Journal de Théorie des Nombres de Bordeaux

Christine Huyghe

Comparison theorem between Fourier transform and Fourier transform with compact support

Victoria Zhuravleva

Diophantine approximations with Fibonacci numbers

Matilde N. Lalín

Equations for Mahler measure and isogenies

Xavier Xarles

Trivial points on towers of curves

Thomas Christ

Justas Kalpokas

Lower bounds of discrete moments of the derivatives of the Riemann zeta-function on the critical line

Paul Mercat

Construction de fractions continues périodiques uniformément bornées

Víctor Cuauhtemoc García

On the distribution of sparse sequences in prime fields and applications

Jean-Claude Douai

Jean-Claude DOUAI
2-Cohomology of semi-simple simply connected group-schemes over curves
defined over p-adic fields
Tome 25, no 2 (2013), p. 307-316.
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Journal de Théorie des Nombres
de Bordeaux 25 (2013), 307–316
2-Cohomology of semi-simple simply connected
group-schemes over curves defined over
p-adic fields
par Jean-Claude DOUAI
Résumé. Soit X une courbe propre, lisse, géométriquement con-
nexe, définie sur un corps p-adique k. Lichtenbaum a prouvé l’exis-
tence d’une dualité parfaite :
Br(X)× Pic(X) −→ Q/Z
entre le groupe de Brauer et le groupe de Picard de X et en a
déduit l’existence d’une injection de Br(X) dans le produit des
Br(kP ) où P décrit les point fermés de X et kP désigne le corps
résiduel du point P . Le but cet article est de montrer que si,
G = G̃ est un Xet-schéma en groupes semi-simples simplement
connexes (groupes s.s.s.c), alors le résultat de Lichtenbaum im-
plique la neutralité de chaque Xet-gerbe qui est localement liée
par G̃. En particulier, si X est un modèle de X sur l’anneau O des
entiers de k, i.e X = X ×O k, alors chaque Xet-gerbe localement
liée par un X-groupe s.s.s.c est neutre (ceci étant une application
du théorème de changement propre).
Plus généralement, reprenant un procédé du à Colliot-Thélène
et Saito, nous pouvons montrer que si X est une k-variété propre,
lisse, de dimension strictement plus grande que 1, alors chaque
classe du quotient H2(Xet,L)H2(Xet,L) est neutre où X est
un O-modèle de X et L un X-lien localement représentable par
un schéma en groupes s.s.s.c sous la condition mineure que le
cardinal de son centre soit premier à p. Nous donnerons ensuite
des applications.
Abstract. Let X be a proper, smooth, geometrically connected
curve over a p-adic field k. Lichtenbaum proved that there exists
a perfect duality:
Br(X)× Pic(X) −→ Q/Z
between the Brauer and the Picard group of X, from which he
deduced the existence of an injection of Br(X) in
∏
P∈X
Br(kP )
where P ∈ X and kP denotes the residual field of the point P .
The aim of this paper is to prove that if G = G̃ is an Xet- scheme
of semi-simple simply connected groups (s.s.s.c groups), then we
308
can deduce from Lichtenbaum’s results the neutrality of everyXet-
gerb which is locally tied by G̃. In particular, if X is a model of
X over the ring of integers O in k, i.e X = X ×O k, then every
Xet-gerb which is locally tied by a s.s.s.c X-group is neutral (this
being a variant of the proper base change theorem).
More generally, using a technique of Colliot-Thélène and Saito,
we can prove that, if X is a proper smooth k-variety of dimen-
