Hodge cycles on abelian varieties

89 Pages·2003·0.62 MB·English

Hodge cycles on abelian varieties
P. Deligne (notes by J.S. Milne)
July 4, 2003
Abstract
ThisisaTeXedcopyof
– Hodge cycles on abelian varieties (the notes of most of the seminar
“Pe´riodesdesInte´gralesAbe´liennes”givenbyP.DeligneatI.H.E.S.,
1978–79;pp9–100ofDeligneetal. 1982),and
– Motives for absolute Hodge cycles (§6 of “Tannakian categories”,
ibid. pp196–220)
somewhatrevisedandupdated. Seetheendnotes1 formoredetails.
Contents
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1 Reviewofcohomology 6
Topologicalmanifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
Differentiablemanifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
Complexmanifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
Completesmoothvarieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
Applicationstoperiods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2 AbsoluteHodgecycles;principleB 16
Definitions(k algebraicallyclosedoffinitetranscendencedegree) . . . . . . . . 16
BasicpropertiesofabsoluteHodgecycles . . . . . . . . . . . . . . . . . . . . . 18
Definitions(arbitrary k) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
Statementofthemaintheorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
PrincipleB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
1
CONTENTS 2
3 Mumford-Tategroups;principleA 24
Characterizingsubgroupsbytheirfixedtensors . . . . . . . . . . . . . . . . . . 24
Hodgestructures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
Mumford-Tategroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
PrincipleA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
4 ConstructionofsomeabsoluteHodgecycles 31
Hermitianforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
(cid:86)
Conditionsfor dH1(A,Q)toconsistofabsoluteHodgecycles . . . . . . . . . 32
E
5 CompletionoftheproofforabelianvarietiesofCM-type 39
AbelianvarietiesofCM-type . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
ProofofthemaintheoremforabelianvarietiesofCM-type . . . . . . . . . . . . 40
6 Completionoftheproof;consequences 45
CompletionoftheproofofTheorem2.11 . . . . . . . . . . . . . . . . . . . . . 45
ConsequencesofTheorem2.11 . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
7 Algebraicityofvaluesofthe Γ-function 48
TheFermathypersurface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
Calculationofthecohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
Theactionof Gal(Q/k)onthe e´talecohomology . . . . . . . . . . . . . . . . . 53
Calculationoftheperiods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
Thetheorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
Restatementofthetheorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
8 MotivesforabsoluteHodgecycles 63
ComplementsonabsoluteHodgecycles . . . . . . . . . . . . . . . . . . . . . . 63
Constructionofthecategoryofmotives . . . . . . . . . . . . . . . . . . . . . . 65
Somecalculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
ArtinMotives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
Effectivemotivesofdegree1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
ThemotivicGaloisgroup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
Motivesofabelianvarieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
MotivesofabelianvarietiesofpotentialCM-type . . . . . . . . . . . . . . . . . 76
Introduction
Let X be a smooth projective variety over C. Hodge conjectured that certain cohomology
classesonX arealgebraic. Themainresultprovedinthesenotesshowsthat,whenX isan
abelian variety, the classes considered by Hodge have many of the properties of algebraic
classes.
CONTENTS 3
In more detail, let Xan be the complex analytic manifold associated with X, and con-
sider the singular cohomology groups Hn(Xan,Q). The variety Xan being of Ka¨hler type
(anyprojectiveembeddingdefinesaKa¨hlerstructure),itscohomologygroupsHn(Xan,C) ∼=
Hn(Xan,Q)⊗Chavecanonicaldecompositions
(cid:76)
Hn(Xan,C) = Hp,q, Hp,q = Hq(Xan,Ωp ).
Xan
p+q=n
The cohomology class cl(Z) ∈ H2p(Xan,C) of an algebraic subvariety Z of codimension
pinX isrational(i.e.,itliesinH2p(Xan,Q))andisofbidegree(p,p)(i.e.,itliesinHp,p).
TheHodgeconjecturestatesthat,conversely,everyelementof
H2p(Xan,Q)∩Hp,p
is a Q-linear combination of the classes of algebraic subvarieties. Since the conjecture is
unproven,itisconvenienttocalltheserational (p,p)-classesHodgecycles onX.
