitetype over

k

;stillf. Remark 4.5.4(5)). Yetthese motivesshouldgeneratetheheart ofthis weightstruture(sinethe objetsofthisheart should orepresent o-variant exatfuntors from the ategoryof homotopy invariant sheaveswith transfersto

Ab

).

So,weneedaategorythatwouldontainertainhomotopylimitsofobjetsof

DM

ef f

gm

. Wesueedin onstrutingatriangulatedategory

D

(ofomotives) thatallowsustoreahtheobjetiveslisted. Unfortunately,inordertoontrol morphisms between homotopy limits mentioned we have to assume

k

to be ountable. Inthis asethere exists a largeenough triangulatedategory

D

s

(

DM

ef f

gm

⊂

D

s

⊂

D)

endowed with aertain Gersten weight struture

w

; its heartis'generated'byomotivesoffuntionelds.

w

is(left)orthogonaltothe homotopy

t

-struture on

DM

ef f

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thosereaderswhowouldjustwanttohaveaategorythatontainsreasonable homotopy limits of geometri motives(inluding omotivesof funtion elds and ofsmoothsemi-loal shemes),andonsider ohomologytheoriesforthis ategory,mayfreely ignore thisrestrition. Moreover,foran arbitrary

k

one anstillpasstoaountablehomotopylimitintheGysindistinguishedtriangle (asin Proposition3.6.1). Yetforanunountable

k

ountablehomotopylimits don't seem to be interesting; in partiular, they denitely do not allow to onstrutaGerstenweightstruture (inthisase).

So, weonsider aertain triangulated ategory

D

⊃

DM

ef f

gm

that (roughly!) 'onsists of' (ovariant) homologial funtors

DM

ef f

gm

→

Ab

. In partiular, objets of

D

dene ovariant funtors

SmV ar

→

Ab

(whereas another 'big' motivi ategory

DM

ef f

−

dened by Voevodsky is onstruted from ertain sheaves i.e. ontravariant funtors

SmV ar

→

Ab

; this is also true for all motivihomotopyategoriesofVoevodskyandMorel). Besides,

DM

ef f

gm

yields afamilyof(weak)oompatogeneratorsfor

D.

Thisiswhyweallobjetsof

D

omotives.Yetnotethattheembedding

DM

ef f

gm

→

D

isovariant(atually, we invert the arrows in the orresponding 'ategory of funtors' in order to make the Yoneda embedding funtor ovariant), as well as the funtor that sendsasmoothsheme

U

(not neessarilyofnitetypeover

k

)to itsomotif (whih oinideswithitsmotifif

U

isasmoothvariety).

Wealsoreallthe Chowweightstruture

w

′

Chow

introduedin [6℄; the orre-sponding Chow-weight spetral sequenes are isomorphito the lassial(i.e. Deligne's)weightspetralsequeneswhenthelatteraredened.

w

′

Chow

ould be naturally extended to a weight struture

w

Chow

for

D.

We always have a naturalomparison morphism from the Chow-weightspetralsequene for

(H, X)

to the orrespondingoniveauone; itis anisomorphismfor any bira-tional ohomology theory. We onsider the ategory of birational omotives

D

bir

i.e. theloalizationof

D

by

D

(1)

(thatontainstheategoryofbirational geometrimotivesintroduedin[15℄;thoughsomeoftheresultsofthis unpub-lished preprintare erroneous,thismakesnodierene fortheurrentpaper). Itturnsourthat

w

and

w

Chow

induethesameweightstruture

w

′

bir

on

D

bir

. Conversely,startingfrom

w

′

bir

onean'glue'(fromslies)theweightstrutures induedby

w

and

w

Chow

on

D

/

D

(n)

forall

n >

0

. Moreover,thesestrutures belongtoaninterestingfamilyofweightstruturesindexedbyasingleintegral parameter! Itouldbeinterestingtoonsiderothermembersofthisfamily. We relatebrieythese observationswiththoseofA. Beilinson(in[3℄ heproposed a'geometri'haraterizationoftheonjeturalmotivi

t

-struture).

NowwedesribetheonnetionofourresultswithrelatedresultsofF.Deglise (see[9℄,[10℄,and[11℄; notethatthetwolatterpapersarenotpublishedatthe moment yet). He onsiders a ertain ategoryof pro-motives whose objets arenaiveinverselimitsofobjetsof

DM

ef f

gm

(thisategoryisnottriangulated, thoughit is pro-triangulated in aertain sense). This approah allowsto ob-tain(in auniversalway)lassialoniveauspetralsequenesforohomology ofmotivesofsmoothvarieties;Deglisealsoprovestheirrelationwiththe homo-topy

t

-trunationsforohomologyrepresentedbyanobjetof

DM

ef f

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ohomologytheoriesnotomingfrommotiviomplexes,thismethoddoesnot seem to extendto (spetral sequenes for ohomology of) arbitrarymotives; motivifuntorialityis notobviousalso. Moreover,Deglise didn'tprovethat thepro-motifofa(smoothonneted)semi-loalshemeisadiretsummand ofthepro-motifofitsgeneripoint(though thisistrue,atleastintheaseof aountable

k

). Wewilltellmuhmoreaboutourstrategyandontherelation ofourresultswiththoseofDeglisein 1.5below. Notealsothat ourmethods are muhmore onvenientfor studying funtoriality(of oniveauspetral se-quenes)thanthemethods appliedbyM.Rostin therelatedontextofyle modules(see[24℄and4of[10℄).

The author would like to indiate the interdependeniesof the parts of this text (in order to simplify reading for those who are not interested in all of it). Those readers whoarenot (verymuh) interestedin (oniveau) spetral sequenes,mayavoidmostofsetion2andreadonly2.12.2(Remark2.2.2 ouldalsobeignored). Moreover,inordertoproveourdiretsummandsresults (i.e. Theorem 4.2.1, Corollary4.2.2,and Proposition4.3.1) oneneedsonly a small portion of the theory of weight strutures; so a reader very relutant to study this theory may tryto derivethem from theresults of3 'by hand' without reading 2at all. Still,for motivifuntorialityof oniveauspetral sequenes and ltrations (see Proposition 4.4.1 and Remark 4.4.2)one needs more of weight strutures. On the other hand, those readers who are more interestedin the(general)theory oftriangulatedategoriesmayrestrittheir attentionto1.11.2and2;yetnotethat therest ofthepaperdesribesin detailanimportant(andquitenon-trivial)exampleofaweightstruturewhih is orthogonal to a

t

-struture with respet to a nie duality (of triangulated ategories). Moreover,muh ofsetion4ouldalsobeextended toageneral setting of atriangulated ategorysatisfyingpropertiessimilar to those listed in Proposition 3.1.1;yettheauthor hose notto dothis inorder tomakethe papersomewhatlessabstrat.

Now we list the ontents of the paper. More details ould be found at the beginningsofsetions.

Westart1withthereolletionof

t

-strutures,idempotentompletions,and Postnikovtowersfortriangulatedategories. Wedesribeamethodfor extend-ing ohomologialfuntors from afull triangulated subategoryto thewhole

C

(afterH. Krause). Nextwereall someresultsand denitions for Voevod-sky's motives (thisinludes ertain properties of Tate twists for motivesand ohomologialfuntors). Lastly,wedenepro-motives(followingDeglise)and omparethem with ourtriangulatedategory

D

of omotives. Thisallowsto explainourstrategystepbystep.