sion greater than 1, then every class of H2(Xet,L)H2(Xet,L) is
neutral whenever L is a X-band that is locally represented by a
s.s.s.c group under the condition that the cardinality of its center
is coprime to p. We will then give some applications.
Je tiens à remercier le referee pour ses riches observations. Je tiens aussi
à remercier tout particulièrement Benaouda Djamai pour son aide dans la
réalisation de ce papier.
1. Moduli field, definition field
Let k be a p-adic field (= finite extension of Qp), k/k a separable closure
of k, X a projective, smooth, geometrically irreducible curve over k and
X = X
⊗
k k.
Theorem 1.1. Let G̃ be a scheme of semi-simple, simply connected (or
s.s.s.c) groups over X/k, G̃ = G̃
⊗
k k. Then, there are the isomorphisms:
H1(X, G̃) ' H0(k,H1(Xet, G̃)) ' H0(k,H1(XZar, G̃))
where H1(X,G) means H1et(X,G), i.e each X-torsor under G̃ of moduli
field k is defined over k and the descent from k is unique.
Remark. The second isomorphism comes from Nisnevich’s isomorphism
(Corollaire 4.4 de [9]):
H1(Xet, G̃) ' H1(XZar, G̃)
Proof. Leray’s spectral sequence
Hp(k,Hq(X, G̃))⇒ Hp+q(X, G̃)
does not exist but we get the exact sequence (Prop. 3.1.3-Chap.V, p.323
of Giraud [4]):
1 // H1(k, G̃) // H1(X, G̃) // H0(k,H1(X̄, ¯̃G)) .
To each class [Z] of H0(k,H1(X̄, ¯̃G)), we associate the k-gerb GZ whose
objects over the open set
U = (SpecK −→ Spec k) ∈ Ob(Spec(k)et)
2-cohomology of schemes. . . 309
are the X ⊗k K-torsors under G̃ which are isomorphic to Z/X⊗kK . The
k-gerb GZ belongs to Z2(k,L), where the k-band L is locally (for the etale
topology) represented by G̃. 
Because k is a p-adic field and G̃ semi-simple, GZ is neutral (cf. [3],[3′],[1])
and by the Prop.3.1.6 (i) of Chap.V, p.325 of Giraud [4], [Z] belongs to
the image of H1(X, G̃).
Because G̃ is s.s.s.c, H1(k, G̃) = 0 by Kneser’s theorem 1 of [5]. From
this, it results that H1(X, G̃) ' H0(k,H1(X̄, ¯̃G)).
Remark. If G is only semi-simple (s.s), then we get the exact sequence:
1 // H1(k,G) // H1(X,G) // // H0(k,H1(X̄Zar, Ḡ)) // ∗
i.e the descent is no longer unique.
2. The main Theorem
Let k be a p-adic field, X a projective, smooth, geometrically irreducible
curve over k. Lichtenbaum [6] proved that there exists a perfect duality:
Br(X)× Pic(X) −→ Q/Z
between the Brauer and the Picard group of X, from which he deduced the
existence of an injection
Br(X) ↪→
∏
P∈X
Br(kP )
where kP denotes the residual field of the point P .
Theorem 2.1. Let k be a p-adic field, X a projective, smooth, geometri-
cally irreducible curve over k, G̃ being a s.s.s.c X-group, L a X-band that
is locally (for the etale topology) representable by G̃. Then each class of
H2et(X,L) is neutral.
First, we shall prove Theor.2.1. In the §3, we shall see (Rem. a))
that theor.3.1 gives again theor.2.1 but only modulo the restriction
(|Z(L)|, p) = 1, where Z(L)= center of L.
Proof of theor. 2.1. As in [3], we can suppose that L has the form lien(G̃)
and then establish the neutrality of each class of H2(X, G̃) with s.s.s.c and
quasi-split G̃. Let (B̃, T̃ ) be a Killing pair of G̃: by the prop.3-13 exposé
XXIV of [8], T̃ is isomorphic to an induced torus
∏
X′/X
GmX′ , where X
′ is
the X-scheme of Dynkin of G̃. The injectivity of
Br(X ′) ↪→
∏
P ′∈X′
Br(kP )
310 Jean-Claude Douai
induces the injectivity of
H2(X, T̃ ) ↪→
∏
P∈X
H2(kP , T̃ ).
Consider the diagram
0