NowconsiderasmoothprojectivevarietyX overafieldk thatisofcharacteristiczero,
algebraically closed, and small enough to be embeddable in C. The algebraic de Rham
cohomologygroupsHn (X/k)havethepropertythat,foranyembeddingσ: k (cid:44)→ C,there
dR
arecanonicalisomorphisms
Hn (X/k)⊗ C →∼= Hn (Xan,C) ∼= Hn(Xan,C).
dR k,σ dR
It is natural to say that t ∈ H2p(X/k) is a Hodge cycle on X relative to σ if its image in
dR
H2p(Xan,C) is (2πi)p times a Hodge cycle on X ⊗ C. The arguments in these notes
k,σ
show that, if X is an abelian variety, then an element of H2p(X/k) that is a Hodge cycle
dR
on X relative to one embedding of k into C is a Hodge cycle relative to all embeddings;
further, for any embedding, (2πi)p times a Hodge cycle in H2p(Xan,C) always lies in the
image of H2p(X/k).2 Thus the notion of a Hodge cycle on an abelian variety is intrinsic
dR
to the variety: it is a purely algebraic notion. In the case that k = C the theorem shows
that the image of a Hodge cycle under an automorphism of C is again a Hodge cycle;
equivalently, the notion of a Hodge cycle on an abelian variety over C does not depend on
the map X → SpecC. Of course, all this would be obvious if only one knew the Hodge
conjecture.
Infact,astrongerresultisprovedinwhichaHodgecycleisdefinedtobeanelementof
(cid:81)
Hn (X)× Hn(X ,Q ). Asthetitleoftheoriginalseminarsuggests,thestrongerresult
dR l et l
has consequences for the algebraicity of the periods of abelian integrals: briefly, it allows
one to prove all arithmetic properties of abelian periods that would follow from knowing
theHodgeconjectureforabelianvarieties.3
—————————————————-
In more detail, the main theorem proved in these notes is that any Hodge cycle on
an abelian variety (in characteristic zero) is an absolute Hodge cycle — see §2 for the
definitionsandTheorem2.11foraprecisestatementoftheresult.
Theproofisbasedonthefollowingtwoprinciples.
CONTENTS 4
A. Let t ,...,t be absolute Hodge cycles on a smooth projective variety X and let G
1 N
be the largest algebraic subgroup of GL(H∗(X,Q))×GL(Q(1)) fixing the t ; then
i
everycohomologyclass tonX fixedby GisanabsoluteHodgecycle(see3.8).
B. If (X ) is an algebraic family of smooth projective varieties with S connected and
s s∈S
smooth and (t ) is a family of rational cycles (i.e., a global section of ...) such
s s∈S
thatt isanabsoluteHodgecycleforones,thent isanabsoluteHodgecycleforall
s s
s(see2.12,2.15).
EveryabelianvarietyAwithaHodgecycletiscontainedinasmoothalgebraicfamily
in which t remains Hodge and which contains an abelian variety of CM-type. Therefore,
Principle B shows that it suffices to prove the main theorem for A an abelian variety of
CM-type (see §6). Fix a CM-field E, which we can assume to be Galois over Q, and let
Σ be a set of representatives for the E-isogeny classes over C of abelian varieties with
complexmultiplicationbyE. PrincipleBisusedtoconstructsomeabsoluteHodgeclasses
on ⊕ A — the principle allows us to replace ⊕A by an abelian variety of the form
A∈Σ
A ⊗ O (see§4). LetG ⊂ GL(⊕ H (A,Q))×GL(Q(1))bethesubgroupfixingthe
0 Z E A∈Σ 1
absolute Hodge cycles just constructed plus some other (obvious) absolute Hodge cycles.
It is shown that G fixes every Hodge cycle on A, and Principle A therefore completes the
proof(see §5).