2is dediatedtoweightstrutures. Firstweremindthebasisofthis theory (developed in [6℄). Next we reall that aohomologial funtor

H

from an (arbitrarytriangulatedategory)

C

endowedwithaweightstruture

w

ould be'trunated'asifitbelongedtosometriangulatedategoryoffuntors(from

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(intro-dues in ibid.). Weprovethat thederivedexatouple foraweightspetral sequene ouldbedesribed in termsof virtual

t

-trunations. Nextwe intro-duethedenitiona(nie)duality

Φ :

C

op

×

D

→

A

(here

D

istriangulated,

A

isabelian),andoforthogonalweightand

t

-strutures(withrespetto

Φ

). If

w

isorthogonalto

t

,thenthevirtual

t

-trunations(orrespondingto

w

)of fun-torsofthetype

Φ(−, Y

), Y

∈

ObjD

,areexatlythefuntors'representedvia

Φ

'bytheatual

t

-trunationsof

Y

(orrespondingto

t

). Heneif

w

and

t

are orthogonalwithrespettoanieduality,theweightspetralsequene onverg-ing to

Φ(X, Y

)

(for

X

∈

ObjC, Y

∈

ObjD

)is naturallyisomorphi(starting from

E

2

) to the oneomingfrom

t

-trunations of

Y

. We alsomention some alternativesandpredeessorsofourresults. Lastlyweompareweight deom-positions, virtual

t

-trunations, and weight spetral sequenes orresponding to distintweightstrutures(inpossiblydistinttriangulatedategories). In3wedesribethemainpropertiesof

D

⊃

DM

ef f

gm

. Theexathoieof

D

is notimportantformostofthispaper;sowejustlist themainpropertiesof

D

(anditsertainenhanement

D

′

)in3.1. Weonstrut

D

usingtheformalism ofdierentialgradedmodulesin5later. Nextwedeneomotivesfor(ertain) shemesandind-shemesofinnitetypeover

k

(weallthempro-shemes). We reall the notionof aprimitivesheme. All (smooth) semi-loal pro-shemes areprimitive;primitiveshemeshaveallnie'motivi'propertiesofsemi-loal pro-shemes. We prove that there are no

D-morphisms

of positive degrees betweenomotivesofprimitiveshemes(andalsobetweenertainTate twists of those). In 3.6weprovethat the Gysin distinguishedtriangle for motives of smooth varieties (in

DM

ef f

gm

) ould benaturally extended to omotivesof pro-shemes. This allowsto onstrutertain Postnikovtowersforomotives ofpro-shemes;thesetowersareloselyrelatedwithlassialoniveauspetral sequenesforohomology.

4 is entral in this paper. We introdue a ertain Gersten weight struture for a ertain triangulated ategory

D

s

(

DM

ef f

gm

⊂

D

s

⊂

D).

We provethat Postnikovtowersonstrutedin3.6areatuallyweightPostnikovtowerswith respetto

w

. Wededueour(interesting)resultsondiretsummandsof omo-tivesoffuntionelds. Wetranslatetheseresultstoohomologyintheobvious way.

Nextweprovethatweightspetralsequenesfortheohomologyof

X

(orre-sponding to the Gerstenweightstruture) are naturallyisomorphi (starting from

E

2

) to the lassial oniveau spetral sequenes if

X

is the motif of a smoothvariety;soweallthesespetralsequeneoniveauonesinthegeneral ase also. Wealso prove that the Gerstenweight struture

w

(on

D

s

) is or-thogonalto the homotopy

t

-struture

t

on

DM

ef f

−

(with respetto a ertain

Φ

). It followsthat for anarbitrary

X

∈

ObjDM

s

, for a ohomology theory representedby

Y

∈

ObjDM

ef f

−

(anyhoieof)theoniveauspetralsequene that onvergesto

Φ(X, Y

)

ouldbedesribedin termsof the

t

-trunationsof

Y

(startingfrom

E

2

).

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sequenesoverountableperfetsubeldsofdenition. Thisdenitionis om-patiblewiththelassialone;soweestablishmotivifuntorialityofoniveau spetralsequenesin thisasealso.

Wealsoprovethat theChowweight struturefor

DM

ef f

gm

(introduedin6of [6℄)ouldbeextendedtoaweightstruture

w

Chow

on

D.

Theorresponding Chow-weightspetralsequenesare isomorphito thelassial(i.e. Deligne's) ones whenthelatteraredened(thiswasprovedin [6℄and[7℄). Weompare oniveauspetralsequeneswithChow-weightones: wealwayshavea ompar-ison morphism; it is anisomorphism fora birational ohomology theory. We onsidertheategoryofbirationalomotives

D

bir

i.e. theloalizationof

D

by

D

(1)

.

w

and

w

Chow

induethesameweightstruture

w

′

bir

on

D

bir

;onealmost an glue

w

and

w

Chow

from opies of

w

′

bir

(one may say that these weight struturesouldalmostbegluedfrom thesameslieswithdistintshifts). 5 is dediated to the onstrutionof

D

and theproof of its properties. We applytheformalismofdierentialgradedategories,modulesoverthem,andof theorrespondingderivedategories. A readernotinterestedin these details may skip (most of) this setion. In fat, the author is not sure that there existsonlyone

D

suitableforourpurposes;yetthehoieof

D

doesnotaet ohomologyof(omotivesof)pro-shemesandofVoevodsky'smotives. Wealsoexplainhowthedierentialgradedmodulesformalismanbeusedto dene base hange (extensionand restritionof salars) for omotives. This allowstoextendourresultsondiretsummandsofomotives(andohomology) offuntioneldstopro-shemesobtainedfromthemviabasehange. Wealso dene tensoringof omotivesby motives(in partiular, this yieldsTatetwist for

D),

as wellasaertainointernalHom(i.e. theorrespondingleftadjoint funtor).

6 isdediated to propertiesof omotivesthat arenot (diretly)relatedwith themain resultsof thepaper;wealsomakeseveralomments. Wereall the denitionoftheadditiveategory

D

gen

ofgenerimotives(studiedin [9℄). We provethat theexatonservativeweight omplex funtororrespondingto

w

(that exists by the generaltheory of weightstrutures) ould bemodiedto an exatonservative

W C

:

D

s

→

K

b

(

D

gen

)

. Next weprove that a ofun-tor

Hw

→

Ab

is representable by a homotopy invariant sheaf with transfers wheneverisonvertsallprodutsinto diretsums.

Wealsonotethatourtheoryouldbeeasilyextended to(o)motiveswith o-eientsin an arbitraryring. Next wenote (after B. Kahn)that reasonable motivesofpro-shemeswith ompatsupport doexist in

DM

ef f

−

; this obser-vationouldbeusedfortheonstrutionofanalternativemodelfor

D.

Lastly wedesribewhihparts ofourargumentdonotwork (andwhih dowork)in theaseofanunountable

k

.

A aution: the notion of a weight struture is quite a general formalismfor triangulated ategories. In partiular, onetriangulated ategoryansupport several distint weight strutures (note that there is a similar situation with

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struture

w

for

D

s

and aChowweightstruture

w

Chow

for

D.

Moreover,we showin 4.9 that these weight struturesare ompatible withertain weight struturesdenedontheloalizations

D

/

D

(n)

(forall

n >

0

). Thesetwoseries ofweightstruturesaredenitelydistint: notethat

w

yieldsoniveauspetral sequenes,whereas

w

Chow

yieldsChow-weightspetralsequenes,that general-izeDeligne'sweightspetralsequenesforétaleandmixedHodgeohomology (see [6℄ and [7℄). Also,the weightomplex funtoronstruted in [7℄ and [6℄ isquitedistintfromtheoneonsideredin6.1below(eventhetargetsofthe funtorsmentionedareompletelydistint).

The author is deeply grateful to prof. F. Deglise, prof. B. Kahn, prof. M. Rovinsky, prof. A. Suslin, prof. V. Voevodsky, and to the referee for their interesting remarks. The author gratefully aknowledges the support from Deligne 2004 Balzan prize in mathematis. The work is also supported by RFBR (grantsno. 08-01-00777aand10-01-00287a).

Notation.

