Motivically Functorial Coniveau Spectral Sequences;

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) triangulatedate-

gories;thispieeofhomologialalgebraouldbeinterestingforitself.

We onstrut a ertain Gersten weight struture for a triangulated

ategoryofomotivesthat ontains

DM _gm ^{ef f}

^{aswellas(o)motivesof}

funtioneldsover

k

^{. Itturnsoutthatthe}orrespondingweightspe- tralsequenesgeneralizethelassialoniveauones(toohomologyof

arbitrarymotives). Whenaohomologialfuntorisrepresentedbya

Y ∈ Obj DM ₋ ^{ef f}

^{, the}orrespondingoniveauspetralsequenesan beexpressedin termsofthe(homotopy)

t

-trunations of

Y

^{; thisex-}

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.

Keywords and Phrases: Motives, oniveau, weight struture, t-

struture,triangulatedategory,semi-loalsheme,ohomology.

Contents

1 Some preliminaries on triangulated categories and motives 43

1.1

t

-strutures, Postnikovtowers,idempotentompletions,and an

embeddingtheorem ofMithell . . . 43

1.2 Extendingohomologialfuntorsfrom atriangulatedsubate- gory . . . 46

1.3 SomedenitionsofVoevodsky: reminder. . . 47

1.4 SomepropertiesofTatetwists . . . 49

1.5 Pro-motivesvs. omotives;thedesriptionofourstrategy . . . 50

2 Weight structures: reminder, truncations, weight spectral sequences, and duality with t-structures 53

2.1 Weightstrutures: basidenitions . . . 54

2.2 Basipropertiesofweightstrutures . . . 56

2.3 Virtual

t

-trunationsof(ohomologial)funtors . . . 62

2.4 Weight spetral sequenes and ltrations; relation with virtual

t

-trunations . . . 68

2.5 Dualities of triangulated ategories; orthogonal weight and

t

^- strutures . . . 71

2.6 Comparisonofweightspetralsequeneswiththoseomingfrom (orthogonal)

t

-trunations . . . 74

2.7 'Changeofweightstrutures';omparingweightspetralsequenes 76

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 _gm ^{ef f}

^{. . . . . . . . . 89}

4.2 Diretsummand resultsforomotives . . . 91

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

4.4 Coniveau spetralsequenesforohomologyof(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

4.9 Birationalmotives;onstrutingtheGerstenweightstrutureby gluing;otherpossibleweightstrutures. . . 101

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

;motiveswithompatsupportofpro- 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 byanythe-

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 Ÿ4.4}

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).

A relatedobjetiveisto larifythenatureoftheinjetivityresultmentioned;

it turnedourthat (in theaseofaountable

k

^{)theohomologyof asmooth}

onneted semi-loal (more generally, primitive) sheme is atually a diret

summand oftheohomology ofitsgeneri point. Moreover,the(twisted) o-

homologyofaresidueeldofafuntioneld

K/k

^{(foranygeometrivaluation}

of

K

^{)is adiretsummand of theohomology of}

K

^{. We atuallyprovemore}

in Ÿ4.3.

Ourmainhomologialalgebratoolisthetheoryofweightstrutures(intrian-

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 Ÿ4.4of [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 omparingsimilarspetralse-

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}

introduedinŸ2.5of[6℄andisstudiedfurther intheurrentpaper.

Now,weexplainwhywereallyneedaertainnewategoryofomotives(on-

taining Voevodsky's

DM _gm ^{ef f}

^{), and so the theory of adjaent strutures (i.e.}

orthogonalstruturesinthease

C = D

^,

Φ = C(−, −)

^{)isnotsuientforour}

purposes. Itwasalreadyprovedin[6℄thatweightstruturesprovideapower-

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 orepresento-

variant exatfuntors from the ategoryof homotopy invariant sheaveswith

transfersto

Ab

^).