On analyzing which properties of absolute Hodge cycles are used in the above proof,
one arrives at a slightly stronger result. Call a rational cohomology class c on a smooth
projective complex variety X accessible if it belongs to the smallest family of rational
cohomologyclassessuchthat:
(a) thecohomologyclassofeveryalgebraiccycleisaccessible;
(b) thepull-backbyamapofvarietiesofanaccessibleclassisaccessible;
(c) if t ,...,t ∈ H∗(X,Q) are accessible, and if a rational class t in some H2p(X,Q)
1 N
isfixedbyanalgebraicsubgroupGofAut(H∗(X,Q))(automorphismsofH∗(X,Q)
asagradedalgebra)fixingthe t ,thentisaccessible;
i
(d) PrincipleB holdswith“absoluteHodge”replacedby“accessible”.
Sections 4,5,6 of these notes can be interpreted as proving that, when X is an abelian
variety, every Hodge cycle is accessible.4 Sections 2,3 define the notion of an absolute
Hodge cycle and show that the family of absolute Hodge cycles satisfies (a), (b), (c), and
(d);5 therefore,anaccessibleclassisabsolutelyHodge. Wehavetheimplications:
abelianvarieties trivial
Hodge========⇒accessible==⇒absolutelyHodge ==⇒Hodge.
Onlythefirstimplicationisrestrictedtoabelianvarieties.
The remaining three sections, §1, §7, and §8, serve respectively to review the different
cohomology theories, to give some applications of the main results to the algebraicity of
products of special values of the Γ-function, and to explain the theory of motives that can
bebuiltonabsoluteHodgecycles.
CONTENTS 5
Notations:
We define C to be the algebraic closure of R and i ∈ C to be a square root of −1; thus i is
only defined up to sign. A choice of i determines an orientation of C as a real manifold —
we take that for which 1∧i > 0 — and hence an orientation of every complex manifold.
Complexconjugationon Cisdenotedby ιorby z (cid:55)→ z.
Recallthatthecategoryofabelianvarietiesuptoisogenyisobtainedfromthecategory
of abelian varieties by taking the same class of objects but replacing Hom(A,B) with
Hom(A,B)⊗Q. Weshallalwaysregardanabelianvarietyasan objectinthecategoryof
abelianvarietiesuptoisogeny: thus Hom(A,B)isavectorspaceover Q.
If(V )isafamilyofrationalrepresentationsofanalgebraicgroupGoverk andt ∈
α α,β
V , then the subgroup of G fixing the t is the algebraic subgroup H of G such that, for
α α,β
allk-algebrasR,
H(R) = {g ∈ G(R) | g(t ⊗1) = t ⊗1,allα,β}.
α,β α,β
Linear duals are denoted by ∨. If X is a variety over a field k and σ is a homomorphism
σ: k (cid:44)→ k(cid:48),thenσX denotesthevarietyX ⊗ k(cid:48) (= X × Spec(k(cid:48))).
k,σ Spec(k)
1 REVIEWOFCOHOMOLOGY 6
1 Review of cohomology
Topological manifolds
LetX beatopologicalmanifoldand F asheafofabeliangroupson X. Weset
Hn(X,F) = Hn(Γ(X,F•))
where F → F• is any acyclic resolution of F. This defines Hn(X,F) uniquely up to a
uniqueisomorphism.
When F is the constant sheaf defined by a field K, these groups can be identified with
singularcohomologygroupsasfollows. LetS (X,K)bethecomplexinwhichS (X,K)
• n
istheK-vectorspacewithbasisthesingularn-simplicesinX andtheboundarymapsends
asimplextothe(usual)alternatingsumofitsfaces. Set
S•(X,K) = Hom(S (X,K),K)
•
withtheboundarymapforwhich
(α,σ) (cid:55)→ α(σ): S•(X,K)⊗S (X,K) → K
•
isamorphismofcomplexes,namely,thatdefinedby
(dα)(σ) = (−1)deg(α)+1α(dσ).
PROPOSITION 1.1. Thereisacanonicalisomorphism Hn(S•(X,K)) → Hn(X,K).
PROOF. If U is the unit ball, then H0(S•(U,K)) = K and Hn(S•(U,K)) = 0 for n > 0.
