Chern classes of automorphic vector bundles

0.23 MB·

by  Hélène Esnault

CHERN CLASSES OF AUTOMORPHIC VECTOR
BUNDLES
7
1 HE´LE`NE ESNAULT AND MICHAEL HARRIS
0
2

n
a Dedicated to Yuri Ivanovich Manin on the occasion of his 80-th
J birthday

1
3
Abstract. We prove that Chern classes in continuous ℓ-adic co-

homology of automorphic bundles associatedto representations of
]
G G on a projective Shimura variety with data (G,X) are trivial ra-
A tionally. It is a consequence of Beilinson’s conjectures which pre-
dict that the Chern classes in the Chow groups vanish rationally.
.
h
t
a
m
Introduction
[

Let X be a smooth projective variety defined over a number field
1
v k. Beilinson [Bei85, Conj. 2.4.2.1] conjectures that the rational Chow
3 ring CH(X) injects into Deligne cohomology of X ⊗ C. Concretely,
Q k
7
if a class in CHn(X) vanishes in H2n(X,Q(n)), it is expected to be 0.
0 Q D
9 There is not a single example with dimension X ≥ 2 with large Chow
0 ring CH(X ⊗ C) for which this conjecture has been verified.
. k
1
0 On the other hand, there are Chern classes reflecting the fact that
7
X is defined over a number field. On proper models X over a non-
1 U
: trivial open U of Spec(Z), one has Chern classes in ℓ-adic cohomology
v
i H2n(XU,Qℓ(n)). By taking the inductive limit over such XU, these
X
yieldtheChern classes incontinuous ℓ-adiccohomologyH2n (X,Q(n))
r cont
a ([Jan87, Section 2]). Each space H2n (X,Q(n)) is filtered by the abut-
cont
ment of the Hochshild-Serre spectral sequence, which, by Deligne’s ar-
gument [Del68, Thm. 1.5] using the strong Lefschetz theorem [Del80,
Thm. 4.4.1], degenerates at E . Given that H1(k,H2n−1(X ,Q (n))),
2 k¯ ℓ
the first graded piece of the filtration, can be interpreted as the ex-
tension group of Q (0) by H2n−1(X ,Q (n)) in the category of Galois
ℓ k¯ ℓ
modules – just as H2n−1(X ,Q(n)) can be interpreted as the extension
D C
TheresearchleadingtotheseresultshasreceivedfundingfromtheEuropeanRe-
search Council under the European Community’s Seventh Framework Programme
(FP7/2007-2013)/ERCAdvancedGrants22224001and290766,theEinsteinFoun-
dation, the NSF grant DMS- 1404769.
1
2 HE´LE`NEESNAULTANDMICHAEL HARRIS
group of Q(0) by H2n−1(X ,Q(n)) in the category of Hodge structures
C
over Q – Beilinson’s conjecture predicts that
Conjecture 0.1. With notation as above, if a class in CHn(X) van-
Q
ishes in Hi(k,H2n−i(X ,Q (n)) for i = 0,1, then it does for i = 2.
k¯ ℓ
This conjecture seems to be more modest than the general motivic
one above. It is a fascinating problem in Galois cohomology, and it
hasn’t been studied at all. An analogous question for function fields
over finite fields has been considered and proved to be true for 0-cycles
in [Ras95, Thm. 0.1].
On the other hand, the Chern classes of flat bundles on a smooth
projective variety X defined over C vanish in Deligne cohomology
H2n(X,Q(n)) for n ≥ 2, due to Reznikov’s theorem [Rez94, Thm. 1.1],
D
giving a positive answer to Bloch’s conjecture [Blo77, Intro.]. In par-
ticular, Beilinson’s conjecture implies that the Chern classes of flat
bundles on a smooth projective variety X defined over a number field
vanish in the Chow groups CHn(X) for n ≥ 2.
Q
In addition, in [Esn96, 4.7] (in a vague form) and in [EV02, Intro.]
(in a more precise form) the problem is posed whether Chern classes
of Gauß-Manin bundles on a smooth variety X defined over a field of
characteristic 0 vanish in the Chow groups CHn(X) for n ≥ 2. It is
Q
proved to be the case for those of weight 1; that is, for the Gauß-Manin
bundles of relative first de Rham cohomology of an abelianscheme over
X ([vdG99, Thm. 1.1], [EV02, Thm. 1.1]). In fact the weight 1 Gauß-
Manin bundle is defined on A , which is defined over Q, and in [EV02,
g
loc. cit.] it is proved that the Chern classes in the Chow groups of the
Deligne extension on the toroidal compactification of A vanish. Thus
g
this example confirms (in a weak sense) Beilinson’s conjecture as well.
