Motives and the Standard Conjectures

Neantro Saavedra Rivano

Translation last revised October 2, 2019

Abstract

This is a translation of Chapitre VI, �4 and Appendice, of

Saavedra Rivano, Neantro. Categories Tannakiennes. Lecture Notesin Mathematics, Vol. 265. Springer-Verlag, Berlin-New York, 1972,

where Saavedra states the standard conjectures and applies them to the tan-nakian category of motives. It is available at www.jmilne.org/math/ underDocuments. Corrections should be sent to the email address on that page. Theoriginal (somewhat chaotic) numbering has been retained. All footnotes havebeen added by the translator.

Contents4 Motives . . . . . . . . . . . . . . . . . . . . . . . 1

4.1 Motives without conjectures . . . . . . . . . . . . 24.2 The tannakian category of motives . . . . . . . . . . 104.3 Levels (niveaux) . . . . . . . . . . . . . . . . . 154.4 The canonical polarization . . . . . . . . . . . . . 174.5 Motives in characteristic zero . . . . . . . . . . . . 184.6 Motives in nonzero characteristic . . . . . . . . . . . 19

Appendix: Conjectures in Algebraic Geometry . . . . . . . . . . . 22A.0 Equivalence relations for algebraic cycles . . . . . . . . 22A.1 Cohomology theories . . . . . . . . . . . . . . . 26A.2 The standard conjectures . . . . . . . . . . . . . . 32A.3 Consequences of the standard conjectures . . . . . . . 35A.4 The Tate conjecture. . . . . . . . . . . . . . . . 36A.5 The Hodge conjecture . . . . . . . . . . . . . . . 38

4 MotivesIn numbers 4.1 to 4.4, we fix a field k. We make constant use of the notation anddefinitions of the Appendix. The constructions made in 4.1 can also be found inManin [2], Kleiman [2], and are included here for the convenience of the reader. From4.2 to the end, the material is conjectural.

1

4 MOTIVES 2

4.1 Motives without conjectures1

4.1.0 Preliminaries.

We fix an admissible equivalence relation � for the algebraic cycles (A 0.2.1), whichin practice will be alg or num (A 0.2.2). The graded Q-algebra C�.X/ will be denotedsimply as C.X/.

In this number we shall construct a˝-category ACU2 PM.k/ (or more simply M.k//which, up to the commutativity constraint, is the category of motives. We proceed insteps starting from V.k/ and adjoining each time, either objects or morphisms. Thefirst step in this process is the category of correspondences of degree zero CV0.k/defined in A 0.3.3.

4.1.1 The category CV0.k/ of correspondences

We make this category, defined in A 0.3.3.3, a˝-category ACU in the following way.If X , Y are objects of CV.k/, i.e., smooth projective k-schemes, we put

X˝Y DX �Y I

if f 2 C.X �X 0/, g 2 C.Y ˝ Y 0/ are algebraic correspondences, then f ˝ g 2C.X � Y �X 0 � Y 0/ is the algebraic cycle deduced from f � g by the canonicalisomorphism

X �X 0�Y �Y 0 'X �Y �X 0�Y 0:

The law ˝ thus defined is equipped with an obvious ACU constraint (I 2.4.1), and itis Q-linear. If, with the preceding notation, f (resp. g/ has of degree i (resp. j ), thenf ˝g has degree iCj . This implies that CV0.k/ is a˝-subcategory of CV.k/.

Let 'WY !X be a morphism of smooth projective k-schemes, and let � .'/ bethe class in C.X �Y / of the graph3 � t' �X �Y of '. It is a correspondence from X

to Y of degree zero (the order X;Y is important here), and so defines a (contravariant)functor

V.k/opp! CV0.k/; (4.1.1.1)

which is even a strict ˝-functor ACU (I 4.1.1). This allows us to interpret CV0.k/as the category with the same objects as V.k/ in which we have adjoined to themorphisms of k-schemes the correspondences having the appropriate codimension.

The ˝-category ACU which we obtain in this way, namely CV0.k/, has theadvantage over V.k/ of being linear, and even Q-linear. However, it is not abelian,and there are even endomorphisms p with the property p2 D p (i.e., projectors) that

1The heading was omitted in the original.2I.e., a category with an ˝ functor satisfying compatible associativity, commutativity, and unit

constraints; in other words, a symmetric monoidal category. In addition to Saavedra, the necessarybackground in tannakian categories can also be found in Deligne and Milne 1982.

3The original writes �' for � t' , but for me (and the Wikipedia) the graph �' of 'WX ! Y is theimage of x 7! .x;'.x//WX !X �Y .

4 MOTIVES 3

do not have a kernel. We remedy the nonexistence of the kernels of projectors byintroducing them in a formal way (see 4.1.2), and we shall see in the following numberhow the standard conjectures imply that this suffices to make the category abelian.

4.1.2 The category MC.k/ of effective motives

4.1.2.1. Recall some definitions and constructions (SGA 4, IV 7.5). A category C iskaroubian if for every object X in C and every projector p of X , the kernel of the pair

idX ;pWX �X

is representable. Equivalently, the cokernel of the preceding pair is representable, andin this case Ker.idX ;p/ and Coker.idX ;p/ are canonically isomorphic. For everycategory C, there exists a karoubian category kar.C/ and a functor 'WC! kar.C/universal among functors from C to karoubian categories, i.e., such that if C0 is akaroubian category, the functor

Hom.kar.C/;C0/! Hom.C;C0/; f 7! f ı'

is an equivalence of categories. Here is a construction of kar.C/ and '. The objectsof kar.C/ are the pairs .X;p/ with X an object of C and p a projector of X , and if.X;p/, .Y;q/ are two such objects

Hom..X;p/; .Y;q/g D ff WX ! Y j qfp D f g: (4.1.2.1)

The functor 'WC! kar.C/ is defined by

'.X/D .X; idX /: (4.1.2.2)

It is fully faithful. For this choice of kar.C/, we have for categories C, C0 a canonicalisomorphism

kar.C�C0/' kar.C/�kar.C0/:

We deduce that if C is a ˝-category ACU, then kar.C/ is equipped in a naturalway with a structure of the same type.

Let C be a linear category, and let p be a projector of an object X of C. ThenidX �p is also a projector, and if C is karoubian, then canonically

X D Ker.p/˚Ker.idX �p/:

If C is not karoubian, then its karoubian envelope kar.C/ is again linear and we have

.X; idX /D .X; idX �p/˚ .X;p/:

In this expression, .X; idX �p/ corresponds to the kernel of p, .X;p/ to the image ofp. More generally, if X is an object of a linear category C, then every decomposition

idX DnXiD1

pi

4 MOTIVES 4

of the identity into a sum of projectors, orthogonal in pairs, defines a decompositionof X D .X; idX /

X D

nMiD1

.X;pi /; (4.1.2.3)

i.e., a decomposition into a sum of the images of the projectors. When C is a linear˝-category ACU, we make explicit the direct sum and tensor product of objects inkar.C/, �

.X;p/˚ .Y;q/D .X˚Y;p˚q/

.X;p/˝ .Y;q/D .X˝Y;p˝q/.(4.1.2.4)

4.1.2.2. The category of effective motives MC.k/ is defined by

MC.k/D kar.CV0.k//: (4.1.2.5)

An effective motive is therefore a pair .X;p/ consisting of a smooth projectivek-scheme and of an idempotent element p in the ring of correspondences of degree 0from X into X . We have a strict Q-linear˝-functor ACU

CV0.k/!MC.k/

which, composed with (4.1.1.1), gives a˝-functor ACU

kWV.k/opp!MC.k/:

If X 2 obV.k/, then h.X/ is called the motive of X , or also the motivic cohomologyof X . The equality

h.X �Y /D h.X/˝h.Y / (4.1.2.6)

can be interpreted as the Kunneth formula in motivic cohomology.

4.1.2.3. Let X be a connected smooth projective k-scheme of dimension n, and x aclosed point of X of degree d.x/. The element

eX D d.x/�1x 2 C n.X/

is independent of the choice of the closed point x, and defines an isomorphismC n.X/'Q. One checks immediately that the endomorphisms of h.X/,

pX D pr�1.eX /; qX D pr�2.eX / 2 Cn.X �X/

are projectors. More generally, if Y and Z are also connected of dimension n, and ifwe put

pY;X D pr�X .eX / 2 Cn.X �Y /

qY;X D pr�Y .eY / 2 Cn.X �Y /;

then we have4 �pZ;Y ıpY;X D pZ;XqZ;Y ıqY;X D qZ;X :

(4.1.2.7)

4Misprint fixed.

4 MOTIVES 5

This says that the pY;X (resp. qY;X / form a transitive system of isomorphisms betweenthe effective motives .X;pX / (resp. .X;qX // forX connected of dimension n: indeed,we have id.X;p/ D p for an effective motive .X;p/.

In fact, in defining pY;X as a morphism in CV0.k/, we did not use the conditiondimY D dimX , condition which is however indispensable in the definition of qY;X .We see therefore that the effective motives .X;pX / are canonically isomorphic; wemay in particular take X D Spec.k/, whence

.X;pX /' 11 (4.1.2.8)

where 11 denotes as usual the unit object h.Spec.k//.

4.1.2.4. It remains to calculate .X;qX / for X of dimension n. We can take X D.P1/n. The effective motive

LD .P1;qP1/ (4.1.2.9)

is called the Lefschetz motive. We deduce immediately that

.X;qX /' L˝n; (4.1.2.10)

and so there is a canonical decomposition of the motive of a connected smoothprojective k-scheme of dimension n¤ 0,

h.X/D 11˚hC.X/˚L˝n (4.1.2.11)

where hC.X/D .X; idX �pX �qX / (this notation is legitimate if X is geometricallyconnected). If X D P1, then idX D pX CqX , and so

h.P1/D 11˚L: (4.1.2.12)

Knowing C.Pn/ and C.Pn�X/ as C.X/-algebras (A 0.2.3) allows us to prove thiseasily.

PROPOSITION 4.1.2.5. We have a decomposition

h.Pn/D 11˚L˚�� �˚L˝nI

on the other hand, the motive L is 1-regular (I 0.1.3), i.e., the map

Hom.M;N /! Hom.M ˝L;N ˝L/; f 7! f ˝ idL

is bijective. In particular, End.L˝n/'Q.