So,weneedaategorythatwouldontainertainhomotopylimitsofobjetsof

DM _gm ^{ef f}

^{. Wesueedin onstrutinga}triangulatedategory

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 _gm ^{ef f} ⊂ 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}

^{and (so) is loselyonneted with oniveau}

spetralsequenesand Gerstenresolutions for sheaves. Note still: we need

k

to be ountable only in order to onstrut the Gersten weightstruture. So

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, we onsider aertain triangulated ategory

D ⊃ DM _gm ^{ef f}

^{that (roughly!)}

'onsists of' (ovariant) homologial funtors

DM _gm ^{ef f} → 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 _gm ^{ef f}

^yields

afamilyof(weak)oompatogeneratorsfor

D

. Thisiswhyweallobjetsof

D

omotives.Yetnotethattheembedding

DM _gm ^{ef f} → 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℄; theorre-}

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℄;thoughsomeoftheresultsofthisunpub-

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 _gm ^{ef f}

^{(thisategoryisnot}triangulated, thoughit is pro-triangulated in aertain sense). This approah allowsto ob-

tain(in auniversalway)lassialoniveauspetralsequenesforohomology

ofmotivesofsmoothvarieties;Deglisealsoprovestheirrelationwiththehomo-

topy

t

-trunationsforohomologyrepresentedbyanobjetof

DM ₋ ^{ef f}

^{. Yetfor}

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 Ÿ1.5below. Notealsothat ourmethods

are muhmore onvenientfor studying funtoriality(of oniveauspetralse-

quenes)thanthemethods appliedbyM.Rostin therelatedontextofyle

modules(see[24℄andŸ4of[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,mayavoidmostofsetion2andreadonlyŸŸ2.12.2(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 ofŸ3 'by hand'

without reading Ÿ2at 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

attentiontoŸŸ1.11.2andŸ2;yetnotethat therest ofthepaperdesribesin

detailanimportant(andquitenon-trivial)exampleofaweightstruturewhih

is orthogonal to a

t

^{-struture with respet to a nie duality (of} triangulated ategories). Moreover,muh ofsetionŸ4ouldalsobeextended 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.

WestartŸ1withthereolletionof

t

-strutures,idempotentompletions,and Postnikovtowersfortriangulatedategories. Wedesribeamethodforextend-

ing ohomologialfuntors from afull triangulated subategoryto thewhole

C

^{(afterH. Krause). Nextwereall someresultsand denitions forVoevod-}

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.

Ÿ2is dediatedtoweightstrutures. Firstweremindthebasisofthis theory

(developed in Ÿ[6℄). Next we reall that aohomologial funtor

H

^{from an}

(arbitrarytriangulatedategory)

C

^{endowedwithaweightstruture}

w

^ould

be'trunated'asifitbelongedtosometriangulatedategoryoffuntors(from

C

^{)thatisendowedwitha}

t

^{-struture;weallthe}orrespondingpieesof

H

^its

virtual

t

-trunations. Wereallthenotionofaweightspetralsequene(intro-

dues in ibid.). Weprovethat thederivedexatouple foraweightspetral

sequene ouldbedesribed in termsof virtual

t

-trunations. Nextweintro- 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

^)offun-

torsofthetype

Φ(−, Y ), Y ∈ ObjD

^{,areexatlythefuntors}'representedvia

Φ

^'bytheatual

t

-trunationsof

Y

(orrespondingto

t

^{). Heneif}

w

^and

t

^are

orthogonalwithrespettoanieduality,theweightspetralsequeneonverg-

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. Lastlyweompareweightdeom-

positions, virtual

t

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

InŸ3wedesribethemainpropertiesof

D ⊃ DM _gm ^{ef f}

^{. Theexathoieof}

D

is notimportantformostofthispaper;sowejustlist themainpropertiesof

D

(anditsertainenhanement

D ^′

)inŸ3.1. Weonstrut

D

usingtheformalism ofdierentialgradedmodulesinŸ5later. 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 Ÿ3.6weprovethat the Gysin distinguishedtriangle for motives

of smooth varieties (in

DM _gm ^{ef f}

^{) ould benaturally extended to omotivesof}

pro-shemes. This allowsto onstrutertain Postnikovtowersforomotives

ofpro-shemes;thesetowersareloselyrelatedwithlassialoniveauspetral

sequenesforohomology.