The moduli space A is a quasi-projective Shimura variety and the
g
weight 1 Gauß-Manin bundle on it is an automorphic bundle associ-
ated to the tautological representation of Sp(2g). A Shimura variety
S(G,X) (notation explained below) has a canonical model over its
K
reflex field E(G,X), which is a number field [Mil05, Section 14]. It car-
ries a natural family of automorphic vector bundles that are defined on
this model and themselves have models over explicit finite extensions
of E(G,X) [Har85, Thm. 4.8] (for unexplained notation see Section 1.1
below). An automorphic vector bundle [E] that comes from a repre-
K
sentation of G is endowed canonically with a flat connection, and its
Chern classes in Deligne cohomology H2n( S(G,X),Q(n)) vanish for
D K
n ≥ 1 (Theorem 2.3, Remark 2.5 1)). Thus, at least when S(G,X) is
K
projective, Beilinson’s conjecture implies that the Chern classes vanish
CHERN CLASSES OF AUTOMORPHIC VECTOR BUNDLES 3
even in CHn( S(G,X)) for n ≥ 1. Unfortunately, we can not prove
K Q
this. Instead we prove
Theorem 0.2. If S(G,X) is projective, the Chern classes of an auto-
K
morphic bundle attached to a representation of G vanish in continuous
ℓ-adic cohomology H2n ( S(G,X),Q (n)) for n ≥ 1.
cont K ℓ
Stated differently, we prove Conjecture 0.1 in this particular case.
The proof relies strongly on the purely algebraic definition of the au-
tomorphic bundles, as being associated to a representation of G. In-
deed, all automorphic bundles, seen in the category of vector bun-
dles on the Shimura variety, are eigenvectors for the so-called volume
character of the Hecke algebra. The Hecke algebra acts semi-simply
on (continuous) ℓ-adic cohomology, and the corresponding eigenspace
Hj( S(G,X) ,Q ) in ℓ-adic cohomology identifies with ℓ-adic coho-
K Q¯ ℓ v
mology Hj(Xˆ ,Q ) of the compact dual Xˆ of X, which itself is gen-
Q¯ ℓ
erated by algebraic cycles. This allows us to compute the invariant in
i-th Galois cohomology of H2n−i( S(X,G) ,Q (n)) (Theorem 3.3).
K Q¯ ℓ
Finally we remark that if every Q-simple factor of G has real rank at
least 2, then the super-rigidity theorem of Margulis, applied to the con-
nected components of S(G,X), implies that every flat vector bundle
K
over S(G,X) becomes isomorphic to an automorphic vector bundle
K
after replacing K by an appropriate subgroup of finite index. Since the
non-vanishing of Chern classes in the cohomology theories considered
here is stable under finite coverings, this implies that for most Shimura
varieties the vanishing holds for all flat vector bundles.
Acknowledgements: It is a pleasure to thank Alexander Beilinson
for email exchanges on his conjecture [Bei85, Conj. 2.4.2.1], most par-
ticularly on the much weaker version Conjecture 0.1. This has been
a guiding line for formulating the problem on automorphic bundles.
We thank Spencer Bloch for discussions on Conjecture 0.1. We thank
Ben Moonen for enlightening exchanges on Conjecture 1.10, which is
a particular instance of Beilinson’s conjecture, and Bruno Klingler for
Lemma 2.2.
Conventions. If G is a reductive algebraic group over Q, by an ad-
missible irreducible representation of G(A) we will mean an irreducible
admissible (g,K) × G(A )-module, where g is the complexified Lie
f
algebra of G, K ⊂ G(R) is a connected subgroup generated by the
center of G(R) and a maximal compact connected subgroup, G(A ) is
f
the group of finite ad`eles of G. If π is such a representation then we
will write
π ≃ π ⊗π
∞ f
4 HE´LE`NEESNAULTANDMICHAEL HARRIS
where π is an irreducible admissible (g,K)-module and π is an irre-
∞ f
ducible admissible representation of G(A ).
f
1. Automorphic vector bundles and flag varieties
1.1. Review of automorphic vector bundles. Let (G,X) be a
Shimura datum, in other words a datum defining a Shimura variety.