Thus, K → S•(U,K) is a resolution of the group K. Let Sn be the sheaf of X associated
with the presheaf V (cid:55)→ Sn(V,K). The last remark shows that K → S• is a resolution of
the sheaf K. As each Sn is fine (Warner 1971, 5.32), Hn(X,K) ∼= Hn(Γ(X,S•)). But
theobviousmapS•(X,K) → Γ(X,S•)issurjectivewithanexactcomplexaskernel(loc.
cit.),andso
Hn(S•(X,K)) →∼= Hn(Γ(X,S•)) ∼= Hn(X,K).
Differentiable manifolds
Now assume X is a differentiable manifold. On replacing “singular n-simplex” by “dif-
ferentiablesingularn-simplex”intheabovedefinitions,oneobtainscomplexesS∞(X,K)
•
andS• (X,K). Thesameargumentshowsthatthereisacanonicalisomorphism
∞
Hn(X,K) =df Hn(S∞(X,K)) →∼= Hn(X,K)
∞ •
(loc. cit.).
1 REVIEWOFCOHOMOLOGY 7
LetO bethesheafofC∞ real-valuedfunctionsonX,letΩn betheO -module
X∞ X∞ X∞
ofC∞ differentialn-formson X,andletΩ• bethecomplex
X∞
O →d Ω1 →d Ω2 →d ··· .
X∞ X∞ X∞
ThedeRhamcohomologygroupsof X aredefinedtobe
{closedn-forms}
Hn (X) = Hn(Γ(X,Ω• )) = .
dR X∞ {exactn-forms}
IfU istheunitball,Poincare´’slemmashowsthatH0 (U) = RandHn (U) = 0forn > 0.
dR dR
Thus, R → Ω• is a resolution of the constant sheaf R, and as the sheaves Ωn are fine
X∞ X∞
(Warner1971,5.28),wehaveHn(X,R) ∼= Hn (X).
dR
Forω ∈ Γ(X,Ωn )andσ ∈ S∞(X,R),define
X∞ n (cid:90)
n(n+1)
(cid:104)ω,σ(cid:105) = (−1) 2 ω ∈ R.
(cid:82) (cid:82) σ
Stokes’stheoremstatesthat dω = ω,andso
σ dσ
(cid:104)dω,σ(cid:105)+(−1)n(cid:104)ω,dσ(cid:105) = 0.
Thepairing (cid:104),(cid:105)thereforedefinesamapofcomplexes
f: Γ(X,Ω• ) → S• (X,R).
X∞ ∞
THEOREM 1.2 (DE RHAM). The map Hn (X) → Hn(X,R) defined by f is an isomor-
dR ∞
phismforall n.
PROOF. Themapisinversetothemap
Hn(X,R) →∼= Hn(X,R) ∼= Hn (X)
∞ dR
defined in the previous two paragraphs (Warner 1971, 5.36). (Our signs differ from the
usualsignsbecausethestandardsignconventions
(cid:90) (cid:90) (cid:90) (cid:90) (cid:90)
dω = ω, pr∗ω ∧pr∗η = ω · η, etc.
1 2
σ dσ X×Y X Y
violatethesignconventionsforcomplexes.)
(cid:82)
A number ω, σ ∈ H (X,Q), is called a period of ω. The map in (1.2) identifies
σ n
Hn(X,Q)withthespaceofclassesofclosedformswhoseperiodsareallrational. Theorem
1.2 can be restated as follows: a closed differential form is exact if all its periods are zero;
thereexistsacloseddifferentialformhavingarbitrarilyassignedperiodsonanindependent
setofcycles.
REMARK 1.3 (SINGER AND THORPE 1967, 6.2). If X is compact, then it has a smooth
triangulation T. Define S (X,T,K) and S•(X,T,K) as before, but using only simplices
•
inT. Thenthemap
Γ(X,Ω• ) → S•(X,T,K)
X∞
definedbythesameformulasas f aboveinducesisomorphisms
Hn (X) → Hn(S•(X,T,K)).
dR
1 REVIEWOFCOHOMOLOGY 8
Complex manifolds
Nowlet X beacomplexmanifold,andwrite Ω• forthecomplex
Xan
O →d Ω1 →d Ω2 →d ···
Xan Xan Xan
in which Ωn is the sheaf of holomorphic differential n-forms. Thus, locally a section of
Xan
Ωn isoftheform
Xan (cid:88)
ω = α dz ∧...∧dz
i1...in i1 in
with α a holomorphic function and the z complex local coordinates. The complex
i1...in i
form of Poincare´’s lemma shows that C → Ω• is a resolution of the constant sheaf C,
Xan
andsothereisacanonicalisomorphism
Hn(X,C) → Hn(X,Ω• ) (hypercohomology).