4.1.3 The category of motives

4.1.3.1. We first present a general construction in ˝-categories, that of makinginvertible a 1-regular (I 0.1.3) object. Let C be a˝-category ACU and L a 1-regularobject of C. We want to construct a˝-category ACU CŒL�1� and a˝-functor ACU

'WC! CŒL�1�

4 MOTIVES 6

such that '.L/ is invertible, and which is universal among tensor categories havingthis property. Here is a construction: the objects of CŒL�1� are the pairs .M;m/consisting of an object M of C and an integer m 2 Z. If .M;m/, .N;n/ are two suchobjects, then

Hom..M;m/;.N;n//D lim�!

N�n;m

Hom.M ˝LN�m;N ˝LN�n/; (4.1.3.1)

the transition arrows, and the composition of morphisms. being obvious. The˝ lawis defined by

.M;m/˝ .N;n/D .M ˝N;mCn/; (4.1.3.2)

and the˝-functor ' by'.M/D .M;0/:

We have a canonical isomorphism '.L/' .11;�1/; and we see that the object T D.11;1/ is an inverse for '.L/. We adopt the following notation: if M is an object of C,m 2 Z, we denote by M.m/ the object .M;m/; when we identify C to a subcategoryof CŒL�1� by ', we have a canonical isomorphism

M.m/'M ˝T˝m:

4.1.3.2. The category of motives PM.k/ is defined by

PM.k/DMC.k/ŒL�1�: (4.1.3.3)

where L is the Lefschetz motive. A motive M can therefore be written in the form

M DM 0˝T˝n

where M 0 is effective and T , the inverse of the Lefschetz motive, is called the Tatemotive.5

We have a fully faithful Q-linear˝-functor ACU

MC.k/! PM.k/;

and so a˝-functor ACU, again called motivic cohomology,

hWV.k/opp! PM.k/:

REMARK 4.1.3.3. The terminology “category of motives” used for PM.k/ is provi-sional. In the next number, we modify the commutativity constraint of PM.k/ in anontrivial fashion. The ˝-category ACU obtained, denoted M.k/, will be the truecategory of motives.

4.1.3.4. From the˝-category ACU of motives, equipped with the Tate motive, wecan reconstruct the ˝-category CV.k/ of correspondences. For this we have to showthat, if X;Y are smooth projective k-schemes, then we can reconstruct the group

5The original continues with “where M 00 DM 0˝L.”, which makes no sense to me.

4 MOTIVES 7

Homi .X;Y / of correspondences of degree i , for i 2 Z. Suppose, to simplify, that Xis connected of dimension n; then, for i;j 2 Z, we can define a canonical bijection,6

Hom.h.X/.i/;h.Y /.j // C nCj�i .X �Y /

Homj�i .X;Y /:

'

(4.1.3.4)

To do this, we first remark that we may suppose that either i D 0 and j � 0, or thati � 0 and j D 0, according as j � i is negative or positive. We treat the first case, thesecond being similar: We have to define a bijection, for j � 0,

C nCj .X �Y /' Hom.h.X/;h.Y /˝L˝�j /:

This is induced by the commutative diagram (see 4.1.2.5 and A 0.2.3)

C n.X �Y �P�j / Hom.h.X/;h.Y ˝P�j //

�jMrD0

C n�r.X �Y /

�jMrD0

Hom.h.X/;h.Y /˝L˝r/:

' '

'

PROPOSITION 4.1.3.5. The category of motives is a rigid˝-category (I 5.1.1). If Xis a connected smooth projective k-scheme of dimension n, then there is a canonicalisomorphism

h.X/_ ' h.X/.n/D h.X/˝T˝nI

the evaluation map h.X/_˝h.X/! 11 (I 3.1.1) can be deduced from

h.X �X/h.X/�! h.X/! L˝n;

where h.X/! L˝n is the canonical projection (4.1.2.11).

4.1.3.6. PROOF: We define a functor

PM.k/opp! PM.k/; M 7!M_

and canonical morphismsevM WM_˝M ! 11

by extending in several steps the functor

V0.k/opp! PM.k/; X 7! h.X/.dimX/

6More generally,Hom..X;e; i/; .Y;f;j //D f CnCj�i .X �Y /e:

This allows us to define the category of motives to have as objects triples .X;e; i/ and morphisms givenby this equality.

4 MOTIVES 8

equipped with the explicit evaluation morphism in 4.1.3.5. For this, we use theuniversal properties of the functors V0.k/! V.k/ (for direct sums), CV0.k/.k/!MC.k/, and MC.k/! PM.k/.

When M;N are motives, we put

Hom.M;N /DM_˝N

equipped with the obvious evaluation map Hom.M;N /˝M !N . In order to checkthat this is a Hom object, it suffices to prove that if X;Y;Z are smooth projectiveschemes, where X (resp.Y ) is connected of dimension n (resp.m), then

Hom.h.X/;h.Y /.m/˝h.Z//��! Hom.h.X �Y /;h.Z//:

Taking account of 4.1.3.4, this bijection is none other than

C nCm.X �Y �Z/D C nCm.X �Y �Z/:

The rest of the proposition is now trivial.

4.1.4 Motives and cohomology theories

4.1.4.1. In this number we suppose that the equivalence relation � for cycles, fixedin 4.1.0, is algebraic equivalence. We let MD PM.k/.

4.1.4.2. Let .C;w;TC/ be a triple as in A 1.1.1, and .H; ;Tr/ a cohomology theorywith values in this triple. We attach to H D .H; ;Tr/ a pair .!H ; �H / consisting of arigid˝-functor ACU

!H WM! C

and an isomorphism�H W!H .T /

��! TC

in the following way: the functor

H WV.k/opp! CC! C

extends to CV.k/ by using the maps induced by and Tr (see A 1.3.2)

C i .X �Y /! Hom.H.X/;H.Y /.i �n//

where nD dim.X/. Since C is by hypothesis karoubian, the tensor functor CV0.k/!C extends to MC.k/, and so we have a commutative triangle

V.k/opp

C

MC.k/:

h

!

4 MOTIVES 9

On the other hand, the axioms of the cohomology theory give a canonical isomorphism

H.P1/D 11˚T˝�1C ;

whence, by (4.1.2.12), an isomorphism

� 0W!.L/��! T˝�1C I

in particular, !.L/ is invertible, and so, by 4.1.3.1, ! defines a˝-functor ACU

!H WM! C

and � 0 defines an isomorphism

�H W!H .T /' TC:

PROPOSITION 4.1.4.3. The correspondence .H; ;Tr/ 7! .!H ; �H / defines an equiv-alence of the category TC.k; .C;w;TC// of cohomology theories over k with valuesin .C;w;TC/ with the category of pairs .!;�/ consisting of a rigid ˝-functor ACU!WM! C and of an isomorphism �W!.T /! TC such that ! transforms effectivemotives into objects of C with positive degree.

4.1.4.4. Here is how .!;�/ defines a cohomology theory .H; ;Tr/. Put H D ! ıh,and if X is connected of dimension n, put h.X/.n/! 11 equal to the canonicalmorphism deduced from (4.1.2.11); TrX is the image by ! of this morphism. Finally,in order to define , use the isomorphisms (4.1.3.4)

Hom.h.X/.i/;h.Y /.j //' C nCj�i .X �Y /

and �Wh.T /' TC.

4.1.5 Effective motives of degree 0;1.

4.1.5.1. An effective motive M is of degree 0 if it is a direct factor of the motiveof a k-scheme of dimension zero. The category of effective motives of degree zero,denoted MC0.k/, is the karoubian subcategory of MC.k/ generated by the h.X/ withX of dimension zero; it is a tensor subcategory of MC.k/.

Choose an algebraic closure xk of k, and let � D Gal.xk=k/. We have a˝-functorACU

MC0.k/! Repcont0.� /

from the effective motives of degree 0 to the category of Q-vector spaces of finitedimension equipped with a continuous action of the profinite group � (i.e., factoringthrough a finite quotient), whose value on h.X/ is

QX.xk/

equipped with the obvious continuous action of � . It can be checked without difficultythat this is a tensor equivalence, identifying an effective motive of degree 0 to a Q-vector space of finite dimension with a continuous action of � D Gal.xk=k/.

4 MOTIVES 10

4.1.5.2. The category MC1.k/ of effective motives of degree 1 is the additivekaroubian subcategory of MC.k/ generated by the motives hC.X/ (4.1.2.4) for X acurve (i.e., geometrically connected of dimension 1). It is proved in Manin [2], �10,that there is a canonical equivalence of Q-linear categories

MC1.k/! Isab.k/;

where Isab.k/ is the semisimple abelian category of abelian varieties over k up toisogeny. The equivalence attaches to hC.X/ the Jacobian of X , JX .

4.2 The tannakian category of motives

4.2.0

We consider here the category of motives constructed using numerical equivalence ofcycles, i.e., (4.1.0)

PM.k/D PMnum.k/:

We assume moreover the existence of a cohomology theory with values in a fieldof characteristic zero satisfying the standard conjectures (A 2.4.1); for example, wecould assume the validity of the standard conjectures for an `-adic cohomology (A1.4.1).

If TD .C;w;TC/ is a Tate triple (V 3.1.1), by a cohomology theory with values inT we mean a cohomology theory with values in T satisfying the standard conjectures(A 1.1.5, A 2.4.4).

We keep these conventions in 4.3–4.6.

4.2.1

We shall construct a natural˝-gradation on the category PM.k/. Recall that this meansthat each motive M is equipped with a canonical decomposition

M DMn2Z

M n

compatible in an obvious sense with˝.The construction requires several steps.

4.2.1.1. Let X be a connected smooth projective k-scheme of dimension n. After A3.2.1, we have a decomposition in the ring C n.X �X/ of endomorphisms in CV0.k/

idX DXi2Z

� iX ; (4.2.1.1)

where � iX D 0 for i … Œ0;2n�. On the other hand, if Y has dimension m, we have7

�kX�Y DX

iCjDk

� iX ˝�jY : (4.2.1.2)

7Misprint fixed.