We recall that this means that G is a connected reductive group over
QandthatX isaG(R)-conjugacyclassofhomomorphisms h : S → G
R
of real groups, where S = R G is C× viewed as an algebraic group
C/R m,C
over R. Thepair(G,X)must satisfyalist offamiliaraxiomsthatguar-
antee that X is a G(R)-equivariant finite union of hermitian symmetric
spaces for the identity component of the derived subgroup of G(R); see
[Mil05]. In particular, we include the axiom that guarantees that the
maximal R-split torus in the center of G is also split over Q; without
this hypothesis the construction of automorphic vector bundles, as in
(1.5), is not strictly true as stated, although there are ways to fix this.
Then for any open compact subgroup K ⊂ G(A ), the double coset
f
space
S(G,X) = G(Q)\X ×G(A )/K
K f
is canonically the set of complex points of a quasiprojective algebraic
variety that has a canonical model over a number field, usually denoted
E(G,X) and called the reflex field of (G,K). If K′ ⊂ K is a subgroup
of finite index then the natural map
πK,K′ :K′ S(G,X) → KS(G,X)
is finite; if K′ is a normal subgroup of K then πK,K′ is the quotient
map for the action of the group K/K′ on the right. Moreover, if K is
sufficiently small (neat, in the sense of [Pin90]), then the map πK,K′ is
finite ´etale.
The precise nature of the canonical model will not be considered in
this paper; we will be concerned with S(G,X) as a complex algebraic
K
variety, and our aim is to study the Chern classes of a class of vector
bundles on S(G,X) that are defined canonically by reference to the
K
origin of the variety in linear algebra. To this end, choose a base point
h ∈ X and let K ⊂ G(R) be its stabilizer. Then K is the group
h h
of real points of a reductive subgroup of G, which we also denote K .
h
We make the useful assumption that K is defined over a number field
h
E ; this is always possible, and we may even assume that E is a CM
h h
field and that every irreducible representation of K is defined over a
h
CM field. In any case, K is reductive and there is a natural maximal
h
parabolic subgroup P ⊂ G that contains K as a Levi factor. Let
h C h,C
CHERN CLASSES OF AUTOMORPHIC VECTOR BUNDLES 5
Xˆ = G/P be the corresponding flag variety. We view X as an analytic
h
open subset of Xˆ by means of the Borel embedding β (this determines
the choice of P among the two maximal parabolics containing K ). In
h h
particular, the complex dimension of this analytic variety is the same
ˆ
as that of X, which is the same as that of S(G,X). Moreover, h may
K
be viewed as a point of Xˆ.
Let E′ ⊃ E denote a finite extension over which K becomes a split
h h h
reductive group. Then every representation of K has a model over E′.
h h
For any variety Z over C, let Vect(Z) denote the exact category
of complex vector bundles on Z. Let Vect (Xˆ) be the category of
G
G-equivariant vector bundles on Xˆ with coefficients in C; let
ˆ ˆ
f : Vect (X) → Vect(X)
G
be the forgetful functor. Let Vectss(Xˆ) ⊂ VectXˆ denote the subcate-
G G
gory of semisimple G-equivariant bundles. If H is an algebraic group
over a ring R and k ⊃ R is another ring, let Rep (H) denote the cate-
k
gory of algebraic representations (of finite type) of H on free modules
over k. There is an equivalence of symmetric monoidal categories
(1.1) r : Vect (Xˆ) ≃ Rep (P )
P G Q¯ h
ˆ
given by taking a vector bundle B/X to its fiber B at h, with the
h
isotropy representation of the stabilizer P . Similarly, let Vectss(Xˆ) ⊂
h G
VectXˆ denote the subcategory of semisimple G-equivariant vector bun-
G
dlesonXˆ. Then(1.1)restrictstoanequivalenceofsymmetricmonoidal
categories
(1.2) r : Vectss(Xˆ) ≃ Rep (K )
G Q¯ h
Evidently we have canonical isomorphisms
(1.3)
K (Vectss(Xˆ)) −∼→ K (Vect (Xˆ)); K (Rep (K )) −∼→ K (Rep (P ))
0 G 0 G 0 Q¯ h 0 Q¯ h
compatible with the isomorphisms (1.1) and (1.2).
Lemma 1.4. Every simple object in Vect (Xˆ) has a model over E′.