Foraategory

C, A, B

∈

ObjC

, wedenoteby

C(A, B)

thesetof

A

-morphismsfrom

A

into

B

.

Forategories

C, D

wewrite

C

⊂

D

if

C

is afullsubategoryof

D

.

Foradditive

C, D

wedenoteby

AddFun(C, D)

theategoryofadditivefuntors from

C

to

D

(wewillignoreset-theoretidiultiesheresinetheydonotaet ourargumentsseriously).

Ab

istheategoryofabeliangroups. Foranadditive

B

wewilldenote by

B

∗

theategory

AddFun(B, Ab)

andby

B

∗

theategory

AddFun(B

op

, Ab)

. Note thatbothoftheseareabelian. Besides,Yoneda'slemmagivesfullembeddings of

B

into

B

∗

andof

B

op

into

B

∗

(thesesend

X

∈

ObjB

to

X

∗

=

B(−, X)

and to

X

∗

=

B(X,

−)

,respetively).

Foraategory

C, X, Y

∈

ObjC

, we saythat

X

is aretrat of

Y

if

id

X

ould be fatorized through

Y

. Note that when

C

is triangulated orabelian then

X

is aretrat of

Y

if and onlyif

X

is itsdiret summand. For any

D

⊂

C

the subategory

D

is alled Karoubi-losed in

C

if it ontains all retrats of its objets in

C

. We will all the smallest Karoubi-losed subategoryof

C

ontaining

D

the Karoubization of

D

in

C

; sometimes we will use the same term for the lass of objetsof the Karoubization of afull subategory of

C

(orrespondingtosomesublassof

ObjC

).

Foraategory

C

wedenoteby

C

op

itsoppositeategory. Foranadditive

C

anobjet

X

∈

ObjC

isalledoompatif

C(

Q

i∈I

Y

i

, X) =

L

i∈I

C(Y

i

, X)

for anyset

I

and any

Y

i

∈

ObjC

suh that theprodutexists (herewedon'tneedtodemandallprodutstoexist,thoughtheyatuallywill exist below).

For

X, Y

∈

ObjC

wewillwrite

X

⊥

Y

if

C(X, Y

) =

{0}

. For

D, E

⊂

ObjC

we will write

D

⊥

E

if

X

⊥

Y

forall

X

∈

D, Y

∈

E

. For

D

⊂

C

wewilldenote by

D

⊥

thelass

{Y

∈

ObjC

:

X

⊥

Y

∀X

∈

D}.

Sometimes we will denote by

D

⊥

the orresponding full subategory of

C

. Dually,

⊥

D

(10)

oppositetotheoneof9.1of[21℄.

Inthispaperallomplexeswillbeohomologiali.e. thedegreeofall dieren-tialsis

+1

;respetively,wewilluseohomologialnotationfortheirterms. For anadditiveategory

B

wedenote by

C(B)

the ategoryof (unbounded) omplexes overit.

K(B)

will denotethehomotopy ategoryof omplexes. If

B

is alsoabelian,wewilldenote by

D(B)

thederivedategoryof

B

. Wewill also need ertain bounded analoguesof these ategories (i.e.

C

b

(B)

,

K

b

(B)

,

D

−

(B)

).

C

and

D

will usually denote some triangulated ategories. We will use the term 'exat funtor' for afuntor of triangulated ategories (i.e. for a for a funtorthatpreservesthestruturesoftriangulatedategories).

A

willusuallydenote someabelianategory. Wewill allaovariantadditive funtor

C

→

A

for an abelian

A

homologial if it onverts distinguished tri-anglesinto longexatsequenes;homologialfuntors

C

op

→

A

will bealled ohomologial whenonsideredasontravariantfuntors

C

→

A

.

H

:

C

op

→

A

willalwaysbeadditive;itwillusuallybeohomologial.

For

f

∈

C(X, Y

)

,

X, Y

∈

ObjC

, wewill allthe third vertex of(any) distin-guishedtriangle

X

f

→

Y

→

Z

aoneof

f

. Notethatdierenthoiesof ones areonnetedbynon-uniqueisomorphisms,f. IV.1.7of[13℄. Besides,in

C(B

)

wehaveanonialonesofmorphisms(seesetionIII.3ofibid.).

Wewilloftenspeifyadistinguishedtrianglebytwoofitsmorphisms. When dealing with triangulated ategories we (mostly) use onventions and auxiliary statements of [13℄. For a set of objets

C

i

∈

ObjC

,

i

∈

I

, we will denoteby

hC

i

thesmalleststritlyfulltriangulatedsubategoryontainingall

C

i

;for

D

⊂

C

wewill write

hDi

insteadof

hObjDi

.

We will saythat

C

i

generate

C

if

C

equals

hC

i

. We will saythat

C

i

weakly ogenerate

C

iffor

X

∈

ObjC

wehave

C(X, C

i

[j]) =

{0} ∀i

∈

I, j

∈

Z

=

⇒

X

= 0

(i.e. if

⊥

{C

i

[j]}

ontainsonlyzeroobjets).

We will all a partially ordered set

L

a(ltered) projetive system iffor any

x, y

∈

L

thereexistssomemaximumi.e. a

z

∈

L

suhthat

z

≥

x

and

z

≥

y

. By abuseofnotation,wewillidentify

L

withthefollowingategory

D

:

ObjD

=

L

;

D(l

′

, l)

isemptywhenever

l

′

< l

,and onsistsofasinglemorphismotherwise; the omposition of morphisms is the only one possible. If

L

is a projetive system,

C

is someategory,

X

:

L

→

C

isaovariantfuntor,wewilldenote

X

(l)

for

l

∈

L

by

X

l

. We will write

Y

= lim

←−

l∈L

X

l

for the limit of this funtor; we will all it the inverse limit of

X

l

. We will denote the olimitof a ontravariant funtor

Y

:

L

→

C

by

lim

−→

l∈L

Y

l

and all it the diret limit. Besides,wewillsometimesalltheategorialimageof

L

withrespettosuh an

Y

anindutivesystem.

Below

I, L

will often be projetive systems; we will usually require

I

to be ountable.

A subsystem

L

′

of

L

is apartially ordered subset in whih maximums exist (wewillalsoonsidertheorrespondingfullsubategoryof

L

). Wewillall

L

′

unboundedin

L

ifforany

l

∈

L

thereexistsan

l

′

∈

L

′

suhthat

l

′

≥

l

(11)

k

willbeourperfetbaseeld. Belowwewillusuallydemand

k

tobeountable. Note: thisyieldsthatforanyvarietythesetofitslosed(oropen)subshemes isountable.

Wealsolistentraldenitions andmainnotationofthispaper.

Firstwelistthemain(general)homologialalgebradenitions.

t

-strutures,

t

-trunations,andPostnikovtowersintriangulatedategoriesaredenedin1.1; weightstrutures,weightdeompositions,weighttrunations,weightPostnikov towers,andweightomplexesareonsideredin 2.1;virtual

t

-trunationsand nieexatomplexesoffuntorsaredenedin2.3;weightspetralsequenes arestudiedin2.4;(nie)dualitiesandorthogonalweightand

t

-struturesare dened in Denition 2.5.1;rightand left weight-exat funtorsare dened in Denition 2.7.1.

Nowwelist notation (andsome denitions) formotives.

DM

ef f

gm

⊂

DM

ef f

−

,

HI

andthehomotopy

t

-struturefor

DM

ef f

gm

aredenedin1.3;Tatetwistsare onsideredin1.4;

D

naive

isdenedin 1.5;omotives(Dand

D

′

)aredened in3.1;in3.2wedisusspro-shemesandtheiromotives;in3.3wereallthe denitionofaprimitivesheme;in4.1wedenetheGerstenweightstruture

w

onaertaintriangulated

D

s

; weonsider

w

Chow

in 4.7;

D

bir

and

w

′

bir

are dened in 4.9; several dierential graded onstrutions (inludingextension and restritionof salarsfor omotives) areonsidered in 5; wedene

D

gen

and

W C

:

D

s

→

K

b

(

D

gen

)

in6.1.