Xan
IfX isacompactKa¨hlermanifold,thenthespectralsequence
Ep,q = Hq(X,Ωp ) =⇒ Hp+q(X,Ω• )
1 Xan Xan
degenerates,andsoprovidesacanonicalsplitting6
(cid:76)
Hn(X,C) = Hq(X,Ωp ) (theHodgedecomposition)
Xan
p+q=n
as Hp,q = Hq(X,Ωp ) is the complex conjugate of Hq,p relative to the real struc-
df Xan
ture Hn(X,R) ⊗ C ∼= Hn(X,C) (Weil 1958). The decomposition has the following
explicit description: the complex Ω• ⊗ C of sheaves of complex-valued differential
X∞
forms on the underlying differentiable manifold is an acyclic resolution of C, and so
Hn(X,C) = Hn(Γ(X,Ω• ⊗C)); Hodge theory shows that each element of the second
X∞
group is represented by a unique harmonic n-form, and the decomposition corresponds to
thedecompositionofharmonic n-formsintosumsofharmonic (p,q)-forms,p+q = n.7
Complete smooth varieties
Finally, let X be a complete smooth variety over a field k of characteristic zero. If k = C,
thenX definesacompactcomplexmanifoldXan,andtherearethereforegroupsHn(Xan,Q),
depending on the map X → Spec(C), that we shall write Hn(X) (here B abbreviates
B
Betti). If X is projective, then the choice of a projective embedding determines a Ka¨hler
structure on Xan, and hence a Hodge decomposition (which is independent of the choice
oftheembeddingbecauseitisdeterminedbytheHodgefiltration,andtheHodgefiltration
dependsonlyonX;seeTheorem1.4below). Inthegeneralcase,werefertoDeligne1968,
5.3,5.5,fortheexistenceofthedecomposition.
For an arbitrary field k and an embedding σ: k (cid:44)→ C, we write Hn(X) for Hn(σX)
σ B
and Hp,q(X) for Hp,q(σX). As ι defines a homeomorphism σXan → ισXan, it induces an
σ
1 REVIEWOFCOHOMOLOGY 9
isomorphismHn(X) → Hn(X). Sometimes,whenk isgivenasasubfieldofC,wewrite
ισ σ
Hn(X)forHn(X ).
B B C
Let Ω• denote the complex in which Ωn is the sheaf of algebraic differential n-
X/k X/k
forms,anddefinethe(algebraic)deRhamcohomologygroupHn (X/k)tobeHn(X ,Ω• )
dR Zar X/k
(hypercohomology with respect to the Zariski cohomology). For any homomorphism
σ: k (cid:44)→ k(cid:48),thereisacanonicalisomorphism
Hn (X/k)⊗ k(cid:48) → Hn (X ⊗ k(cid:48)/k(cid:48)).
dR k,σ dR k
Thespectralsequence
Ep,q = Hq(X ,Ωp ) =⇒ Hp+q(X ,Ω• )
1 Zar X/k Zar X/k
definesafiltration(theHodgefiltration)FpHn (X)onHn (X)whichisstableunderbase
dR dR
change.
THEOREM 1.4. Ifk = C,theobviousmaps
Xan → X , Ω• ← Ω• ,
Zar Xan X
induceisomorphisms
Hn (X) → Hn (Xan) ∼= Hn(Xan,C)
dR dR
(cid:76)
underwhichFpHn (X)correspondsto FpHn(Xan,C) = Hp(cid:48),q(cid:48).
dR df
p(cid:48)≥p,p(cid:48)+q(cid:48)=n
PROOF. Theinitialtermsofthespectralsequences
Ep,q = Hq(X ,Ωp ) =⇒ Hp+q(X ,Ω• )
1 Zar X/k Zar X/k
Ep,q = Hq(X,Ωp ) =⇒ Hp+q(X,Ω• )
1 Xan Xan
are isomorphic — see Serre 1956 for the projective case and Grothendieck 1966 for the
general case. The theorem follows from this because, by definition of the Hodge decom-
position, the filtration of Hn (Xan) defined by the above spectral sequence is equal to the
dR
filtrationof Hn(Xan,C)definedinthestatementofthetheorem.