G h
Proof. Thefiberfunctor(1.2)athisevidently rationalover thenumber
fieldE , sotheclaimcomesdowntotheassertionthatevery irreducible
h
representationofK hasamodeloverE′, whichwehavealreadynoted.
h h
(cid:3)
On the other hand, for any K ⊂ G(A ) as above, there is a functor
f
ˆ
(1.5) E 7→ [E] : Vect (X) → Vect( S(G,X))
G K
6 HE´LE`NEESNAULTANDMICHAEL HARRIS
defined algebraically in [Har85, Thm. 4.8]. As a functor on complex
vector bundles we have the explicit construction:
[E] = [E] = G(Q)\E ×G(A )/K.
K f
This is a monoidal functor and it satisfies the following property with
respect to change of group: if K′ ⊂ K then there are canonical isomor-
phisms
(1.6) πK∗,K′([E]K) −∼→ [E]K′; πK,K′,∗([E]K′) −∼→ [E]K ⊗IKK′1,
where IK1 is the representation of K induced from the trivial repre-
K′
sentation of K′. On the left, this is by definition, and on the right, this
is the projection formula. In particular,
(1.7) πK,K′,∗([E]K′) −∼→ ([E]K)[K:K′]
as vector bundles.
In this paper we work systematically with Chow groups CH and the
Grothendieck groupK oflocally freesheaves with rationalcoefficients.
0
For H and k as above, we let K (Rep (H)) denote the Grothendieck
0 k
group of Rep (H), tensored with Q.
k
ch : Vect(Xˆ) → CH(Xˆ) , ch : Vect( S(G,X)) → CH( S(G,X))
Xˆ Q K K K Q
denote the respective Chern characters. We shall use the following
proposition:
Proposition 1.8. 1) The map
ch ◦◦f ◦r−1 : Rep (K ) → Vect(Xˆ) → CH(Xˆ)
Xˆ Q¯ h Q Q
factors through the composite homomorphism
K (Rep (K )) → K (Xˆ) → CH(Xˆ) .
0 Q h Q 0 Q Q
2) The restriction of ch to Vect (Xˆ) generates CH(Xˆ) .
Xˆ G Q
3) If we let Rep (G) → Rep (K ) denote the restriction functor,
Q Q h
then ch ◦r−1 induces an isomorphism
Xˆ
(1.9) K (Rep (K ))⊗ Q −∼→ CH(Xˆ) .
Q Q h KQ(RepQ(G)) Q
Here the map K (Rep (G)) → Q is given by the augmentation,
Q Q
that is by the rank of a representation.
Proof. Point 1) is essentially a tautology: the Chern character obvi-
ously factors through K (Xˆ) and r is an exact tensor functor. Point 2)
0
is the main theorem of [Mar76]. Suppose V is a representation of G;
thenthecorrespondinghomogeneousbundleonXˆ isjustV×Xˆ, withG
acting diagonally. In particular, as a vector bundle it is a sum of dimV
copies of the trivial bundle, hence the restriction to K (Rep (G)) of
Q Q
CHERN CLASSES OF AUTOMORPHIC VECTOR BUNDLES 7
the Chern character factors through the augmentation map. Thus the
surjection ch ◦ r−1 factors through the left-hand side of (1.9). Now
Xˆ
it follows from the main theorem of [Mar76] that this left hand side is
of dimension [W : W ], where W (resp. W ) is the absolute Weyl
G Kh G Kh
group of G (resp. K ) relative to a common maximal torus. On the
h
other hand, the Schubert cells form a basis for the right-hand side, and
there are [W : W ] of them (cf. [Bri05, 3.4.2 (2)]. So the surjection
G Kh
(cid:3)
is an isomorphism by comparing dimensions.
The purpose of the present note is to provide some evidence for
the following conjecture, which is an analogue of Proposition 1.8 for
Shimura varieties.
Conjecture 1.10. The map c : Rep (K ) → CH( S(G,X)) de-
K Q¯ h K Q
fined by
c (W) = ch ([r−1(W)])
K K
induces an injective ring homomorphism
K (Rep (K ))⊗ Q ֒→ CH( S(G,X)) .
Q Q h KQ(RepQ(G)) K Q
We denote by c>0 the composite of c with the projection
K K
CH( S(G,X)) = ⊕ CHn( S(G,X)) → ⊕ CHn( S(G,X)) .
K Q n≥0 K Q n>0 K Q
Claim 1.11. Conjecture 1.10 is equivalent to
c>0| = 0.