1

Some preliminaries on triangulated categories and motives

1.1wereallthenotionofa

t

-struture(andintroduesomenotationforit), reallthenotionofanidempotentompletion ofanadditiveategory;wealso reallthatanysmallabelianategoryouldbefaithfullyembeddedinto

Ab

(a well-knownresultbyMithell).

In1.2 wedesribe(followingH.Krause)anaturalmethod forextending o-homologialfuntorsfromafulltriangulated

C

′

⊂

C

to

C

. In1.3wereallsomedenitionsandresultsofVoevodsky.

In 1.4 we reall thenotion of aTate twist; we study the properties of Tate twistsformotivesandhomotopyinvariantsheaves.

In1.5wedene pro-motives(following[9℄and[10℄). Thesearenotneessary for ourmain result; yet theyallow to explainour methods stepby step. We alsodesribeindetailtherelationofouronstrutionsandresultswiththose ofDeglise.

1.1

t

-structures, Postnikov towers, idempotent completions, and

an embedding theorem of Mitchell

Toxthenotationwereallthedenitionofa

t

-struture.

Definition

1.1.1

.

Apairofsublasses

C

t≥

0

, C

t≤

0

⊂

ObjC

foratriangulated ategory

C

will be said to dene a

t

-struture

t

if

(ii)

C

t≥

. (iii)Orthogonality.

C

t≤

0

[1]

⊥

C

t≥

0

.

(iv)

t

-deomposition. Forany

X

∈

ObjC

thereexistsadistinguishedtriangle

A

→

X

→

B[−1]→A[1]

(1)

suhthat

A

∈

C

t≤

0

, B

∈

C

t≥

0

.

Wewillneedsomemorenotationfor

t

-strutures.

Definition

1.1.2

.

1. A ategory

Ht

whoseobjetsare

C

t

=0

=

C

t≥

0

∩

C

t≤

0

,

Ht(X, Y

) =

C(X, Y

)

for

X, Y

∈

C

t

=0

,will bealledtheheartof

t

. Reall(f. Theorem 1.3.6 of [2℄) that

Ht

is abelian (short exat sequenes in

Ht

ome fromdistinguishedtrianglesin

C

).

2.

C

t≥l

(resp.

C

t≤l

)willdenote

C

t≥

0

[−l]

(resp.

C

t≤

0

[−l]

).

Remark 1.1.3. 1. The axiomatisof

t

-strutures is self-dual: if

D

=

C

op

(so

ObjC

=

ObjD

)thenoneandenethe(opposite)weightstruture

t

′

on

D

,respetively.

3. (1)willbealledthet-deompositionof

X

. If

X

=

Y

[i]

forsome

Y

∈

ObjC

,

i

∈

Z

, then we will denote

A

by

Y

t≤i

(itbelongsto

C

t≤

0

)and

B

by

Y

t≥i

+1

(itbelongsto

C

t≥

0

),respetively. Sometimeswewilldenote

Y

t

≤

i

[−i]

by

t

≤i

Y

;

t

≥i

+1

Y

=

Y

t

≥

i

+1

[−i

−

1]

. Objetsofthetype

Y

t

≤

i

[j]

and

Y

t

≥

i

[j]

(for

i, j

∈

Z

) willbealled

t

-trunationsof

Y

.

4. Wedenoteby

X

t

=

i

the

i

-thohomologyof

X

withrespetto

t

i.e.

(Y

t≤i

)

t≥

0

(f. part10ofIV.4of[13℄).

5. The following statements are obvious (and well-known):

C

t≤

0

=

⊥

C

t≥

1

;

C

t≥

0

=

C

t≤−

1

⊥

.

Nowwereallthenotionofidempotentompletion.

Definition

1.1.4

.

An additiveategory

B

is said tobeidempotent omplete iffor any

X

∈

ObjB

and anyidempotent

p

∈

B(X, X)

there exists a deom-position

X

=

Y

L

Z

suhthat

p

=

i

◦

j

, where

i

istheinlusion

Y

→

Y

L

Z

,

j

istheprojetion

Y

L

Z

→

Y

.

Reallthatanyadditive

B

anbeanoniallyidempotentompleted. Its idem-potentompletion is (by denition) theategory

B

′

whose objetsare

(X, p)

for

X

∈

ObjB

and

p

∈

B(X, X) :

p

2

=

p

;wedene

(13)

Itanbeeasilyhekedthatthisategoryisadditiveandidempotentomplete, and for any idempotent omplete

C

⊃

B

we have anatural full embedding

B

′

→

C

.

The main result of [1℄ (Theorem 1.5) states that an idempotent ompletion of atriangulated ategory

C

has anatural triangulation (with distinguished trianglesbeingallretratsofdistinguishedtrianglesof

C

).

Belowwewill needthenotionofaPostnikovtowerinatriangulatedategory severaltimes(f. IV2of[13℄)).

Definition

1.1.5

.

Let

C

beatriangulatedategory. 1. Let

l

≤

m

∈

Z

.

We will all a bounded Postnikov tower for

X

∈

ObjC

the following data: a sequene of

C

-morphisms

(0 =)Y

l

→

Y

l

+1

→ · · · →

Y

m

=

X

along with distinguishedtriangles

Y

i

→

Y

i

+1

→

X

i

(2)

forsome

X

i

∈

ObjC

;here

l

≤

i < m

.

2. An unbounded Postnikovtowerfor

X

is a olletionof

Y

i

for

i

∈

Z

that is equipped (for all

i

∈

Z

) with: onneting arrows

Y

i

→

Y

i

+1

(for

i

∈

Z

), morphisms

Y

i

→

X

suh that all the orresponding triangles ommute, and distinguishedtriangles(2).

Inbothases,wewilldenote

X

−p

[p]

by

X

p

;wewillall

X

p

thefatorsofout Postnikovtower.

Remark 1.1.6. 1. Composing (andshifting) arrowsfrom triangles(2) fortwo subsequent

i

oneanonstrutaomplexwhosetermsare

X

p

(itiseasilyseen that this is aomplexindeed, f. Proposition 2.2.2 of [6℄). This observation will beimportant forus belowwhen we willonsider ertain weightomplex funtors.

2. Certainly,abounded Postnikovtowerould beeasily ompleted to an un-boundedone. Forexample,oneouldtake

Y

i

= 0

for

i < l

,

Y

i

=

X

for

i > m

; then

X

i

= 0

if

i < l

or

i

≥

m

.

Lastly,wereallthefollowing(well-known)result.

Proposition

1.1.7

.

For any small abelian ategory

A

there exists an exat faithfulfuntor

A

→

Ab

.

Proof. BytheFreyd-Mithell'sembeddingtheorem,anysmall

A

ouldbefully faithfully embedded into

R

−

mod

for some (assoiative unital) ring

R

. It remainstoapplytheforgetfulfuntor

R

−

mod

→

Ab

.

Remark 1.1.8. 1. Wewill needthis statementbelowin order to assumethat objets of

A

'have elements'; this will onsiderably simplify diagram hase. Note thatweanassumetheexisteneof elementsforanotneessarilysmall

A

intheasewhenareasoningdealsonlywithanitenumberofobjetsof

A

at atime.