4 MOTIVES 11

We know also (A 3.2) that the � iX are independent of the cohomology theorychosen. The constructions that follow depend only on the � iX .

4.2.1.2. The � iX are orthogonal projectors, as one sees immediately, and the formula(4.2.1.1) defines a decomposition of the motive h.X/ in MC.k/ (4.1.2.1)

h.X/DMi2Z

hi .X/ (4.2.1.3)

where hi .X/D .X;� iX / with the notation of 4.1.2. On the other hand, the decompo-sition (4.2.1.1) is functorial on the category CV0.k/, which means the following: iff 2 C n.X �Y /, i.e., f WX ! Y is in CV0.k/, then

� iY ıf D f ı�iX ;

which is only another way of saying that H.f /WH.X/! H.Y / is homogeneousof degree 0. In fact, the commutant of � iX (0 � i � 2n/ in the ring C.X �X/ ofalgebraic correspondences is exactly C n.X �X/.

From the above and (4.2.1.2), we deduce a˝-gradation of MC.k/, characterizedby being the only one for which

h.X/i D hi .X/:

We note that the motive L is homogeneous of degree 2.

4.2.1.3. The˝-gradation of MC.k/ extends directly to a˝-gradation of PM.k/. Wehave

.M;n/i D .M iC2n;n/; i 2 Z: (4.2.1.4)

4.2.1.4. We use this gradation to modify the commutativity constraint of PM.k/.Equipped with the new˝-structure, PM.k/ will be denoted M.k/, and called the truecategory of motives, which we will study in what follows. Here is how we proceedto modify the constraint: if M and M 0 are motives, and P WM ˝N 'N ˝M is thecommutativity isomorphism, then

P DMP p;q

where P p;qWMp˝M q ' N q˝Mp; we let denote the new commutativity iso-morphism defined by �

DL p;q

p;q D .�1/pq P p;q:(4.2.1.5)

We remark that, for this modification to make sense, it has been essential tosuppose the validity of the conjectures of type Lefschetz. This modification is, inits turn, essential for the study of the category of motives from the point of view oftannakian categories.

4 MOTIVES 12

4.2.2 Theorem

M.k/ is a semisimple tannakian category over Q.

4.2.2.1. PROOF. Let .H; ;Tr/ be a cohomology theory over k with values in a fieldK of characteristic zero. It defines a rigid˝-functor

!H W PM.k/! Gradf˙.K/:

In modifying the commutativity constraint of PM.k/, we have done exactly whatwas necessary to interpret !H as a rigid˝-functor

!H WM.k/! Gradf.K/

where in the commutativity constraint of Gradf.K/ there is no sign. Moreover, by4.2.1, giving the preceding functor is equivalent to giving a rigid ˝-functor, againdenoted

!H WM.k/!Modf.K/:

This˝-functor is Q-linear, and also faithful, as follows from A 2.4.1 (c). It followsthat the Krull-Remak-Schmidt theorem holds in M.k/: every motiveM can be writtenas a finite sum of indecomposable motives. In order to prove Theorem 4.2.2, it sufficesto prove that M.k/ is a semisimple abelian category, or again, by the preceding, thatan indecomposable motive M is simple: let N ,!M be a simple subobject of M ,N ¤ 0; it is clear that M is homogeneous, say, of degree p. We may then suppose,after tensoring with a power of T , that M and N are effective, which implies that M(resp. N/ is a direct factor of an effective motive hp.X/ (resp. hp.Y /), say,

M ˚M 0 D hp.X/

N ˚N 0 D hp.Y /:

Let uWhp.Y /! hp.X/ be the morphism given by the matrix

uD

�i 0

0 0

�and let

u0 D

�a b

c d

�be its transpose u0Whp.X/! hp.Y / defined in A 3.3 (we assume that X and Y arepolarized). Then (loc. cit.)

Tr.u0u/D Tr.ai/ > 0;

and soai ¤ 0:

Because N is simple, it follows that ai is an automorphism, and now the morphism

j D .a ı i/�1 ıaWM !N

is a retraction for i , and so M DN .

REMARK 4.2.2.2. The ˝-category ACU PM.k/ is not tannakian. Indeed. if it were,we could deduce immediately that Gradf˙.k/ is tannakian, which is obviously false.

4 MOTIVES 13

4.2.3

We saw while proving 4.2.2 that a cohomology theoryH with values inK determinesa fibre functor (III 3.2.1.2)

!H WM.k/!Modf.K/

equipped with an isomorphism

�H W!H .T /��!K:

This remains true when we replace K with a Q-scheme S (A 1.1.4).More generally, if we take a cohomology theory H with values in a tannakian

category C (A 1.1.5.1), then we get a pair .!H ; �H / consisting of a morphism

!H WM.k/! C

and an isomorphism�H W!H .T /

��! 11:

More generally still, if we take a cohomology theoryH with values in a Tate tripleTD .C;w;T / (A 1.1.5.1), then we get a pair .!H ; �H / consisting of a morphism oftannakian categories compatible with the˝-gradations

!H WM.k/! C

and an isomorphism�H W!H .T /

��! T:

The next three statements follow from the above and 4.1.4.2.

PROPOSITION 4.2.3.1. Let S be a Q-scheme. The correspondence H 7! .!H ; �H /

defines an equivalence of categories from TC.k;S/ to the category of pairs .!;�/consisting of a fibre functor !WM.k/! LocLib.S/ and an isomorphism �W!.T /

��!

OS .

PROPOSITION 4.2.3.2. Let C be a tannakian category over a field of characteristiczero. The correspondence H 7! .!H ; �H / defines an equivalence of categories fromTC.k;C/ to the category of pairs .!;�/ consisting of a morphism !WM.k/! C andof an isomorphism �W!.T /

��! 11.

PROPOSITION 4.2.3.3. Let T D .C;w;TC/ be a Tate triple over a field of charac-teristic zero. The correspondence H 7! .!H ; �H / defines an equivalence of cat-egories of TC.k;T/ with the category of pairs .!;�/ consisting of a morphism!WM.k/! C compatible with the central˝-gradations on M.k/ and C and an isomor-phism �W!.T /

��! TC.

4 MOTIVES 14

4.2.4 Scholium

The category M.k/ of motives is a semisimple tannakian category over Q, correspond-ing therefore to a tannakian Q-gerbe G (III 2.2.2) whose band L is represented, locallyfor the fpqc topology, by a pro-reductive group (III 3.3.3 (b)). It is equipped with acanonical˝-gradation, which corresponds to a morphism of central gerbes

TORS.Gm/! G;

or again (IV 1.3) to a morphism of abelian Q-groups

Gm! Cent.L/:

The Tate object T , which is invertible, and therefore corresponds to a morphism ofgerbes

G! TORS.Gm/;

is of degree �2, i.e., the composite

TORS.Gm/! G! TORS.Gm/

corresponds to the morphism

� 7! ��2WGm!Gm.

Finally, if G0 is the kernel of G! TORS.Gm/, then we have a canonical isomor-phism of gerbes,

G0 ' TC.k/;

where TC.k/ is the stack over Sch=Q of cohomology theories (A 1.1.4).

4.2.5

Assume that the standard conjectures are valid for the `-adic topology, where ` is afixed prime number different from the characteristic of k. We deduce a morphism oftannakian categories (A 4.3)

M.k/! GradTate.k/; (4.2.5.1)

or, again, a morphism of tannakian categories over Q`,

M.k/Q`! GradTate.k/ (4.2.5.2)

(see III 3.2.4 for the process of extending scalars in tannakian categories.) Whenwe let G` denote the Q`-group of automorphisms of the fibre functor with values inQ` defined by `-adic cohomology, and we keep the notation A 4.3, then we get acommutative diagram of Q`-groups

G�Gm G`

G �Q`

pr (4.2.5.3)

4 MOTIVES 15

where G�Gm! G` is induced by (4.2.5.2), G! �Q`is the canonical projection

(loc. cit.), and G`! �Q`is the epimorphism defined by the inclusion in M.k/ of the

effective motives of degree zero (see 4.1.5.1).If we admit the Tate conjecture (A 4), (4.2.5.2) is fully faithful, i.e.,G�Gm!G`

is an epimorphism, and it follows from the diagram (4.2.5.3) that �Q`can be identified

with the group of connected components of G`. In particular,

PROPOSITION 4.2.5.1. If k is algebraically closed, the band of M.k/ is connected.

4.3 Levels (niveaux)

4.3.0

Let M denote the tannakian category M.k/ defined in 4.2; for n 2 Z, Mn denotes thefull subcategory of motives of degree n, and MCn the full subcategory of effectivemotives of degree n. If n < 0, then MCn D 0.

4.3.1

Let M be a motive of degree n; for i 2 Z, we say that M is of level (niveau) � n�2iif the motive M.i/ is effective. If k 2 Z, we say also that M is of level � k (resp.< k) if M is of level � k or � k�1 (resp. � k�1 or � k�2) according to the parityof n�k; therefore, if M is of level � k and k � k0, M is of level � k0.

The following statements are immediate:

(a) level of M < 0 ” M D 0;(b) level of M � k ” level of M.i/� k� i (i 2 Z);(c) if M is of degree n, M has level � n ” M is effective;(d) if M is of degree n and of level � n�2i , then every subobject of M is of level� n�2i .

The level defines an increasing filtration of M n by abelian (semisimple) subcate-gories.

4.3.2

Let M be a motive of degree n, and let i 2 Z. We let MnC2i denote the largestsubmotive of M of level � nC2i . We define in this way an increasing filtration ofthe motive M , called the filtration by level

� � � �MnC2iC2 �MnC2i � �� � I (4.3.2.1)

if M is effective, then Mn DM . We say that an effective motive M of degree n ispurely of level n if Mn�2 D 0, i.e., if the preceding filtration has only one step. Welet MŒn� denote the full subcategory of MCn of pure motives of level n. We have anobvious functor M

i2Z

�L˝i ˝MŒn�2i�

�!M n (4.3.2.2)

defined by.Mn�2i /i2Z 7!