K RepQ¯(G)
Proof. Indeed this condition is equivalent to saying that c induces a
K
homomorphism
K (Rep (K ))⊗ Q → CH( S(G,X)) .
Q Q h KQ(RepQ(G)) K Q
On the other hand, given the Grothendieck-Riemann-Roch theorem, to
say that it is injective is equivalent to saying that the ring homomor-
phism
K (Rep (K )) ⊗ Q → K ( S(G,X))
0 Q¯ h Q KQ(RepQ(G)) 0 K Q
induced by the functor K (r−1) is injective. This is true, as follows
0
(cid:3)
from Proposition 1.20 and point 3) of Proposition 1.8.
Remarks 1.12. 1) The only instance for which one knows that Con-
jecture 1.10 is true is when S(G,X) is the Siegel domain A and
K g
the representation of G = Sp(2g) is the tautological one (see [EV02,
Thm. 1.1], [vdG99, Thm. 1.1]). Then the flat vector bundle [E] on A
g
is the Gauß-Manin bundle of the relative de Rham cohomology H1 of
the universal abelian scheme. The family is defined only over a level
structure, but the Gauß-Manin bundle, together with its Gauß-Manin
8 HE´LE`NEESNAULTANDMICHAEL HARRIS
connection, descends to A . In this case the vanishing is even stronger:
g
on the finite cover over which the local monodromies are unipotent,
the Deligne extension of the Gauß-Manin bundle has vanishing Chern
classes in the rational Chow groups.
2) To tie up with the conjecture on flat bundles alluded to in the
introduction, we remark that the automorphic vector bundles [E] in
K
the conjecture are those coming from Rep (G) which have a G(A )-
Q¯ f
equivariant integrable connection [Har85, Lemma 3.6].
3) Let c : CH∗( S(G,X)) → H2∗( S(G,X),∗) be the cycle ho-
D K Q D K
momorphism into Deligne cohomology [EV88, Section 7]. We prove in
Theorem 2.3 that
c ◦c>0| = 0.
D K RepQ¯(G)
Using Claim 1.11 in addition, one sees that Conjecture 1.10 when
S(G,X) is projective is a special case of Beilinson’s motivic con-
K
jecture discussed in the introduction.
4) We are not able to prove Conjecture 1.10. In fact, apart from the
example mentioned in 1), we can not prove vanishing in any other
example.
Let us denote by S(G,X) the model of the Shimura variety
K E(G,X)
over its reflex field. If
c : CH∗( S(G,X) ) → H2∗ ( S(G,X) ,Q (∗))
ℓ K E(G,X) Q cont K E(G,X) ℓ
denotesthecyclehomomorphismtocontinuousℓ-adiccohomology[Jan87,
Section 2], we prove
c ◦c>0| = 0.
ℓ K RepQ¯(G)
in Theorem 3.3. (More precisely, we prove this after replacing E(G,X)
by the finite extension E′ of Lemma 1.4.) In particular, we verify
h
Conjecture 0.1 in this case. To our knowledge, the examples treated in
this note are the first that confirm this prediction for cycle classes of
flat bundles that do not depend on knowing in advance that the Chow
class itself vanishes.
1.2. Hecke operators. Fix K ⊂ G(A ). Let g ∈ G(A ) and consider
f f
K = K ∩gKg−1 ⊂ K. Let
g
π = π : S(G,X) → S(G,X),
1,g K,Kg Kg K
defined as above. Right multiplication by g defines an isomorphism
∼
rg :gKg−1 S(G,X) −→ KS(G,X).
Let
π2,g = rg ◦πgKg−1,Kg :Kg S(G,X) → KS(G,X).
CHERN CLASSES OF AUTOMORPHIC VECTOR BUNDLES 9
We first observe that
(1.13) T(g)[E] ∼= ⊕[K:Kg][E]
K 1 K
as vector bundles, where T(g) = π ◦ π∗ . Indeed, the definition
2,g∗ 1,g
implies rg∗[E]K ∼= [E]g−1Kg, and formula (1.6) implies formula (1.13).
Both π an π are finite´etale morphisms. So for any contravariant
1,g 2,g
cohomology theory H which has push-downs for proper (or even only
finite ´etale) morphisms, one can define the Hecke operator
(1.14) T(g) : H( S(G,X)) −π−2,−g∗−◦π−1∗→,g H( S(G,X)).