(14)

1.2

Extending cohomological functors from a triangulated

sub-category

Wedesribeamethod forextendingohomologialfuntorsfrom afull trian-gulated

C

′

⊂

C

to

C

(afterH.Krause). Notethatbelowwewillapplysomeof theresultsof [17℄in thedual form. Theonstrutionrequires

C

′

to be skele-tallysmalli.e. thereshould exista(proper) subset

D

⊂

ObjC

′

suh thatany objetof

C

′

isisomorphitosomeelementof

D

. Forsimpliity,wewill some-times(whenwritingsumsover

ObjC

′

)assumethat

ObjC

′

isasetitself. Sine thedistintionbetweensmallandskeletallysmallategorieswillnotaetour argumentsandresults,wewillignoreitintherestofthepaper.

If

A

isanabelianategory,then

AddFun(C

′op

, A)

isabelianalso;omplexesin itareexatwhenevertheyareexatomponentwisely.

Supposethat

A

satisesAB5i.e. itislosedwithrespettoallsmall oprod-uts,andltereddiretlimitsofexatsequenesin

A

are exat.

Let

H

′

∈

AddFun(C

′op

, A)

beanadditivefuntor(it willusually be ohomo-logial).

Proposition

1.2.1

.

ILet

A, H

′

bexed. 1. There existsan extension of

H

′

to an additive funtor

H

:

C

→

A

. It is ohomologial whenever

H

is. Theorrespondene

H

′

→

H

denesanadditive funtor

AddFun(C

′op

, A)

→

AddFun(C

op

, A)

.

2. Moreover,supposethatin

C

wehaveaprojetivesystem

X

l

, l

∈

L

,equipped with a ompatible system of morphisms

X

→

X

l

, suh that the latter system for any

Y

∈

ObjC

′

indues an isomorphism

C(X, Y

)

∼

= lim

−→

C(X

l

, Y

)

. Then wehave

H(X

)

∼

= lim

−→

H(X

l

)

. IILet

X

∈

ObjC

bexed.

1. One an hoose a family of

X

l

∈

ObjC

and

f

l

∈

C(X, X

l

)

suh that

(f

l

)

indue a surjetion

⊕H

′

(X

l

)

→

H(X

)

for any

H

′

, A

, and

H

as in assertion I1.

2. Let

F

′

→

f

′

G

′

→

g

′

H

′

be a (three-term) omplex in

AddFun(C

′op

, A)

that is exat in the middle; suppose that

H

′

is ohomologial. Then the omplex

F

→

f

G

→

g

H

(here

F, G, H, f, g

are the orresponding extensions) isexat in the middlealso.

Proof. I1. Following1.2of[17℄(anddualizingit),weonsidertheabelian at-egory

C

=

C

′∗

= AddFun(C

′

, Ab)

(thisis

Mod

C

′

op

inthenotationofKrause). Thedenitioneasilyimpliesthatdiretlimitsin

C

areexatlyomponentwise diretlimitsoffuntors. WehavetheYoneda'sfuntor

i

′

:

C

op

→

C

thatsends

X

∈

ObjC

to thefuntor

X

∗

= (Y

7→

C(X, Y

), Y

∈

ObjC

′

)

; it isobviously ohomologial. Wedenoteby

i

therestritionof

i

′

to

C

′

(

i

isoppositetoafull embedding).

ByLemma2.2of[17℄(appliedtotheategory

C

′op

)weobtainthatthereexists an exatfuntor

G

:

C

→

A

that preservesallsmall oproduts andsatises

G

◦

i

=

H

′

(15)

exatsequene(in

C

)

⊕

j∈J

X

j

∗

→ ⊕

l∈L

X

l

∗

→

X

∗

→

0

(3)

for

X

j

, X

l

∈

C

′

,thenweset

G(X

) = Coker

⊕

j∈J

H

′

(X

j

)

→ ⊕

l∈L

H

′

(X

l

).

(4)

We dene

H

=

G

◦

i

′

; itwasprovedin lo.it. that weobtaina well-dened funtor thisway. As was also provedin lo.it.,the orrespondene

H

′

7→

H

yieldsafuntor;

H

isohomologialif

H

′

is.

2. The proofoflo.it. shows(andmentions) that

G

respets(small)ltered inverselimits. Nownotethat ourassertionsimply:

X

∗

= lim

−→

X

l

∗

in

C

. II1. Thisisimmediatefrom(4).

2. Note that the assertion is obviously valid if

X

∈

ObjC

′

. We redue the generalstatementtothisase.

Applying Yoneda's lemma to (3) is weobtain (anonially) some morphisms

f

l

:

X

→

X

l

forall

l

∈

L

and

g

lj

:

X

l

→

X

j

forall

l

∈

L

,

j

∈

J

,suhthat: for any

l

∈

L

almostall

g

lj

are

0

; forany

j

∈

J

almost all

g

lj

is

0

;for any

j

∈

J

wehave

P

l∈L

g

lj

◦

f

l

= 0

.

Now,by Proposition 1.1.7, wemayassumethat

A

=

Ab

(see Remark 1.1.8). Weshould hek: iffor

a

∈

G(X)

wehave

g

∗

(a) = 0

, then

a

=

f

∗

(b)

forsome

b

∈

F

(X

)

.

Usingadditivityof

C

′

and

C

,weangathernitesetsof

X

l

and

X

j

intosingle objets. Hene we an assume that

a

=

G(f

l

0

)(c)

for some

c

∈

G(X

l

) (=

G

′

(X

l

)), l

0

∈

L

and that

g

∗

(c)

∈

H

(g

∈

J

, whereas

g

l

0

j

0

◦

f

l

0

= 0

. We omplete

X

l

0

→

X

j

0

to a distinguished triangle

Y

α

→

X

l

0

g

l

0

j

0

→

X

j

0

; we anassume that

B

∈

ObjC

′

. Weobtain that

f

l

0

ould be

presentedas

α

◦

β

forsome

β

∈

C(X, Y

)

. Sine

H

′

isohomologial,weobtain that

H

(α)(g

∗

(c)) = 0

. Sine

Y

∈

ObjC

, theomplex

F

(Y

)

→

G(Y

)

→

H

(Y

)

is exat in the middle; hene

G(α)(c) =

f

∗

(d)

for some

d

∈

F

(Y

)

. Then we antake

b

=

F

(β)(d)

.

1.3

Some definitions of Voevodsky: reminder

Weusemuhnotationfrom[25℄. Wereall(someof)itherefortheonveniene ofthereader,andintroduesomenotationof ourown.

V ar

⊃

SmV ar

⊃

SmP rV ar

willdenote thelassof allvarietiesover

k

, resp. ofsmoothvarieties,resp. ofsmoothprojetivevarieties.

Wereallthatforategoriesofgeometriorigin(inpartiular,for

SmCor

de-nedbelow)theadditionofobjetsisdenedviathedisjointunionofvarieties operation.

We dene the ategory

SmCor

of smooth orrespondenes.

ObjSmCor

=

SmV ar

,

SmCor(X, Y

) =

L

(16)

oforrespondenesisdenedintheusualwayviaintersetions(yet,wedonot needtoonsider orrespondenesupto anequivalenerelation).

We will write

· · · →

X

i−

1

→

X

i

→

X

i

+1

→

. . .

, for

X

l

∈

SmV ar

, for the orrespondingomplexover

SmCor

.

P reShv(SmCor)

will denote the (abelian) ategory of additive ofuntors

SmCor

→

Ab

; itsobjetsareusually alledpresheaves withtransfers.

Shv(SmCor) =

Shv(SmCor)

N is

⊂

P reShv(SmCor)

is theabelianategory ofadditiveofuntors

SmCor

→

Ab

thataresheavesintheNisnevihtopology (whenrestritedtotheategoryofsmoothvarieties);thesesheavesareusually alledsheaves with transfers.

D

−

(Shv(SmCor))

will be the bounded above derived ategory of

Shv(SmCor)

.