Ÿ4 is entral in this paper. We introdue a ertain Gersten weight struture

for a ertain triangulated ategory

D _s

(

DM _gm ^{ef f} ⊂ D _s ⊂ D

). We provethat PostnikovtowersonstrutedinŸ3.6areatuallyweightPostnikovtowerswith

respetto

w

^{. Wededueour}(interesting)resultsondiretsummandsofomo- 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

^).

We also dene oniveau spetral sequenes for ohomology of motives over

unountable base elds as the limits of the orresponding oniveau spetral

sequenesoverountableperfetsubeldsofdenition. Thisdenitionisom-

patiblewiththelassialone;soweestablishmotivifuntorialityofoniveau

spetralsequenesin thisasealso.

Wealsoprovethat theChowweight struturefor

DM _gm ^{ef f}

^{(introduedinŸ6of}

[6℄)ouldbeextendedtoaweightstruture

w Chow

^on

D

. Theorresponding Chow-weightspetralsequenesare isomorphito thelassial(i.e. Deligne's)

ones whenthelatteraredened(thiswasprovedin [6℄and[7℄). Weompare

oniveauspetralsequeneswithChow-weightones: wealwayshaveaompar-

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).

Ÿ5 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).

Ÿ6 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)motiveswitho-

eientsin an arbitraryring. Next wenote (after B. Kahn)that reasonable

motivesofpro-shemeswith ompatsupport doexist in

DM ₋ ^{ef f}

^{; thisobser-}

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

t

-strutures). In fat, we onstrut an example for suh a situation in this paper(ertainly, muh simplerexamplesexist): wedene theGerstenweight

struture

w

^for

D _s

and aChowweightstruture

w Chow

^for

D

. Moreover,we showin Ÿ4.9 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,thatgeneral- izeDeligne'sweightspetralsequenesforétaleandmixedHodgeohomology

(see [6℄ and [7℄). Also,the weightomplex funtoronstruted in [7℄ and [6℄

isquitedistintfromtheoneonsideredinŸ6.1below(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

^{(wewillignore}set-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

^{is the lass}

{Y ∈ ObjC : Y ⊥ X ∀X ∈ D}

^{. This onventionis}

oppositetotheoneofŸ9.1of[21℄.

Inthispaperallomplexeswillbeohomologiali.e. thedegreeofalldieren-

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;itwillusuallybe}ohomologial.

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(seesetionŸIII.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 i

^{thesmalleststritlyfull}triangulatedsubategoryontainingall

C i

^;for

D ⊂ C

^{wewill write}

hDi

^insteadof

hObjDi

^.

We will saythat

C i

^generate

C

^if

C

^equals

hC i 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

^.

k

^{willbeourperfetbaseeld. Belowwewillusuallydemand}

k

^{tobeountable.}

Note: thisyieldsthatforanyvarietythesetofitslosed(oropen)subshemes

isountable.

Wealsolistentraldenitions andmainnotationofthispaper.

Firstwelistthemain(general)homologialalgebradenitions.

t

-strutures,

t

^-

trunations,andPostnikovtowersintriangulatedategoriesaredenedinŸ1.1;

weightstrutures,weightdeompositions,weighttrunations,weightPostnikov

towers,andweightomplexesareonsideredin Ÿ2.1;virtual

t

-trunationsand nieexatomplexesoffuntorsaredenedinŸ2.3;weightspetralsequenes

arestudiedinŸ2.4;(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 _gm ^{ef f} ⊂ DM − ^{ef f}

^,

HI

^{andthehomotopy}

t

^-struturefor

DM _gm ^{ef f}

^{aredenedinŸ1.3;Tatetwistsare}

onsideredinŸ1.4;

D ^naive

isdenedin Ÿ1.5;omotives(

D

and

D ^′

)aredened inŸ3.1;inŸ3.2wedisusspro-shemesandtheiromotives;inŸ3.3wereallthe

denitionofaprimitivesheme;inŸ4.1wedenetheGerstenweightstruture

w

^onaertaintriangulated

D _s

; weonsider

w Chow

^{in Ÿ4.7;}

D _bir

and

w ^′ _bir

^are

dened in Ÿ4.9; several dierential graded onstrutions (inludingextension

and restritionof salarsfor omotives) areonsidered in Ÿ5; wedene

D ^gen

and

W C : D _s → K ^b ( D ^gen )

^inŸ6.1.