MiL˝i ˝Mn�2i :

4 MOTIVES 16

4.3.3 Proposition

The functor (4.3.2.2) is an equivalence of categories; in particular, it induces anequivalence of categoriesM

i�0

�L˝i ˝MŒn�2i�

�!MCn: (4.3.3.1)

4.3.3.1. PROOF: It suffices to show that if M is a motive of degree n, then thefiltration by level (4.3.2.1) splits canonically. It is clear that it splits, because M is asemisimple category. It is canonical because in the inclusion

MnC2i �MnC2iC2

there exists a unique direct complement ofMnC2i : indeed, ifN DMnC2iC2=MnC2i ,then Hom.N;MnC2i /D 0, as follows immediately from the definition of the filtrationby level.

4.3.4

Let M be a motive of degree n. If n is even, the filtration by the levels has the form

MnC2i � �� � �M2 �M0 � 0I

if n is odd, it has the form

MnC2i � �� � �M3 �M1 � 0:

In the first case, nD 2k, and M0.k/ is an effective motive of degree 0, correspondingtherefore (4.1.5) to a finite-dimensional vector space equipped with an action ofGal.xk=k/. Its formation is functorial in M .

In the second case, n D 2kC 1, and M1.k/ is an effective motive of degree1, corresponding therefore (4.1.5) to an abelian variety up to isogeny over k. Itsformation is also functorial in M . If X is a smooth projective k-scheme and M Dh2kC1.X/, we can regard M1.k/ as the “algebraic part of the intermediate jacobianof order k of X”.

4.3.5

Let X be a connected smooth projective k-scheme of dimension n, equipped with asmooth hyperplane section as in A 2.1.2. We again denote by `, the image by �X of` 2 C 1.X/,

`D�X .`/ 2 CnC1.X �X/I

it is therefore (4.1.3.4) a morphism

`Wh.X/! h.X/.i/I

the mapsC i .X/! C iC1.X/

4 MOTIVES 17

that it induces are those denoted by ` in A 2.1.2. Let i � n; the conjectures ofLefschetz type (A 2.2) imply that

`n�i Whi .X/! h2n�i .X/.n� i/

is an isomorphism of motives. In particular, they imply: if j 2 N, then the motivehj .X/ has level � inf.j;2n�j /� n.

4.4 The canonical polarization

4.4.1

The category of motives M.k/, equipped with its central˝-gradation and Tate motiveT , is a Tate triple over Q in the sense of V 3.1.1. In this subsection, we construct acanonical polarization on this Tate triple.

4.4.2

Let X be a connected smooth projective k-scheme of dimension n equipped with a“polarization” ` 2 C 1.X/. The polarization of X defines morphisms of motives (A2.2.2 and A 2.2.4)

�Whi .X/! h2n�i .X/.n� i/:

We let '` denote the bilinear form

'`Whi .X/˝hi .X/! T˝�i

equal to the composite

hi .X/˝hi .X/id˝��! hi .X/˝h2n�i .X/.n� i/! h2n.X/.n� i/

Tr�! T˝�i :

It is .�1/i -symmetric, and it follows from A 3.3 that the forms '` are Weil formscompatible in pairs (V 2.3.1). Moreover, if X , Y are connected smooth projectivek-schemes equipped with polarizations `X , `Y , the form 'X˝'Y on hi .X/˝hj .Y /with values in T˝�i�j is deduced (up to a positive factor, see Kleiman [1], 3.11) fromthe form 'X�Y on hiCj .X �Y / when one polarizes X �Y by `X ˝1Y C1X ˝`Y .

We deduce from this discussion that the set of Weil forms of the preceding typedefine a polarization on the Tate triple of motives (V 3.2.1).

4.4.3

Assume the standard conjectures and the Tate conjecture for `-adic cohomology (A4), where ` is a prime number different from the characteristic. Suppose first that kis algebraically closed; then the band of the tannakian category M 0.k/ of motivesof degree zero (not necessarily effective) is connected by 2.2.5. On the other hand,the polarization of the Tate triple of motives defined above induces on M 0.k/ asymmetric polarization in the sense of V 2.4.1. It follows, after V 2.4.5.1.4 thatM 0.k/ is “ind-neutral” over the real number field: on each tannakian subcategory

4 MOTIVES 18

of M 0.k/ ˝-generated by a finite number (or more generally an infinite countablenumber) of objects, there exists a fibre functor with values in R. After loc. cit. thereeven exists a fibre functor, unique up to a nonunique isomorphism, with the propertythat it transforms the forms of the polarization into positive definite forms.

The conclusions of ind-neutrality over R are valid also if k is not algebraicallyclosed; we have, in fact, a morphism of tannakian categories

M.k/!M.xk/:

We summarize the discussion.

PROPOSITION 4.4.3.1. Let N be an algebraic tannakian subcategory (III 3.3.1) ofthe category M 2:.k/ of motives of even degree. There exists on N a fibre functorwith values in R.

REMARK 4.4.3.2. We can interpret this as saying that there exists an even piece of acohomology theory with values in R. It well-known, after Serre, that in characteristic¤ 0, there does not exist a cohomology theory with values in R.8

4.5 Motives in characteristic zero

4.5.0

Let M denote the category of motives over a field k of characteristic zero. Thecohomology theories envisaged are supposed to satisfy the standard conjectures (A2).

4.5.1

The Hodge and de Rham cohomologies define fibre functors on the category ofmotives over k (A 1.4.2, 1.4.3)

hHdg;hdRWM!Modf.k/:

The Hodge filtration on the de Rham cohomology (loc. cit.) defines an exact˝-filtration F on hdR (IV 2.1.1.1) and by A 1.4.3 we have a canonical isomorphismof fibre functors

grF hdR ' hHdg: (4.5.1.1)

To give a splitting of the filtration F is the same as giving an isomorphismhdR ' hHdg respecting the Hodge filtrations and inducing the identity map on theassociated graded objects (IV 2.2.1). When we restrict hdR to an algebraic subcategoryN of M, we know that there exists such a splitting of the Hodge filtration (IV 2.4).

8This is little misleading. First, Deuring and Hasse proved in the 1930s that the endomorphismalgebra of a supersingular elliptic curve is a quaternion algebra over Q ramified at p and1 and sounable to act on a two-dimensional vector space over Qp or R. It follows that there can be no Qp- orR-valued fibre functor on M.k/. It was Serre who explained this to Grothendieck, probably in the 1950s.Second, while it is true that there is no cohomology theory with values in R, there do exist cohomologytheories with values in a triple over R in the sense of A 1.1.1 (Deligne and Milne 1982, 5.20).

4 MOTIVES 19

4.5.1.1. When we suppose that k D C, Hodge theory provides us with a canonicalsplitting

hdR ' hHdg

of the Hodge filtration, which is of a transcendental nature.

4.5.2

Suppose that k D C. Then the Betti-Hodge cohomology (A 1.4.4) defines a fibrefunctor

hBHWM! Hodge.Q/

compatible with the˝-gradations, and an isomorphism

hBH.T /' T

where T denotes respectively the Tate motive and the Tate Hodge structure. Inother words, we have a morphism from the Tate triple of motives over C (4.4.1)into that defined by Hodge.Q/ (2.1.3.1). By definition of the polarizations on M(4.4) and on Hodge.Q/ (2.1.3.1), and after Weil [1], chap. IV, th. 7, we see that themorphism of Tate triples defined by the Betti-Hodge cohomology is compatible withthe polarizations.

4.5.2.1. It follows easily from (4.1.3.4) and A 5.1 that the simple form of the Hodgeconjecture is equivalent to the full faithfulness of the functor hBH.

Similarly, one checks immediately that the generalized form of the Hodge conjec-ture is equivalent to the following assertion on hBH:

A motive M over C is effective if and only if the Hodge structure hBH.M/ iseffective, i.e., of positive bidegree.

Therefore, if we accept the Hodge conjecture, the functor hBH identifies M.C/with a full subcategory of Hodge.Q/; the˝-gradation of the category of motives isinduced by that on Hodge.Q/; the Tate Hodge structure T is found in M.C/; thepolarization of the category of motives is induced by that of Hodge.Q/; finally, thestructure of levels (4.3) of M.C/ is induced by that of Hodge.Q/ (defined by theHodge level, see A 5.2.1). Of course, we know no characterization of the image ofM.C/ in Hodge.Q/, i.e., of the Hodge structures that are algebraic. In degree 1, theyare all of them, after the theory of abelian varieties (Mumford [2], chap I). AfterGriffiths, if we fix the Hodge numbers ha;b defining a weight � 2, with the exceptionof some cases, the algebraic Hodge structures define a thin (maigre) set in the modulispace of polarizable Hodge structures.9

4.6 Motives in nonzero characteristic

4.6.0

Let M denote the category of motives over a field k of nonzero characteristic p. Thecohomology theories envisaged are supposed to satisfy the standard conjectures (A2).

9See also Deligne 1972 (Invent. Math.), �7.

4 MOTIVES 20

4.6.1

Let ` be a prime number¤ p. The `-adic cohomology (A 1.4.1 and A 4.4). defines afibre functor h` with values in the neutral tannakian category Tate.k/ over Q`,

h`WMQ`! Tate.k/;

or, again, a morphism of tannakian categories

MQ`! GradTate.k/: (4.6.1.1)

If follows from (4,1.3.4) and A 4.4 that the Tate conjecture is equivalent to the fullfaithfulness of the functor (4.6.1.1); this also amounts to saying that the morphism ofgerbes (resp. of bands) induced by (4.6.1.1) is an epimorphism.

4.6.2

If `D p, the “p-adic cohomology” is given by the crystalline cohomology, whichdefines a morphism of tannakian categories

hcrysWMQp! Fcryso.k/:

4.6.3

Put k D Fp. We shall see here how the standard conjectures and the Tate conjecture,joined to recent results of Honda and Tate, allow us to give a description of the tan-nakian category MDM.Fp/ in terms of the theory of groups (analogous descriptionsare valid for k D Fpr , r � 1).

4.6.3.1. The band of M is an abelian Q-group G; we have an extension

0 Gı G Gal.xFp=Fp/Q 0

yZQ

'

where Gı, the identity component of G, can be identified with the band of M.xFp/.