For

Y

∈

SmV ar

(more generally,for

Y

∈

V ar

, see 4.1of [25℄) weonsider

L(Y

) =

SmCor(−, Y

)

∈

Shv(SmCor)

. For a bounded omplex

X

= (X

i

)

(as above) wewill denote by

L(X

)

the omplex

· · · →

L(X

i−

1

)

→

L(X

i

)

→

L(X

i

+1

)

→ · · · ∈

C

b

(Shv(SmCor))

.

S

∈

Shv(SmCor)

is alled homotopy invariant if for any

X

∈

SmV ar

the projetion

A

1

×

X

→

X

givesanisomorphism

S

(X

)

→

S(

A

1

×

X

)

. Wewill denote theategoryofhomotopy invariantsheaves(withtransfers) by

HI

;it isanexatabeliansubategoryof

SmCor

byProposition3.1.13of[25℄.

DM

−

ef f

⊂

D

−

(Shv(SmCor))

isthefullsubategoryofomplexeswhose oho-mology sheavesare homotopyinvariant;it is triangulatedbylo.it. Wewill need the homotopy

t

-struture on

DM

ef f

−

: it is the restritionof the anon-ial

t

-struture on

D

−

(Shv(SmCor))

to

DM

ef f

−

. Below (when dealingwith

DM

−

ef f

)wewill denoteitbyjust by

t

. Wehave

Ht

=

HI

. Wereallthefollowingresultsof[25℄.

Proposition

1.3.1

.

1. There exists an exat funtor

RC

:

D

−

(Shv(SmCor))

→

DM

ef f

−

right adjoint to the embedding

DM

ef f

−

→

D

−

(Shv(SmCor))

. 2.

DM

ef f

−

(M

gm

(Y

)[−i], F

) =

H

i

(F)(Y

)

(the

i

-th Nisnevih hyperohomology of

F

omputedin

Y

)for any

Y

∈

SmV ar

.

3. Denote

RC

◦

L

by

M

gm

. Then the orresponding funtor

K

b

(SmCor)

→

DM

−

ef f

ouldbedesribedasaertain loalization of

K

b

(SmCor)

. Proof. See3of[25℄.

Remark 1.3.2. 1. In[25℄ (Denition 2.1.1)the triangulatedategory

DM

ef f

gm

(ofeetivegeometri motives)wasdened astheidempotentompletionofa ertainloalizationof

K

b

(SmCor)

. Thisdenitionisompatiblewitha dier-entialgradedenhanementfor

DM

ef f

gm

;f. 5.3below. YetinTheorem3.2.6of [25℄ itwasshownthat

DM

ef f

gm

is isomorphitothe idempotentompletionof (the ategorialimage)

M

gm

(C

b

(SmCor))

;this desriptionof

DM

ef f

(17)

2. Infat,

RC

ould be desribedin terms ofso-alledSuslin omplexes(see lo.it.). Wewillnotneedthisbelow. Instead,wewilljustnotethat

RC

sends

D

−

(Shv(SmCor))

t≤

0

to

DM

ef f

−

t≤

0

.

1.4

Some properties of Tate twists

Tate twisting in

DM

ef f

−

⊃

DM

gm

ef f

is given by tensoring by the objet

Z

(1)

(itisoftendenoted justby

−(1)

). Tatetwisthasseveraldesriptionsandnie properties. Wewill onlyneed afewofthem; ourmain soureis3.2of [25℄;a moredetailedexposition ouldbefoundin[20℄(see89).

Inordertoalulatethetensorprodutof

X, Y

∈

ObjDM

ef f

−

oneshouldtake anypreimages

X

′

, Y

′

of

X, Y

in

ObjD

−

(Shv(SmCor))

withrespetto

RC

(for example, oneould take

X

′

=

X

,

Y

′

=

Y

); nextone should resolve

X, Y

by diretsumsof

L(Z

i

)

for

Z

i

∈

SmV ar

;lastlyoneshouldtensortheseresolutions usingtheidentity

L(Z

)⊗L(T

) =

L(Z

×T

)

for

Z, T

∈

SmV ar

,andapply

RC

to theresult. Thistensor produtisompatiblewith thenaturaltensorprodut for

K

b

(SmCor)

.

We note that any objet

D

−

(Shv(SmCor))

t≤

0

hasaresolution onentrated in negativedegrees(the anonialresolutionof thebeginningof 3.2of[25℄). It followsthat

DM

ef f

−

t≤

0

⊗

DM

ef f

−

t≤

0

⊂

DM

ef f

−

t≤

0

(seeRemark 1.3.2(2);in fat,thereisanequalitysine

Z

∈

ObjHI

).

Next,wedenote

A

1

\ {0}

by

G

m

. Themorphisms

pt

→

G

m

→

pt

(the pointis mapped to

1

in

G

m

)indue asplitting

M

gm

(G

m

) =

Z

⊕

Z

(1)[1]

for aertain (Tate) motif

Z

(1)

; see Denition 3.1 of [20℄. For

X

∈

ObjDM

ef f

−

wedenote

X

⊗

Z

(1)

by

X(1)

.

One ould also present

Z

(1)

as

Cone(pt

→

G

m

)[−1]

; hene the Tate twist funtor

X

7→

X

(1)

is ompatible withthe funtor

− ⊗

(Cone(pt

→

G

m

)[−1])

on

C

b

(SmCor)

via

M

gm

. Wealsoobtainthat

DM

ef f

−

t≤

0

(1)

⊂

DM

ef f

−

t≤

1

. Nowwedeneertaintwistsforfuntors.

Definition

1.4.1

.

For an

G

∈

AddFun(DM

ef f

gm

, Ab)

,

n

≥

0

, we dene

G

−n

(X

) =

G(X

(n)[n])

.

Note that this denition is ompatible with those of 3.1of [26℄. Indeed,for

X

∈

SmV ar

we have

G

−

1

(M

gm

(X

)) =

G(M

gm

(X

×

G

m

))/G(M

gm

(X

)) =

Ker(G(M

gm

(X

×

G

m

))

→

G(M

gm

(X

)))

(with respet to natural morphisms

X

×

pt

→

X

×

G

m

→

X

×

pt

);

G

−n

forlarger

n

ouldbedened byiterating

−

1

.

Belowwewill extendthisdenition to(o)motivesof pro-shemes. For

F

∈

ObjDM

ef f

−

wewill denote by

F

∗

the funtor

X

7→

DM

ef f

−

(X, F

) :

DM

ef f

gm

→

Ab

.

Proposition

1.4.2

.

Let

X

∈

SmV ar

,

n

≥

0

,

i

∈

Z

. 1. For any

F

∈

ObjDM

ef f

−

we have:

F

∗−n

(M

gm

(X

)[−i])

is a retrat of

H

i

(F)(X

×

G

×n

(18)

2. There exists a

t

-exat funtor

T

n

:

DM

ef f

−

→

DM

ef f

−

suh that for any

F

∈

ObjDM

−

ef f

wehave

F

∗−n

∼

= (T

n

(F

))

∗

.

Proof. 1. Proposition1.3.1alongwithourdesriptionof

Z

(1)

yieldstheresult. 2. For

F

represented by a omplex of

F

i

∈

ObjShv(SmCor)

(

i

∈

Z

) we dene

T

n

(F

)

as the omplex of

T

n

(F

i

)

, where

T

n

:

P reShv(SmCor)

→

P reShv(SmCor)

is dened similarly to

−

−n

in Denition 1.4.1.

T

n

(F

i

)

are sheavessine

T

n

(F

i

)(X

), X

∈

SmV ar

, isafuntorialretratof

F

i

(X

×

G

n

m

)

. Inorder tohekthatweatuallyobtainawell-deneda

t

-exatfuntorthis way, it sues to notethat therestritionof

T

n

to

Shv(SmCor)

isan exat funtorbyProposition3.4.3of[9℄.