K K
We shall use the Hecke operators on Chow groups, on continuous
ℓ-adic cohomology, on Deligne cohomology, on syntomic cohomology.
All of them are considered rationally. In particular, they are (possibly
infinite) dimensional vector spaces over Q, and the Hecke algebra splits
those cohomologies as a sum of generalized eigenspaces.
A rational prime number q is unramified for K if there exists a K ⊂
q
K with K ⊂ ∩G(Q ) a hyperspecial compact open subgroup, and
q q
ramified otherwise. There is a finite set S(K) of ramified primes. We
letH denotetheQ-subalgebraoftheringtensor Qofcorrespondences
K
generated by the T(g), where g runs through elements of G(Q ) with
q
q ∈/ S(K); this is well-known to be a commutative algebra.
The following is obvious:
Lemma 1.15. Let R be a ring, a ∈ R, and let [a] : S(G,X) → R
K
be the constant function with value a. Then [a] is an eigenfunction for
every T(g), with eigenvalue v(K,g) = [K : K ].
g
ForanycohomologytheoryH∗( S(G,X)),includingtheChowgroups,
K
let H∗( S(G,X)) ⊂ H∗( S(G,X)) be the eigenspace for the volume
K v K
character T(g) 7→ v(K,g) of H . As H∗( S(G,X)) is a possibly infi-
K K
nite dimensional Q-vector space, so is H∗( S(G,X)) .
K v
Lemma 1.16. Let E ∈ Vect (Xˆ). Then for any open compact sub-
G
group K ⊂ G(A ), ch([E] ) ∈ CH( S(G,X)) .
f K K v
Proof. As π is finite ´etale, its Todd class is equal to 1, thus the
2,g
Grothendieck-Riemann-Roch theorem implies
ch(π π∗ [E] ) = π (ch(π∗ [E] )·1),
2,g∗ 1,g K 2,g∗ 1,g K
from which we conclude using formula (1.13)
ch(T(g)[E] ) = π (ch(π∗ [E] )·1) = π π∗ ch([E] ) = T(g)ch([E] ).
K 2,g∗ 1,g K 2,g∗ 1,g K K
10 HE´LE`NEESNAULTANDMICHAEL HARRIS
Applying (1.6) and (1.7) one concludes
(1.17) [K : K ]ch([E] ) = T(g)ch([E] ).
g K K
(cid:3)
Corollary 1.18. Let H∗ be a cohomology theory which, for any open
compact subgroup K ⊂ G(A ), admits a cycle map
f
cK : CH( S(G,X)) → H( S(G,X))
H K K
which commutes with the action of H . Then for any E ∈ Vect (Xˆ),
K G
cK ◦ch([E] ) ∈ H∗( S(G,X)) .
H K K v
ThecohomologytheoriesHi(−,j)withcoefficientsinacharacteristic
0 field F considered in this note are all functorial. Thus to check that
the cycle map commutes with the action of H , it suffices to verify
K
compatibility with the push-down via π . In all those cohomology
2,g∗
theories, the cycle map can be defined via purity: for Z = PimiZi a
codimensionncycleonasmoothY, wherem ∈ Zandthecodimension
i
ncyclesZ areprime, theGysinmorphismexistsandisanisomorphism
i
∼
γ : ⊕K ·[Z ] −→= H2n(Y,n).
i Z
See [EV88, Section 7] for Betti, de Rham and Deligne cohomology,
[Jan88, Thm. 3.23] for continuous ℓ-adic cohomology. Thus compati-
bility reduces to showing
γ(π [Z ]) = γ(deg(k(Z )/k(π(Z )))·[π(Z )]) =
∗ i i i i
π γ([Z ]) ∈ H2n (Y′,n)
∗ i p(Zi)
for a finite surjective morphism p : Y → Y′. In Deligne cohomology,
it follows as the cohomology verifies the Bloch-Ogus axioms. In ℓ-
adic cohomology, for lack of reference, we restrict ourselves to the case
where Y is defined over a number field k. Let Y be a flat smooth model
over a non-trivial open U in the spectrum of the ring of integers of k.
Then the cycle class in H2n(Y,n) is just the standard cycle class from
[SGA4.5], for which the trace properties are known [SGA4.5, 2.3].
Corollary 1.19. Corollary 1.18 holds true for Deligne cohomology and
continuous ℓ-adic cohomology.
In the next section we study H∗( S(G,X)) for Betti and ℓ-adic
K v
cohomology.