The property for a band to be representable by an abelian group is local for thetopology envisaged (Giraud [1], IV 1.2.3), which in this case is the fpqc topology onSchQ. It suffices to see that the band of MQ`

(`¤ p) is abelian, and this follows fromthe Tate conjecture under the form 4.6.1 and from the fact that the band of Tate.k/is an abelian Q`-group, namely, the pro-algebraic `-adic envelope of yZ. The lastassertion of 4.6.3.1 follows immediately from the discussion in 4.2.5.

4.6.3.2. The Q-group Gı is a group of multiplicative type and its module of char-acters M is the multiplicative group of Weil p-numbers equipped with the obviousaction of Gal.xQ=Q/.

4 MOTIVES 21

Recall that a Weil p-number is an algebraic number � with the property that thereexists an integer i 2 Z such that � and its conjugates over Q have absolute value pi=2;the integer i is the weight of the Weil p-number. After Honda [1] (see also Tate [3]),the set of isogeny classes of simple abelian varieties, i.e., the isomorphism classes ofsimple effective motives of degree 1 (4.1.5) is in one-to-one correspondence with theset of conjugates of Weil p-numbers of weight 1.10

It is clear that Gı is of multiplicative type, because it is abelian and pro-reductive;the determination of M as the set of Weil p-numbers is easy starting from the result ofHonda cited above and the Weil conjectures, which imply that, for a simple object Xof M, the Frobenius endomorphism p 2 End.X/ is a Weil p-number whose weight isequal to the degree of X .

4.6.3.3. To render complete the description of M, it is necessary to make explicitthe element � 2H 2.Q;G/ which corresponds to M in the classification of tannakiancategories with band G. The morphism of tannakian categories

MDM.Fp/!M.xFp/

bound by Gı! G, shows that � is the image by H 2.Q;Gı/!H 2.Q;G/ of theelement

�0 2H2.Q;Gı/

corresponding to M.xFp/:

4.6.3.4. After Tate [2], to determine �0 it suffices to determine its image at all thefinite places of Q, i.e., it suffices to determine for all prime numbers `

.�0/` 2H2.Q;

�Gı�`/.

The existence of `-adic cohomology for `¤ p proves that, in this case, .�0/` D 0.It remains to determine .�0/p; we have crystalline cohomology

hcrysWM.xFp/! Fcryso.xFp/;

and the tannakian category Fcryso.xFp/ is well understood (3.3.2): its band is thediagonalizable group D.Q/ and under the isomorphism of local class field theoryBr.Qp/ ' Q=Z, the class � 0 of Fcryso in H 2.Qp;D.Q// D Hom.Q;Q=Z/ is thecanonical projection Q!Q=Z. On the other hand, the band of hcrys is the morphismof Qp-groups

D.Q/!GıQp

whose transpose on the character modules

M !Q

is the p-adic valuation on the Weil p-numbers, normalized by vp.p/D 1.The class .�0/p is simply the image of � 0 under the induced morphism in coho-

mologyH 2.Qp;D.Q//!H 2.Qp;GıQp

/:

10This is not quite correct: the isogeny classes of simple abelian varieties correspond to Weil p-integers, i.e., � is required to be an algebraic integer. Similarly, the definition of a Weil p-number �should include the requirement that pN� is an algebraic integer for some N .

4 MOTIVES 22

REMARK 4.6.3.5. We have obtained at the same time a description of the tannakiancategory M.xFp/:

Appendix: Conjectures in Algebraic GeometryWe fix a field k. Let V.k/ (resp. V0.k/) denote the category of smooth projective(resp. and connected) k-schemes. These are˝-categories ACU, the˝ law being theproduct.

A.0 Equivalence relations for algebraic cycles

0.1 Algebraic cycles

0.1.1. When X is a smooth projective k-scheme, Z.X/ denotes the group of cycleson X , graded by codimension; it is the free abelian group generated by the points(closed or not) of X , or again, by the integral subschemes11 of X . If X D

`Xi is

the decomposition of X into its connected components, then there is a canonicalisomorphism of graded groups

Z.X/'Mi

Z.Xi /: (0.1.1.1)

When X is connected of dimension n, we define the degree map

h iWZ.X/! Z (0.1.1.2)

to be that which takes the value zero on the points of codimension < n, and which ona point x 2X of codimension n (i.e., a closed point) satisfies

hxi D deg.k.x/=k/D Œk.x/=k�: (0.1.1.3)

We define the degree map in general by using (0.1.1.1).

0.1.2. Let X;Y be smooth projective k-schemes. By taking products of integralsubschemes of X and Y , we get a morphism of abelian groups

a˝b 7! a�bWZ.X/˝Z.Y /!Z.X �Y /: (0.1.2.1)

When 'WX ! Y is a morphism of k-schemes, we define a linear map

'�WZ.X/!Z.Y /

in the following way: for an element x 2 X of the basis of Z.X/, we put d.x/Ddimfxg, d.y/D dimfyg, where y D '.x/, and we set

'�.x/D

�0 if d.x/ > d.y/Œk.x/=k.y/�y if d.x/D d.y/:

(0.1.2.2)

For example, if 'WX ! Spec.k/ is the structure morphism, then

'� D h i: (0.1.2.3)11Should be the closed integral subschemes.

4 MOTIVES 23

0.1.3. Let a;b 2 Z.X/ be cycles in a smooth projective k-scheme X . The inter-section product of a;b (when defined, see, for example, Serre [3], chap. V) willbe denoted a � b. If a (resp. b/ is homogeneous of degree i (resp. j /, then a � b ishomogeneous of degree iCj .

Let 'WX ! Y be a morphism of smooth projective k-schemes. We then have alinear “map” which is not everywhere defined

'�WZ.Y /!Z.X/:

For example, if a 2Z.X/, b 2Z.X/, and �WX !X �X is the diagonal map, then

a �b D��.a�b/:

The operations '�, '� are related in the following fashion:12

'�.a �'�.b//D '�.a/ �b; a 2Z.X/; b 2Z.Y /:

This is the projection formula, valid whenever the two terms are defined.Note finally the following formula, valid whenever the two terms are defined

'�.b/D prX�.pr�Y .b/ ��'/

where b 2 Z.Y /, prX , prY are the projections of X �Y onto X;Y and where �' 2Z.X �Y / is the class of the graph of '.

0.1.4. If X'�! X 0

'0

�!X 00 are morphisms in V.k/, then

'0� ı'� D .'0ı'/�

and, when this makes sense,

'� ı'0� D .'0 ı'/�:

The map '� does not respect the degrees nor the intersection product when it is defined;if X (resp. X 0/ is connected of dimension n (resp. n0/, then13 '� is homogeneous ofdegree 0 and commutes with products.

We see that Z.X/ is a covariant functor of X when regarded as an abelian group,and contravariant in X when regarded as a graded ring.

0.2 Equivalence relations

0.2.1. An admissible equivalence relation � for algebraic cycles is a law that toeach X 2 obV.k/ attaches an equivalence relation �X on Z.X/ which is bifunctorialin X (see 0.1.4), compatible with the graded group structure of Z.X/, and whichsatisfies the following condition: if a;b 2Z.X/, then there exist a0 � a, b0 � b suchthat a0 �b0 is defined.

12Misprint fixed.13Misprint fixed.

4 MOTIVES 24

The quotient Z�.X/ of Z.X/ by the relation �X is a commutative graded ringwith 1 which depends contravariantly on X and covariantly if one retains only thestructure of an abelian group. It is called the ring of cycle classes for the equivalencerelation �.

We putC�.X/DQ˝Z�.X/:

0.2.2. Here are some examples of admissible equivalence relations: rational equiva-lence, algebraic equivalence, numerical equivalence, denoted respectively by rat, alg,num. We have

rat> alg> num (0.2.2.1)

where> is the relation of fineness. In fact, one can show that rat is the finest admissibleequivalence relation and num the least fine (Kleiman [2], prop. 3.5).

0.2.3. Let � be an admissible equivalence relation, and let x 2 P1.k/. Then theimage t of x in Z1�.P1/ is independent of x, and we have an isomorphism of gradedrings

Z�.P1/' ZŒt �=.t2/: (0.2.3.1)

More generally, if x is a hyperplane in Pn (rational over k), the image t of x inZ1�.Pn/ is independent of x, and we have an isomorphism of graded rings

Z�.Pn/' ZŒt �=.tnC1/: (0.2.3.2)

The validity of these assertions follow from (0.2.2.1) and their well-known validityfor num, rat (see Grothendieck [2]). By the same method, one proves that if X 2obV.k/, the projection X � Pn ! X makes Z�.X � Pn/ into a Z�.X/-algebraisomorphic to Z�.X/˝Z�.Pn/, i.e.,

Z�.X �Pn/'Z�.X/Œt �=.tnC1/: (0.2.3.3)

0.3 Algebraic correspondences

We fix an admissible equivalence relation �, and we write C.X/ for C�.X/.

0.3.1. For X:Y 2 ob.V.k//, the elements of C.X �Y / are called algebraic cor-respondences of X into Y . If X is connected of dimension n and i 2 Z, then thecorrespondences of degree i ofX into Y are the elements of C nCi .X�Y /. We obtainin this way a gradation of type Z on the group C.X �Y / of algebraic correspondencesfor X connected. In the general case, if X D

`Xi is the decomposition of X into its

connected components, then we define a gradation by degree on C.X �Y / throughthe canonical isomorphism

C.X �Y /'Mi

C.Xi �Y /:

4 MOTIVES 25

0.3.2. Let X;Y;Z 2 ob.V.k//, so14

X �Y X �Y �Z Y �Z

X �Z:

pX�Y pY�Z

pX�Z

We define a Q-bilinear map .a;b/ 7! b ıa, called composition,

C.X �Y /�C.Y �Z/! C.X �Z/

by the formulab ıaD .pX�Z/�.p

�X�Y .a/ �p

�Y�Z.b//:

This composition is associative; moreover, if a (resp. b/ is of degree i (resp. j ),then b ıa is of degree iCj . When we take the triple of objects to be Spec.k/, X , Y ,then we get a Q-bilinear map

C.X/�C.X �Y /! C.Y /;

and hence a Q-linear map

C.X �Y /! HomQ.C.X/;C.Y //:

One checks that if 'WX ! Y (resp. WY ! X/ is a morphism, and if � .'/ (resp.� . /) is the image of its graph in C.X �Y /, then the morphism C.X/! C.Y /

which it determines is none other than '� (resp. �/.15

0.3.3. The category of correspondences, denoted CV.k/, is the category having thesame objects as V.k/ and the algebraic correspondences as morphisms. It is a Q-linearadditive category, which is endowed with the structure of a gradation of type Z: Homsare abelian groups with a gradation of type Z and composition is compatible with thisgradation. If X;Y are objects of CV.k/, then

Hom.X;Y /DMi2Z

Homi .X;Y /;

where Homi .X;Y / is the group of correspondences from X to Y of degree i .The subcategory of CV.k/ with the same objects as CV.k/ and the morphisms of

degree 0 as its morphisms will be denoted by CV0.k/; it is also an additive Q-linearcategory and there is an additive Q-linear inclusion functor

CV0.k/! CV.k/:

Note that in both categories a direct sum is a disjoint union of k-schemes.14Diagram added.15Let �' denote the image of x 7! .x;'.x//WX 7!X �Y . Then � .'/ def

D �' corresponds to '�, and

� . /defD � t corresponds to �.