Now,itsuestohekthat

T

n

denedsatisestheassertionfor

n

= 1

. Inthis asethestatementfollowseasily from Proposition4.34of[26℄(notethat itis notimportantwhether weonsiderZariskiorNisnevihtopologybyTheorem 5.7ofibid.).

1.5

Pro-motives vs. comotives; the description of our strategy

Belowwewillembed

DM

ef f

gm

intoaertaintriangulatedategory

D

of omo-tives. Itsonstrution(andomputations init)is ratherompliated;in fat, the author is not sure whether the main properties of

D

(desribed below) speify itup to an isomorphism. So,before working with o-motiveswewill (following F. Deglise) desribea simplerategory of pro-motives. The latter is not needed for our main results (so the reader may skip this subsetion); yettheomparisonoftheategoriesmentionedwouldlarifythenatureofour methods.

Following3.1 of [9℄, we dene the ategory

D

naive

as the additive ategory of naivei.e. formal(ltered) pro-objetsof

DM

ef f

gm

. Thismeansthat forany

X

:

L

→

DM

ef f

gm

,

Y

:

J

→

DM

ef f

gm

wedene

D

naive

(lim

←−

l∈L

X

l

,

lim

←−

j∈J

Y

j

) = lim

←−

j∈J

(lim

−→

l∈L

DM

ef f

gm

(X

l

, Y

j

)).

(5)

Themaindisadvantageof

D

naive

isthatitisnottriangulated. Still,onehasthe obviousshiftforit;followingDeglise,oneandenepro-distinguishedtriangles as(ltered)inverselimitsofdistinguishedtrianglesin

DM

ef f

gm

. This allowsto onstrutaertainmotivioniveauexatoupleforamotifofasmoothvariety in4.2of[10℄(seealso5.3of[9℄). Thisonstrutionisparalleltothelassial onstrution of oniveau spetral sequenes (see 1 of [8℄). One starts with ertain'geometri'Postnikovtowersin

DM

ef f

gm

(Degliseallsthemtriangulated exat ouples). For

Z

∈

SmV ar

we onsider ltrations

∅

=

Z

d

+1

⊂

Z

d

⊂

Z

d−

1

⊂ · · · ⊂

Z

0

=

Z

;

Z

i

is everywhere of odimension

≥

i

in

Z

for all

i

. Then we have a system of distinguished triangles relating

M

gm

(Z

\

Z

i

)

and

M

gm

(Z

\

Z

i

→

Z

\

Z

i

+1

)

; this yields a Postnikov tower. Then one passes to theinverse limitofthese towersin

D

naive

(19)

are indued by the orresponding open embeddings). Lastly, the funtorial form oftheGysindistinguishedtriangleformotivesallowsDeglise toidentify

X

i

= lim

←−

(M

gm

(Z

\

Z

i

→

Z

\

Z

i

+1

))

withtheprodutofshiftedTatetwistsof pro-motivesof all points of

Z

of odimension

i

. Using the resultsof see 5.2 of[9℄(therelationofpro-motiveswithylemodulesofM.Rost,see[24℄)one analsoomputethemorphismsthatonnet

X

i

with

X

i

+1

. Next,foranyohomologial

H

:

DM

ef f

gm

→

A

,where

A

is anabelianategory satisfyingAB5,oneanextend

H

to

D

naive

viatheorrespondingdiretlimits. Applying

H

tothemotivioniveauexatoupleonegetsthelassialoniveau spetral sequene (that onverges to the

H

-ohomology of

Z

). This allows to extend the seminal results of 6 of [5℄ to a omprehensive desription of the oniveau spetral sequene in the ase when

H

is represented by

Y

∈

ObjDM

−

ef f

(intermsofthehomotopy

t

-trunationsof

Y

;see Theorem6.4of [11℄).

Now suppose that one wants to apply a similar proedure for an arbitrary

X

∈

ObjDM

ef f

gm

; say,

X

=

M

gm

(Z

1

→

f

Z

2

)

for

Z

1

, Z

2

∈

SmV ar

,

f

∈

SmCor(Z

1

, Z

2

)

. Onewould expetthat thedesiredexatouplefor

X

ould beonstrutedfromthosefor

Z

j

,

j

= 1,

2

. Thisisindeedtheasewhen

f

satis-esertainodimensionrestritions;f. 7.4of[6℄. Yetforageneral

f

itseems tobequitediulttorelatetheltrationsofdistint

Z

j

(bytheorresponding

Z

i

j

). Ontheother hand,theformalismofweightstruturesand weight spe-tralsequenes(developedin[6℄)allowsto'glue'ertainweightPostnikovtowers for objetsofatriangulated ategories equipped with aweightstruture; see Remark 4.1.2(3)below.

So, we onstruta ertain triangulatedategory

D

that is somewhat similar to

D

naive

. Certainly, wewant distinguished trianglesin

D

to be ompatible withinverselimitsthatomefrom'geometry'. Awell-knownreipeforthisis: oneshouldonsidersomeategory

D

′

where(ertain)onesofmorphismsare funtorialandpassto(inverse)limitsin

D

′

;

D

should bealoalization of

D

′

. Infat,

D

′

onstrutedin5.3belowouldbeendowedwithaertain(Quillen) modelstruturesuhthat

D

isitshomotopyategory. Wewill neverusethis fatbelow;yetwewillsometimesallinverselimitsomingfrom

D

′

homotopy limits(in

D).

Now, in Proposition 4.3.1 below we will prove that ohomologial funtors

H

:

DM

ef f

gm

→

A

ould be extended to

D

in away that is ompatible with homotopy limits(those oming from

D

′

). So onemay say that objetsof

D

have the same ohomology asthose of

D

naive

. On the other hand, we have to pay the prie for

D

beingtriangulated: (5) doesnot ompute morphisms between homotopy limitsin

D.

The'dierene' ould be desribed in terms of ertain higher projetive limits(of the orresponding morphism groupsin

DM

ef f

gm

).

Unfortunately, the author does not know how to ontrol the orresponding

lim

←−

2

(20)

is unountable (yet see 6.5, espeially the last paragraphof it). Inthe ase of aountable

k

only

lim

←−

1

is non-zero. In this ase the morphisms between homotopy limits in

D

are expressed by the formula (28) below. This allows toprovethat therearenomorphismsofpositivedegreesbetweenertainTate twistsofomotivesoffuntionelds(over

k

). Thisimmediatelyyieldsthatone anonstrutaertainweightstrutureonthetriangulatedsubategory

D

s

of

D

generatedbyprodutsofTatetwistsofomotivesoffuntionelds(infat, wealso idempotent omplete

D

s

). Now, in order to provethat

D

s

ontains

DM

ef f

gm

it sues to provethat the motif of any smooth variety

X

belongs to

D

s

. Tothisend itlearly suesto deompose

M

gm

(X

)

into aPostnikov towerwhosefatorsareprodutsofTatetwistsofomotivesoffuntionelds. So,weliftthemotivioniveauexatouple(onstrutedin[10℄)from

D

naive

to

D.

Sineonesin

D

′

areompatible withinverselimits,weanonstruta towerwhose termsarethehomotopylimitsoftheorrespondingtermsof the geometri towersmentioned. Infat,thisouldbedoneforanunountable

k

also; thediulty is to identify the analoguesof

X

i

in

D.

If

k

is ountable, thehomotopylimitsorrespondingtoourtowerareountablealso. Hene(by an easy well-known result) theisomorphism lassesof these homotopy limits ould be omputed in terms of the orresponding objetsand morphisms in

DM

ef f

gm

. Thismeans: itsuestoompute

X

i

in

D

naive

(aswasdonein[10℄); this yields theresultneeded. Note that weannot (ompletely)ompute the

D-morphisms

X

i

→

X

i

+1

;yet weknowhowtheyat onohomology.