It follows from the theorem and the discussion preceding it that every embedding
σ: k (cid:44)→ Cdefinesanisomorphism
Hn (X)⊗ C −→∼= Hn(X)⊗ C
dR k,σ σ Q
and, in particular, a k-structure on Hn(X) ⊗ C. When k = Q, this structure should be
σ Q
distinguishedfromthe Q-structuredefinedby Hn(X): thetwoarerelatedbytheperiods.
σ
When k is algebraically closed, we write Hn(X,A ), or Hn(X), for Hn(X ,Zˆ) ⊗
f et et Z
Q, where Hn(X ,Zˆ) = lim Hn(X ,Z/mZ) (e´tale cohomology). If X is connected,
et ←−m et
1 REVIEWOFCOHOMOLOGY 10
H0(X,A ) = A , the ring of finite ade`les for Q, which justifies the first notation. By defi-
f f
nition, Hn(X) depends only on X (and not on its structure morphism X → Speck). The
et
mapHn(X) → Hn(X⊗ k(cid:48))definedbyaninclusionk (cid:44)→ k(cid:48) ofalgebraicallyclosedfields
et et k
is an isomorphism (special case of the proper base change theorem Artin, Grothendieck,
andVerdier1973,XII).Thecomparisontheorem(ibid. XI)showsthat,when k = C,there
is a canonical isomorphism Hn(X) ⊗ A → Hn(X). It follows that Hn(X) ⊗ A is in-
B f et B f
dependent of the morphism X → SpecC, and that, over any algebraically closed field of
characteristiczero, Hn(X)isafree A -module.
et f
TheA -moduleHn(X,A )canalsobedescribedastherestrictedproductofthespaces
f f
Hn(X,Q ),l aprimenumber,withrespecttothesubspaces Hn(X,Z )/{torsion}.
l l
Nextwedefinethenotionofthe“Tatetwist”ineachofthethreecohomologytheories.
For this we shall define objects Q(1) and set Hn(X)(m) = Hn(X)⊗Q(1)⊗m. We want
Q(1) to be H2(P1) (realization of the Tate motive in the cohomology theory), but to avoid
thepossibilityofintroducingsignambiguitiesweshalldefineitdirectly,
Q (1) = 2πiQ
B
(cid:161) (cid:162)
Q (1) = A (1) =df lim µ ⊗ Q, µ = {ζ ∈ k | ζr = 1}
et f ←−r r Z r
Q (1) = k,
dR
andso
Hn(X)(m) = Hn(X)⊗ (2πi)mQ = Hn(Xan,(2πi)mQ) (k = C)
B B Q (cid:161) (cid:162)
Hn(X)(m) = Hn(X)⊗ (A (1))⊗m = lim Hn(X ,µ⊗m) ⊗ Q (k algebraicallyclosed)
et et Af f ←−r et r Z
Hn (X)(m) = Hn (X).
dR dR
Thesedefinitionsextendinanobviouswaytonegativem. Forexample,wesetQ (−1) =
et
Hom (A (1),A )anddefine
Af f f
Hn(X)(−m) = Hn(X)⊗Q (−1)⊗m.
et et et
Therearecanonicalisomorphisms
Q (1)⊗ A → Q (1) (k ⊂ C,k algebraicallyclosed)
B Q f et
Q (1)⊗C → Q (1)⊗ C (k ⊂ C)
B dR k
andhencecanonicalisomorphisms(thecomparisonisomorphisms)
Hn(X)(m)⊗ A → Hn(X)(m) (k ⊂ C,k algebraicallyclosed)
B Q f et
Hn(X)(m)⊗ C → Hn (X)(m)⊗ C (k ⊂ C).
B Q dR k
Todefinethefirst,notethat expdefinesanisomorphism
z (cid:55)→ ez: 2πiZ/r2πiZ → µ .
r