4 MOTIVES 26

A.1 Cohomology theories

1.1 Definitions

1.1.1. We consider triples .C;w;TC/ consisting of a Q-linear karoubian (4.1.2) rigid˝-category ACU (I 5.1.1) C, a tensor gradation w of idC of type Z, and an invertibleobject TC (or more simply T ) of degree �2 for w. If V is an object of C we use thenotation

� .V /D Hom.11;V / .I 1.3.1)

V.n/D V ˝T˝n:

We let CC denote the full tensor subcategory of C of objects with positive degree.16

A cohomology theory over k with values in a triple .C;w;TC/ is a triple .H; ;Tr/formed of the following:

1.1.1.1. A˝-functor ACU,

H WV0.k/opp! CC:

Each object X of V0.k/ has the structure of a coalgebra ACU (in the sense of I 6.1)with comultiplication �WX !X �X and counit X ! Spec.k/. Hence H.X/ is analgebra ACU of CC which depends functorially on X . Similarly,17M

� .H 2i .X/.i//

is a graded Q-algebra ACU.

1.1.1.2. A family D . i /i2N of natural transformations

iX WCi .X/˝11!H 2i .X/.i/

where Ci .X/D Cialg.X/ with the notation of 0.2. We can regard the iX as Q-linearmaps

iX WCi .X/! � .H 2i .X/.i//:

1.1.1.3. A family of morphisms

TrX WH.X/.n/! 11

where nD dim.X/. To give TrX is the same as giving a morphism

TrX WH 2n.X/.n/! 11:

These data satisfy the axioms.

16I.e., � 0.17We let H i .X/DH.X/i .

4 MOTIVES 27

1.1.1.4. If X;Y are objects of V0.k/, then the following diagram commutes

C.X/˝QC.Y / H.X/˝H.Y /

C.X �Y / H.X �Y /:

X˝ Y

�

X�Y

We deduce that X WC.X/˝11!H.X/ is a morphisms of algebras ACU.

1.1.1.5. For objects X;Y of V0.k/, we have

TrX�Y D TrX˝TrY

with the obvious identifications.

1.1.1.6. If X is an object of V0.k/ of dimension n, then there is a commutativediagram

C n.X/˝11 H 2n.X/

Q˝11 11

� TrX

'

where Q' C n.X/ is the canonical isomorphism (see for example 4.1.2.3). When Xis geometrically connected, nX is an isomorphism. and so

TrX WH 2n.X/.n/��! 11

1.1.1.7 H 1.P1/D 0.

1.1.1.8. If X has dimension n, then for all i 2 N, the pairing

H i .X/˝H 2n�i .X/!H 2n.X/' T˝�n

defines a duality (the Poincare duality):18

H i .X/'H 2n�i .X/.n/_:

For a morphism 'WX ! Y in V0.k/, this allows us to define the Gysin morphism

'�WHj .X/!H jC2.m�n/.Y /.m�n/;

where m (resp. n) is the dimension of Y (resp. X ), as the transpose of '� DH.'/.

1.1.1.9. If 'WX ! Y is a morphism, then there is a commutative diagram

C i .X/˝11 H 2i .X/.i/

C iCm�n.Y /˝11 H 2.iCm�n/.Y /.iCm�n/:

iX

'� '�.i/

iCm�nY

18Misprints fixed.

4 MOTIVES 28

1.1.2. If .H; ;Tr/, .H 0; 0;Tr0/ are cohomology theories on V.k/ with values in.C;w;T /, a morphism from .H; ;Tr/ to .H 0; 0;Tr0/ is a “unifere” ˝-morphismH !H 0 compatible with and Tr in the obvious sense. We obtain in this way acategory19 TC.k; .C;w;T //.

1.1.3. A morphism from a triple .C;w;T / to a triple .C 0;w0;T 0/ is a pair .u;�/consisting of a rigid˝-functor uWC ! C 0 compatible with the gradations w;w0 andan isomorphism

�Wu.T /' T 0.

Such a morphism defines a functor

TC.k; .C;w;T //! TC.k; .C 0;w0;T 0//:

1.1.4. Let S be a Q-scheme and C D Grad˙Loclib.S/ the rigid ˝-category oflocally free OS -modules equipped with a gradation of type Z; the commutativityconstraint is given by the “Koszul rule”: if E (resp. F ) has degree p (resp. q), thenthe isomorphism

E˝F��! F ˝E

sends x˝y to .�1/pqy˝x. The ˝-category C is obviously equipped with a ˝-gradation w of type Z, and we put

T DOS Œ�2�:

A cohomology theory over k with values in a triple .C;w;T / will be said moresimply to have values in S , or values in A if S D Spec.A/, and the category ofcohomology theories will be denoted TC.k;S/ or TC.k;A/ according to case. Wenote that for S a variable Q-scheme, the TC.k;S/ form a fibred category TC.k/ overSch=Q, which is even a stack for the fpqc topology on Sch=Q as one sees immediately.

1.1.5. Let TD .C;w;T / be a Tate triple over a field of characteristic zero (V 3.1.1).A cohomology theory with values in T is a cohomology theory with values in thetriple . PC;w;T /, where PC is equal to C as a˝-category AU but has as commutativityconstraint

P X;Y WX˝Y��! Y ˝X

the isomorphismP X;Y D

M.�1/pq pq;

where DL p;q is the decomposition of the commutativity isomorphism of C

for the gradation w. We obtain a category TC.k;T/.

1.1.5.1. Let C be a tannakian category over a field of characteristic zero, and formthe Tate triple .Gradf.C/;w;11Œ�2�/ consisting of the tannakian category Gradf.C /whose objects are those of C equipped with a gradation of type Z, the obviousgradation w, and the object 11 placed in degree �2. A cohomology theory with valuesin this triple will be called a cohomology theory with values in C. This examplegeneralizes 1.1.4 in the case where S is the spectrum of a field. We denote thecategory of cohomology theories by TC.k;C/.

19Misprint fixed.

4 MOTIVES 29

1.2 Various remarks

1.2.1. If .H; / is a cohomology theory as in 1.1, then H extends by additivity to a˝-functor ACU

H WV.k/opp! CCI

if X D`Xi is the decomposition of a smooth projective k-scheme into its connected

components, thenH.X/D

MiH.Xi /:

1.2.2. With the notation of 1.1.3, the category TC.k; .C;w;T // is a groupoid, i.e.,every morphism .H; /! .H 0; 0/ of cohomology theories is an isomorphism. Thisis proved using Poincare duality (1.1.1.8) by the same method as I 5.2.3.

1.2.3. If resolution of singularities is available (for example, if k has characteristiczero), then in the definition of a cohomology theory, there is no need to give .

1.3 Cohomological correspondences

1.3.1. Fix a cohomology theory as in 1.1.1. Recall that if 'WX ! Y is a morphismin V0.k/, then we defined in 1.1.1.8 the Gysin morphism

'�WH.X/!H.Y /.m�n/;

where m (resp. n)20 is the dimension of Y (resp. X). It is clear that if '00 D '0 ı',then

'00� D '0�.m�n/ı'�;

and finally that if 'WX ! Spec.k/ is the structure morphism, then

'� D TrX :

1.3.2. Let X;Y be objects of V0.k/. We will sometimes callH.X �Y / the object ofcohomological correspondences because there is a canonical isomorphism, deducedfrom Poincare duality,

H.X �Y /' Hom.H.X/;H.Y //.�n/;

where nD dim.X/. Via this isomorphism, X�Y induces Q-linear maps

C nCi .X �Y /! Hom.H.X/;H.Y /.i//

attaching a cohomological correspondence to an algebraic correspondence. Thesemaps are compatible with composition of correspondences (0.3.2).

20Misprint fixed.

4 MOTIVES 30

1.3.3. We make the above explicit in the case of a cohomology theory with valuesin a field K (1.1.4). The isomorphism

H WH.X/˝H.Y /! HomX .HX;HY /

is defined byH.x˝y/.x0/D .�1/pp

0

TrX .xx0/y;

where x 2 Hp.X/, x0 2 Hp0.X/, y 2 H.Y /. If n D dimX , the isomorphism H

makes the elements of H.X/˝H.Y /, homogeneous of degree 2nC d (d 2 Z/,correspond to the K-linear maps H.X/!H.Y / homogeneous of degree d . Finally,one checks immediately that the isomorphism H can be calculated by the followingmore manageable formula

H WH.X �Y /! HomX .HX;HY /

H.z/.x/D .prY /�.z �pr�X .x//:

1.4 Examples

1.4.1. `-adic cohomology. Let ` be a prime number different from the characteristicof k, and put21 CD Gradf.Q`/ equipped with its obvious ˝-gradation w (A 1.1.4).Fix an algebraic closure xk of k, and let T be the Q`-vector space of dimension one

T DQ`˝Z`lim �n

�`n.xk/

placed in degree �2, where �`n.xk/ is the group of `nth roots of 1 in xk. For X 2obV.k/, the `-adic cohomology of X is by definition

H`.X/DQ`˝Z`lim �n

H. xXet;Z=`nZ/;

where xXet is the etale site of xX DX� xk.22 The `-adic cohomology theory is developedin SGA 4 , SGA 5, which notably prove the axioms A 1.1 (see SGA 4, XVII, XVIII,SGA 5, V).