1 Some preliminaries on triangulated categories and motives

Ÿ1.1wereallthenotionofa

t

^{-struture(andintroduesomenotationforit),}

reallthenotionofanidempotentompletion ofanadditiveategory;wealso

reallthatanysmallabelianategoryouldbefaithfullyembeddedinto

Ab

^(a

well-knownresultbyMithell).

InŸ1.2 wedesribe(followingH.Krause)anaturalmethod forextendingo-

homologialfuntorsfromafulltriangulated

C ^′ ⊂ C

^to

C

^.

InŸ1.3wereallsomedenitionsandresultsofVoevodsky.

In Ÿ1.4 we reall thenotion of aTate twist; we study the properties of Tate

twistsformotivesandhomotopyinvariantsheaves.

InŸ1.5wedene 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

(C ^t≥0 , C ^t≤0 )

^{satisfy the}

followingonditions:

(i)

C ^t≥0 , C ^t≤0

^{are strit i.e. ontain allobjetsof}

C

^{isomorphito their ele-}

ments.

(ii)

C ^t≥0 ⊂ C ^t≥0 [1]

^,

C ^t≤0 [1] ⊂ C ^t≤0

^.

(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

^by

taking

D ^{t ′ ≤0} = C ^t≥0

^and

D ^{t ′ ≥0} = C ^t≤0

^{;seepart(iii)ofExamples1.3.2in[2℄.}

2. Reall (f. Lemma IV.4.5 in [13℄) that (1) denes additive funtors

C → C ^t≤0 : X → A

^and

C → C ^t≥0 : X → B

^{. Wewill denote}

A, B

^by

X ^t≤0

^and

X ^t≥1

^,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. part10ofŸIV.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 adeom-}

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. Itsidem-}

potentompletion is (by denition) theategory

B ^′

^{whose objetsare}

(X, p)

for

X ∈ ObjB

^and

p ∈ B(X, X) : p ² = p

^;wedene

A ^′ ((X, p), (X ^′ , p ^′ )) = {f ∈ B(X, X ^′ ) : p ^′ f = f p = f }.

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. ŸIV2of[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 anun-

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.

2. In the proof it sues to have afaithful embedding

A → R − mod

^{; this}

weakerassertionwasalsoprovedbyMithell.

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 beskele-}

tallysmalli.e. thereshould exista(proper) subset

D ⊂ ObjC ^′

^{suh thatany}

objetof

C ^′

^{isisomorphitosomeelementof}

D

^{. Forsimpliity,wewillsome-}

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. itislosedwithrespettoallsmalloprod-}

uts,andltereddiretlimitsofexatsequenesin

A

^{are exat.}

Let

H ^′ ∈ AddFun(C ^′op , A)

^{beanadditivefuntor(it willusually beohomo-}

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. The}orrespondene

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. FollowingŸ1.2of[17℄(anddualizingit),weonsidertheabelianat-

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 ^′

^{. Itisonstrutedinthefollowingway: iffor}

X ∈ ObjC

^wehavean

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 l 0 j 0 )(H (X j 0 ))

^forsome

j 0 ∈ 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 an assume 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

U Z

^{for all integrallosed}

U ⊂ X × Y

^{that are}

niteover

X

^{anddominantoveraonnetedomponentof}

X

^{;theomposition}

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 Ÿ4.1of [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))

^{isthefullsubategoryofomplexeswhoseoho-}

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. SeeŸ3of[25℄.

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

DM _gm ^{ef f}

(ofeetivegeometri motives)wasdened astheidempotentompletionofa

ertainloalizationof

K ^b (SmCor)

^{. Thisdenitionisompatiblewithadier-}

entialgradedenhanementfor

DM _gm ^{ef f}

^{;f. Ÿ5.3below. YetinTheorem3.2.6of}

[25℄ itwasshownthat

DM _gm ^{ef f}

^{is isomorphitothe idempotentompletionof}

(the ategorialimage)

M gm (C ^b (SmCor))

^{;this desriptionof}

DM _gm ^{ef f}

^willbe

suientforustillŸ5.