Themost interestingappliationof theresultsdesribed isthefollowingone. Weprovethattherearenopositive

D-morphisms

between(ertain)Tatetwists of omotivesof smooth semi-loal shemes(or primitive shemes,see below); this generalizes the orresponding result for funtion elds. It follows that these twists belong to the heart of the weight struture on

D

s

mentioned. Thereforeomotivesof(onneted)primitiveshemesareretratsofomotives of their generi points. Hene the same is true for the ohomology of the omotives mentioned and also for the orresponding pro-motives. Also, the omotifofafuntion eldontainsasretratstwistedomotivesofitsresidue elds(forallgeometrivaluations);thisalsoimpliestheorrespondingresults forohomologyandpro-motives.

Remark 1.5.1. In fat, Deglise mostly onsiders pro-objets for Voevodsky's

DM

gm

and of

DM

ef f

−

; yet the distintions are not important sine the full embeddings

DM

ef f

gm

→

DM

gm

and

DM

ef f

gm

→

DM

ef f

−

obviouslyextendtofull embeddingoftheorrespondingategoriesofpro-objets.Still,theembeddings mentionedallowDeglisetoextendseveralnieresultsforVoevodsky'smotives to pro-motives.

(21)

2

Weight structures: reminder, truncations, weight spectral

se-quences, and duality with

t

-structures

In 2.1 we reall basi denitions of the theory of weight strutures (it was developedin [6℄; theoneptwasalsoindependentlyintroduedin [23℄). Note herethatweightstrutures(usuallydenotedby

w

)arenaturalounterpartsof

t

-strutures. Weightstruturesyieldweighttrunations;those(vastly)generalize stupid trunations in

K(B)

: in partiular, they are not anonial, yet any morphism of objets ould be extended (non-anonially) to a morphism of their weighttrunations. We reall several properties of weightstrutures in 2.2.

Wereallvirtual

t

-trunationsfora(ohomologial)funtor

H

:

C

→

A

(for

C

endowedwithaweightstruture)in2.3(thesetrunationsaredenedinterms of weight trunations). Virtual

t

-trunations were introdued in 2.5 of [6℄; theyyieldawaytopresent

H

(anonially)asanextensionofaohomologial funtorthatispositiveinaertainsensebya'negative'one(asif

H

belonged tosometriangulatedategoryoffuntors

C

→

A

endowedwitha

t

-struture). We study this notionfurther here, and provethat virtual

t

-trunations for a ohomologial

H

ouldbeharaterizedupto auniqueisomorphismby their properties (see Theorem 2.3.1(III4)). In order to givesome haraterization alsoforthe'dimensionshift'(onnetingthepositiveandthenegativevirtual

t

-trunationsof

H

),weintroduethenotionofanie(stronglyexat)omplex of funtors. We provethat omplexes of representable funtors omingfrom distinguishedtrianglesin

C

arenie,aswellasthoseomplexesthatouldbe obtainedfromniestronglyexatomplexesoffuntors

C

′

→

A

forsomesmall triangulated

C

′

⊂

C

(viatheextensionproeduregivenbyProposition1.2.1). In2.4weonsider weightspetralsequenes(introdued in 2.32.4of [6℄). Weprovethatthederivedexatouplefortheweightspetralsequene

T

(H

)

(for

H

:

C

→

A

)ouldbenaturallydesribedintermsofvirtual

t

-trunations of

H

. So,oneanexpress

T

(H

)

startingfrom

E

2

(aswellastheorresponding ltrationof

H

∗

)inthesetermsalso. Thisisanimportantresult,sinethebasi denitionof

T

(H

)

isgivenintermsofweightPostnikovtowersforobjetsof

C

, whereasthelatterarenotanonial. Inpartiular,thisresultyieldsanonial funtorialspetralsequenesinlassialsituations (onsideredbyDeligne;f. Remark 2.4.3of[6℄;notethat wedonotneedrationaloeientshere). In 2.5 we introdue the denition a (nie) duality

Φ :

C

op

×

D

→

A

, and of (left) orthogonal weight and

t

-strutures (with respet to

Φ

). The latter denition generalizes the notion of adjaent strutures introdued in 4.4 of [6℄ (this is the ase

C

=

D

,

A

=

Ab

,

Φ =

C(−,

)

). If

w

is orthogonal to

t

then the virtual

t

-trunations (orresponding to

w

) of funtors of the type

Φ(−, Y

), Y

∈

ObjD

,areexatlythefuntors'representedvia

Φ

'bytheatual

t

-trunations of

Y

(orresponding to

t

). We also prove that (nie) dualities ould be extended from

C

′

(22)

In2.6weprove: if

w

and

t

areorthogonal withrespettoanieduality, the weightspetralsequeneonvergingto

Φ(X, Y

)

(for

X

∈

ObjC, Y

∈

ObjD

)is naturallyisomorphi(startingfrom

E

2

)totheone omingfrom

t

-trunations of

Y

. Moreovereven whenthe dualityis notnie, all

E

pq

r

for

r

≥

2

and the ltrations orresponding to these spetral sequenesare still anonially iso-morphi. Here nienessof aduality(dened in 2.5)is asomewhat tehnial ondition (dened in terms of nie omplexes of funtors). Nieness gener-alizes to pairings (

C

×

D

→

A

) the axiom TR3 (of triangulated ategories: any ommutative square in

C

ould be ompleted to a morphism of distin-guishedtriangles;notethatthisaxiomouldbedesribedintermsofthe fun-tor

C(−,

−) :

C

×C

→

Ab

). Wealsodisusssomealternativesandpredeessors ofourmethodsandresults.

In 2.7 we ompare weight deompositions, virtual

t

-trunations, and weight spetralsequenesorrespondingtodistintweightstrutures(inpossibly dis-tinttriangulatedategories,onnetedbyanexatfuntor).

2.1

Weight structures: basic definitions

We reall the denition of a weight struture (see [6℄; in [23℄ D. Pauksztello introduedweightstruturesindependentlyandalledthem o-t-strutures).

Definition

2.1.1 (Denition of a weight struture)

.

A pair of sublasses

C

w≤

0

, C

w≥

0

⊂

ObjC

for a triangulated ategory

C

will be said to dene a weightstruture

w

for

C

iftheysatisfythefollowingonditions:

(i)

C

w≥

0

, C

w≤

0

are additive and Karoubi-losed (i.e. ontain all retrats of theirobjetsthat belongto

ObjC

).

(ii)"Semi-invariane" with respet to translations.

C

w≥

. (iii)Orthogonality.

C

w≥

0

⊥

C

w≤

0

[1]

.

(iv)Weightdeomposition.

Forany

X

∈

ObjC

thereexistsadistinguishedtriangle

B[−1]

→

X

→

A

→

f

B

(6)

suhthat

A

∈

C

w≤

0

, B

∈

C

w≥

0

.

Asimpleexampleofaategorywithaweightstrutureis

K(B)

forany addi-tive

B

: positiveobjetsareomplexes that arehomotopyequivalent tothose onentratedinpositivedegrees;negativeobjetsareomplexesthatare homo-topyequivalenttothoseonentratedinnegativedegrees. Hereoneouldalso onsider thesubategoriesof omplexesthat are bounded from above,below, orfrombothsides.

The triangle (6) will be alled a weight deomposition of

X

. A weight de-omposition is (almost)neverunique; stillwewillsometimesdenote any pair

Directory UMM :Journals:Journal_of_mathematics:OTHER:

Motivically Functorial Coniveau Spectral Sequences;

Contents

3 Categories of comotives (main properties)

5 The construction of

ObjC, Y

SmV ar

0

ObjC, Y

0

Notation.

hObjDi

naive

-structures, Postnikov towers, idempotent completions, and

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Definition

Proposition

AddFun(C

AddFun(C

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Some definitions of Voevodsky: reminder

SmCor

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Some properties of Tate twists

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AddFun(DM

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