1.4.2. Hodge cohomology. Suppose that k has characteristic zero. The Hodgecohomology HHdg is the cohomology theory with values in k, defined by

HHdg.X/DX

p;qH q.X;˝

p

X=k/:

Poincare duality, as well as the trace morphism

Hn.X;˝nX=k/! k

(nD dimX ) can be deduced from Serre duality.21Gradf.Q`/ is the category of finite-dimensional Q`-vector spaces equipped with a gradation of type

Z.22Misprint fixed.

4 MOTIVES 31

1.4.3. de Rham cohomology. With the same hypotheses as above, de Rham coho-mology HdR is a cohomology theory with values in k, defined by

HdR.X/DH.X;˝X=k/;

where the term at right denotes the hypercohomology of the complex of differentialforms. This cohomology is the abutment of the corresponding spectral sequence ofhypercohomology

Epq1 DH

q.X;˝p

X=k/ H) H

pCqdR .X/:

It follows from Hodge theory that this spectral sequence degenerates, giving anisomorphism of cohomology theories

grFHdR DHHdgI

the filtration F on the abutment HdR.X/ is called the Hodge filtration.

1.4.4. Betti-Hodge cohomology. Suppose that k DC. The Betti-Hodge cohomologyHBH is a cohomology theory with values in the Tate triple (A 1.1.5) defined byHodge.Q/ (VI 2.1.3).23 We have

HBH.X/Q DH.X.C/;Q/I

the bigradation on HnBH.X/C DH

n.X.C/;C/ (n being an integer) comes from thecanonical isomorphism

Hn.X.C/;C/'HndR.X/

using that the Hodge filtration on HndR.X/ is n-opposed to its conjugate, i.e., satisfies

the condition of VI 2.1.1.324 (see Deligne [1], 3.2, for a more detailed discussion).The fact thatHn

BH.X/ is an object of Hodge.Q/, i.e., polarizable, follows from Hodgetheory (Weil [1], chap. IV, th. 7).

Note that if we choose a square root i of �1 in C, then, for each connected objectX of V.C/ of dimension n, i defines an orientation of the manifold X.C/ and a tracemorphism

TrX WH 2n.X.C/;Q/��!Q:

The element i defines also an isomorphism of Q-vector spaces of rank 1, TQ ' Q,where T is the Tate Hodge structure (VI 2.1.2). Once such a choice has been made, wecall Betti cohomology the cohomology theory HBetti with values in Q that it defines.

1.4.5. Crystalline cohomology. Suppose that k has nonzero characteristic and, tosimplify, is perfect. In this case, crystalline cohomology, introduced by Grothendieck[4] and developed by Berthelot [3], defines a cohomology theory Hcrys with values inthe field of fractions K.k/ of the ring W.k/ of Witt vectors over k. In fact, if X isan object of V.k/, the K.k/-vector space Hcrys.X/ is equipped with a structure of agraded F -isocrystal (VI 3.1.3 and 3.2.1); the isomorphism F WHcrys.X/

� !Hcrys.X/

23Hodge.Q/ is the category of polarizable Hodge structures over Q.24For pCq D nC1, the map F pV ˚F qV ! VC is an isomorphism.

4 MOTIVES 32

is given by the relative Frobenius of X=k. Moreover, H 2crys.P1k/ is canonically

isomorphic to the inverse of the Tate F -isocrystal (VI 3.1.3). We see thereforethat crystalline cohomology defines a cohomology theory with values in the triple.C;w;T /, where CD Grad˙Fcryso.k/, w is the obvious gradation, and T is the TateF -isocrystal placed in degree �2.

A.2 The standard conjecturesSee Grothendieck [5], Kleiman [1].

2.1 Notation

2.1.1. We fix a field K of characteristic zero, and a cohomology theory .H; ;Tr/over k with values in K. For X 2 obV.k/, we let C.X/ denote the group, graded (bycodimension), of algebraic cycles for cohomological equivalence on X , tensored withQ,

C.X/DZH .X/˝ZQ:

The map X induces injections

i WC i .X/ ,!H 2i .X/; i 2 Z:

2.1.2. Let X be a connected smooth projective k-scheme of dimension n, and let` 2 C 1.X/ be the class of a smooth hyperplane section (if one wants, we may evensuppose that X is polarized, i.e, equipped with an immersion X ,! PN

k). We again

denote by ` the homogeneous homomorphism of degree 1 (resp. 2)

`WC.X/! C.X/ .resp. `WH.X/!H.X//:

We have a commutative diagram (i 2 Z)25

C i .X/ C iC1.X/

H 2i .X/ H 2iC2.X/:

`

iX

iC1X

`

(2.1.2.1)

When 0� i � n, we put

P i .X/D KerŒ`n�iC1WH i .X/!H 2n�iC2.X/�; (2.1.2.2)

and when p 2 Z, 0� 2p � n, we put26

CpPr.X/D C

p.X/\P 2p.X/: (2.1.2.3)

The elements of P i .X/ are said to be primitive.25Misprints fixed.26Misprint fixed.

4 MOTIVES 33

2.2 Conjectures of Lefschetz type

2.2.1. Suppose that X is polarized; let i WY ,! X be a smooth hyperplane sectionand ` 2 C 1.X/ the class that it defines. The weak Lefschetz “theorem” says that theGysin morphism

H i .Y /!H iC2.X/ (2.2.1.1)

is surjective for i D n� 1 and an isomorphism for i � n. The strong Lefschetz“theorem” says that for i � n,

`n�i WH i .X/!H 2n�i .X/ (2.2.1.2)

is an isomorphism.These “theorems” are effectively proved in characteristic zero while in `-adic

cohomology only the weak Lefschetz theorem has been proved (SGA 4, XIV 3) andneither of them has been proved for crystalline cohomology. They are, however,conjectured.27

In the following, we suppose that .H; ;Tr/ satisfies the two Lefschetz theorems.

2.2.2. We retain the preceding notation. We deduce immediately from the hardLefschetz theorem, a decomposition of the cohomology

H i .X/DX

j�max.i�n;0/

`iP i�2j .X/; i 2 Z (2.2.2.1)

i.e., for each x 2H i .X/, there exist unique xj 2 P i�2j .X/ such that

x DX

`j .xj /: (2.2.2.2)

With the help of this decomposition, we define operators �, � by8<:�x D

P.�1/.i�2j /.i�2jC1/=2`n�iCj /.xj /

�.x/DP

j�max.i�n;1/`j�i .xj /:

(2.2.2.3)

The map �, which is homogeneous of degree �2, can be described more explicitlyas that which makes the following squares commute

H i .X/ H 2n�i .X/

H i�2.X/ H 2n�iC2.X/

�

� `

�

(2.2.2.4)

In particular, we have�ı`D id : (2.2.2.5)

27Deligne (La conjecture de Weil. II, 1980) proved the strong Lefschetz theorem for `-adic etalecohomology, and Katz and Messing 1974 deduced that it then holds for any “reasonable” cohomologytheory, in particular for crystalline cohomology. The weak Lefschetz theorem has also been proved forcrystalline cohomology.

4 MOTIVES 34

We also have the formula�� D id : (2.2.2.6)

The operator �WH i !H 2n�i differs by a factor � 0 from that defined by Weil[1].

2.2.3. Here are the conjectures of Lefschetz type, stated for X equipped with `.

A.X;`/W If 0� 2p � n, then the map

`n�2pWCp.X/! C n�p.X/

is bijective.B.X/W The cohomological correspondence � 2 Hom.H.X/;H.X// comes from

C.X �X/, i.e., it is algebraic.

The conjecture B.X/ implies A.X;`/ and is more manageable than it. As thenotation indicates, it does not depend on the smooth hyperplane chosen. It is stableby products, hyperplane sections, and has been proved for curves (trivial), abelianvarieties, and flag varieties. On the other hand,A.X�X;`X˝1XC1X˝`X / impliesB.X/. We see therefore that there is an equivalence between the validity of A.X/ orof B.X/ for all X .

2.2.4. The validity of the conjecture A.X/ induces a decomposition of C i .X/(i 2 Z/,

C i .X/DX

j�max.2i�n;0/

`jCi�jPr .X/: (2.2.4.1)

We deduce also that �WH 2i .X/!H 2n�2i .X/ (assuming 2i � n) sends C i .X/into C n�i .X/. The conjecture B.X/ implies that this operator is induced by anelement of C.X �X/.

2.3 Conjecture of Hodge type (or Hodge positivity)

2.3.1. Let X 2 obV0.k/ be of dimension n, and equipped with ` 2 C 1.X/ as in2.1.2.

The conjecture states the following:

Hdg.X/W If 0� 2p � n, then the symmetric bilinear form

.x;y/ 7! .�1/p TrX .`n�2pxy/WCpPr.X/�C

pPr.X/!Q

is positive definite.

In characteristic zero, this conjecture follows from Hodge theory (see Weil [1]).

2.4 The conjunction of the standard conjectures

2.4.1. The following statements are equivalent:

(a) A.X/ and Hdg.X/ are true for all X .(b) B.X/ and Hdg.X/ are true for all X:

4 MOTIVES 35

(c) For all X , we have Hdg.X/ and

D.X/: Numerical equivalence and cohomological equivalence (with respect to.H; // coincide for algebraic cycles.

Note thatD.X/ is equivalent to saying that the pairing C i .X/˝QCn�i .X/!Q

is a duality.We call this collection of conjectures the standard conjectures (they include the

two Lefschetz theorems).

2.4.2. Suppose that the standard conjectures are true, and let X be as in 2.3. Wethen have, for 0� 2p � n, a decomposition (2.2.4.1)28

Cp.X/DX

`jCp�jPr .X/

and by transport of the forms (2.3.1) on the Cp�jPr .X/, we obtain a positive definitesymmetric bilinear form

Cp.X/˝QCp.X/!Q; (2.4.2.1)

which explicitly is.x;y/ 7! TrX .x � �y/; (2.4.2.2)

where the operation * was defined in (2.2.2.3).

2.4.3. Suppose that .H; /, .H 0; 0/ are cohomology theories over k with values inK satisfying the Lefschetz theorems, and such that the cohomological equivalencerelations they define on algebraic cycles coincide. Then the standard conjectures aretrue for .H; / if and only if they are true for .H 0; 0/. Moreover, in this case, thealgebraic correspondences in C.X �X/ defined by the operations �, �, are the samefor H and H 0.

2.4.4. Let .H; ;Tr/ be a cohomology theory with values in a triple .C;w;TC/ as in1.1.1. With the obvious modifications, the standard conjectures still make sense inthis context.

A.3 Consequences of the standard conjecturesWe fix a cohomology theory .H; ;Tr/ with values in a field K of characteristic zerofor which we assume the validity of the standard conjectures.

3.1. The consequence that motivated the standard conjectures is the Weil conjectures,for a discussion of which we refer to Grothendieck [5] and Kleiman [1]. Here, we arerather interested in the consequences useful for the study of the category of motives.

3.2. Let X;Y 2 obV.k/; for i 2 Z, we have

C i .X �Y / ,!H 2i .X �Y /DXj2Z

H 2i�j .X/˝H j .Y /:

28Misprints fixed.

4 MOTIVES 36

It follows from the standard conjectures that the Kunneth components of an elementof C i .X �Y / are again in C i .X �Y /. This is equivalent to the following simplerstatement: if X has dimension n, then the Kunneth components � iX 2H

2n�i .X/˝

H i .X/ of the diagonal cycle �X 2 C n.X �X/ are algebraic. From this we get adecomposition in C n.X �X/

�X D

2nXiD0

� iX : (3.2.1)

Moreover, the � iX are independent of the cohomology theory satisfying the standardconjectures.

3.3. Let X;Y 2 obV.k/ be polarized. We have nondegenerate bilinear forms

H.X/˝H.X/!K

H.Y /˝H.Y /!K

given by

.x;x0/ 7! TrX .x � �x0/

.y;y0/ 7! TrY .y � �y0/:

It is easily seen that ifuWH.X/!H.Y /

is defined by an algebraic correspondence in C.X �X/, and if we let u0WH.Y /!H.X/ denote its transpose for these nondegenerate forms, then u0 is algebraic and�

Tr.uu0/D Tr.u0u/ 2Qu¤ 0 H) Tr.uu0/ > 0:

(3.3.1)

We conclude, for example, that the Q-algebra C.X �X/ of algebraic correspon-dences of X into X (see 0.3) is semisimple (see 4.2.2 for a more precise result29).

A.4 The Tate conjecture4.0. We fix a prime number ` different from the characteristic of k, and an algebraic

closure xk of k. If X is an object of V.k/, we let ZH .X/ denote the graded ring ofclasses of algebraic cycles for the equivalence relation defined by `-adic cohomology,and we put

C.X/DQ˝ZH .X/C`.X/DQ`˝ZH .X/.

29There is no 4.2.2. Probably it should be VI 4.2.2 (the category of motives is semisimple), which isactually less precise.

4 MOTIVES 37

4.1. Recall (A 1.4.1) the definition of `-adic cohomology: if X 2 ob.V.k//,

H`.X/DQ`˝Z`lim �n

H. xXet;Z=`nZ/:

From this definition, we see that, by the functoriality of the etale site, the Galoisgroup � D Gal.xk=k/ acts continuously on the finite-dimensional Q`-vector spaceH`.X/, this last being equipped with the `-adic topology. If x 2 Cp.X/, then itscohomology class .x/ 2H 2p

`.X/.p/ is invariant under the action of � , and so we

get an injectionCp.X/!H

2p

`.X/.p/� : (4.1.1)

The Tate conjecture (Tate [1]) makes this statement more precise in the case that k isa field finitely generated over the prime field (see 4.4 for the general case):

If k is finitely generated over the prime field, then the image of the map(4.1.1) generates the Q`-vector space of invariant cohomology classes,i.e.,

Q` �Cp.X/D ŒH2p

`.X/.p/�� :

4.2. If k is finitely generated over the prime field, we denote by Tate.k/ the categoryof finite-dimensional Q`-vector spaces equipped with a continuous action of � DGal.xk=k/; its objects will be called Tate modules. The category Tate.k/ is equippedwith an obvious˝-law ACU, as well as a Q`-linear structure. For these structures, itis a tannakian category over Q` (III 3.2.1), equipped with an obvious fibre functorover Q`, namely forget the action of � .

The `-adic cohomology induces a faithful Q-linear ˝-functor from the categoryCV0.k/ of correspondences of degree zero (A 0.3.3) into the category of graded Tatemodules (see A 1.3.2),

H`WCV0.k/! GradTate.k/

and the Tate conjecture implies that, if X;Y are objects of CV0.k/, then the inclusion

Hom.X;Y / ,! Hom.H`.Y /;H`.X//

induces an equality

Q` �Hom.X;Y /D Hom.H`.Y /;H`.X//:

4.3. Suppose now that k is arbitrary, and put

Tate.k/D lim�!k0

Tate.k0/

where k0 runs over the subfields of k finitely generated over the prime field, thetransition functor

Tate.k0/! Tate.k00/

for k00=k0 being induced by the morphism of groups

Gal.xk00=k00/! Gal.xk0=k/:

4 MOTIVES 38

The category Tate.k/ of Tate modules is again a neutral tannakian category overQ`, which depends functorially on the field k.

If G denotes the Q`-group of automorphisms of the forgetful functor on Tate.k/and Gı is its neutral component, then one sees without difficulty that there is acanonical isomorphism

G=Gı ' �Q`

where �Q`is the profinite Q`-group defined by � D Gal.xk=k/. In particular, if k is

algebraically closed, then G is connected.

4.4. Let X be an object of V.k/; then the H i`.X/ are Tate modules in a natural way.

Indeed, there exists a subfield k0 of k finitely generated over the prime field and anobject X 0 of V.k0/ such that X DX 0�k0 k. We have a canonical isomorphism

H i` .X/DH

i` .X

0/

which defines on H i`.X/ an action of Gal.xk0=k/. The object of Tate.k/ thus obtained

depends only on X . We see therefore, that for a general k, `-adic cohomology definesa faithful Q-linear˝-functor with values in the graded Tate modules,

H`WCV0.k/! GradTate.k/:

The Tate conjecture stated above implies that, if X;Y 2 obCV.k/, then the inclusion

Hom.X;Y / ,! Hom.H`.Y /;H`.X//

induces an equality

Q` �Hom.X;Y /D Hom.H`.Y /;H`.X//:

A.5 The Hodge conjecture5.0. In this number, the field k is the field C of complex numbers. IfX is an object of

V.C/, C.X/ denotes the graded ring of classes of algebraic cycles for the equivalencerelation defined by Hodge cohomology (A 1.4.2) tensored with Q

C.X/DQ˝ZZHdg.X/:

5.1. Let X be a connected smooth projective C-scheme, and let H.X/ denote theBetti-Hodge cohomology

H.X/DHBH.X/DH.X.C/;Q/;

defined in A 1.4.2. We have a Hodge decomposition30

H.X/C DMp;q

Hp;q.X/; Hp;q.X/DH q.X;˝p

X=C/;

30Misprints fixed.

4 MOTIVES 39

and if p 2 N, we know that the class of a cycle of codimension p in X is of type.p;p/; we have therefore an inclusion

Cp.X/ ,!Hp;p.X/\H 2p.X/ (5.1.1)

The Hodge conjecture in its simplest form is

Hodge(2p;p) The inclusion (5.1.1) is an equality.

Note that, as for the Tate conjecture and `-adic cohomology, this characteriza-tion of the algebraic cohomology classes implies the algebraicity of the Kunnethcomponents of the diagonal �X 2 C.X �X/ (see A 3.2).31

5.1.1. The Betti-Hodge cohomology induces a faithful Q-linear˝-functor (see A1.1.2)

HBHWCV0.C/! Hodge.Q/: (5.1.2)

The Hodge conjecture says that this functor is fully faithful.

5.2. We keep the notation of 5.1. For i;p 2 N, we denote by Filt0pH i .X/ the subQ-vector space of H i .X/ D H i .X.C/;Q/ of cohomology classes which are zeroon an open subscheme of the form X XT , where T is a subset Z of codimension� p. In fact, Filt0pH i .X/ is a Hodge substructure of H i .X/, i.e., the subspaceFilt0pH i .X/C of H i .X/C D H

i .X.C/;C/ is stable under the decomposition intotypes p;q. This follows from the fact that Filt0pH i .X/ is also the subspace generatedby the images of the morphisms of Hodge structures

H i�2q.Y /.�q/!H i .X/

induced by the morphisms Y ! X , where Y has dimension n� q � n� p andnD dimX (see Grothendieck [6], �1, and Deligne [6], 8.2.8). It follows also fromthis characterization that we have

Filt0pH i .X/��

FiltpH i .X/C

�\H i .X/; (5.2.1)

where FiltpH i .X/C DPr�pH

r;i�r.X/:

The Hodge conjecture in its generalized form, corrected by Grothendieck (loc.cit.) is

Hodge.i;p/ W The inclusion (5.2.1) makes Filt0pH i .X/ the largest Q-vector sub-space of the right hand term that is a Hodge substructure of H i .X/.

For i D 2p, this conjecture becomes the preceding conjecture. In Grothendieck[6] one can find remarks on this conjecture.

5.2.1 Let V be a Hodge Q-structure of weight 1. One can characterize the largestQ-subspace of Filtp VC\VQ which is a Hodge structure as the largest subobject V 0

of V such that V 0.p/ has positive bidegree: it is that which one would like to call thepart Vi�2p of Hodge level (niveau) � i D 2p, as inspired by 4.3 (the effective Hodgestructures are those with positive bidegrees).

31This, in fact, is not obvious for the Tate conjecture. See Tate 1994 (Seattle), �3, for the proof.

4 MOTIVES 40

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