H1(X, T̃ad)

δ1 // H2(X,Z(G̃))

// H2(X, T̃ ) _
∏
P∈X
H1(kP , T̃ad) //
∏
P∈X
H2(kP , Z(G̃)) //
∏
P∈X
H2(kP , T̃ )
0
∗=
(D2.1)
The quotient of H2(X,Z(G̃)) by Im.δ1 is injected into
∏
P∈X
H2(kP , Z(G̃)).
The set H2(X, G̃) is a homogeneous principal space under H2(X,Z(G̃)).
By analogy, H2(X, G̃)/Im.δ1 is injected into
∏
P∈X
H2(kP , G̃) and we obtain
the following diagram:
H2(X, G̃)/Im.δ1 
 //
∏
P∈X
H2(kP , G̃)
H2(X,Z(G̃))/Im.δ1
	

OO
  //
∏
P∈X
H2(kP , Z(G̃))
	

OO
(D2.2)
where 	
 // is the relation of the non abelian 2-cohomology ([4], Def.
3.1.4 Chap IV p.248).
We know ([3], [1]) that each class of H2(kP , G̃) is neutral, therefore each
class of H2(kP , G̃) is in relation with the class 0 of H2(kP , Z(G̃)) (Cor.
3.3.7 Chap IV, p.258-259 of [4]).
From the previous diagram, we deduce that each class ofH2(X, G̃)/Im.δ1
is in relation with the class 0 of H2(X,Z(G̃))/Im.δ1, hence is neutral.
More, it is easy to see that each class of H2(X, G̃) which is in relation
by 	
 // with a class of Im.δ1 is also neutral (Let α ∈ H1(X, T̃ad): the
∗ T̃ad is also an induced torus
2-cohomology of schemes. . . 311
X-gerb δ1(α) ·Tors G̃ is X-equivalent with the gerb Tors(G̃α
′) where α′
is the image of α in H1(X, G̃ad), then is neutral). The result follows from
these facts. 
Corollary 2.1 (to theorem 2.1). Let L be a Xet-band which is locally
representable by s.s.s.c group. Denote
X2(k(X),L) := Ker
{
H2(k(X),L) −→
∏
P∈X
H2(k(X)P ,L)
}
Then, each class of X2(k(X),L) is neutral.
This result is a consequence of the exactness of the localization sequence
H2et(X,L) −→ H2(k(X),L) −→
∏
P∈X
H2(k(X)P ,L),
each class of H2et(X,L) being neutral ( H2(k(X)P ,L) is pointed by
Tors(G̃L), where G̃L represents the k(X)P -band L, cf. the analogy with
the second line of the diagram D1 of [3′′] p.124 ).
3. Higher dimensional varieties
J.L Colliot-Thélène and S.Saito have generalized in [2] Lichtenbaum’s
duality for higher dimensional varieties : the evaluation map on the closed
points together with the corestriction defines a pairing
(3.1) Br(X)× CH0(X) −→ Br(k)
(cf. the "Φ-Eigenshaft" property of Pop and Wiesend [7] ).
More generally, if X is a smooth, projective, geometrically irreducible k-
variety of dimension > 1, k always p-adic field, they have established ([2],
Cor. 2.4) that
Br(X)/Br(X) ↪→
∏
P∈X
Br(kP ) (modulo p-primary torsion)
where X is a model of X over the ring Ok of integers of k.
Adapting their method we get:
Theorem 3.1. Let X be a projective, smooth, geometrically irredu-
cible scheme of dimension > 1 over a p-adic field. Then each class of
H2et(X,L)/H2et(X,L) is neutral, where L is a X-band which is locally repre-
sentable by a scheme of s.s.s.c groups, (|Z(L)|, p) = 1.
Proof of theor.3.1. We will continue on the lines of the proof of the theor.
2.1.
Consider the following diagram (D3.1) that extends diagram (D2.1) in
the proof of the theor. 2.1.
312 Jean-Claude Douai
H1(X, T̃ad) //

H2(X, Z(G̃)) //

H2(X, T̃ ) = Br(X′)

H1(X, T̃ad)

δ1 // H2(X, Z(G̃))

// H2(X, T̃ ) = Br(X′)

H1(X, T̃ad)
H1(X, T̃ad)
δ1 // H2(X, Z(G̃))
H2(X, Z(G̃))
//

H2(X, T̃ )
H2(X, T̃ ) _
← modulo p-primary
torsion
∏
P∈X
H
1(kP , T̃ad) //
∏
P∈X
H
2(kP , Z(G̃)) //
∏
P∈X
H
2(kP , T̃ ) =
∏
P ′∈X′
Br(kP ′ )
0
=
(D3.1)
where X
′
k is defined by T̃ =
∏
X′X
Gm′X , as in the proof of the theor. 2.1
and X′ is an Ok-model of X ′. 
From the diagram (D3.1), we deduce the following diagram (D3.2) corre-
sponding with (D2.2):
H2(X, G̃)H2(X, G̃)Im.δ1
 
modulo p-primary
torsion
↓ //
∏
P∈X
H2(kP , G̃)
H2(X,Z(G̃))H2(X, Z(G̃))Im.δ1
 
↑
modulo
p-primary
torsion
//
	

OO
∏
P∈X
H2(kP , Z(G̃))
	

OO
(D3.2)
We conlude as in theor.2.1, but modulo the p-torsion: each class
of H2(X, G̃)H2(X, G̃)Im.δ1
is in relation mod. p-torsion with
the class 0 of H2(X,Z(G̃))H2(X, Z(G̃))Im.δ1
and each class of
2-cohomology of schemes. . . 313
H2(X, G̃)H2(X, G̃)Im.δ1
which is in relation by 	
 // with a class
of Im.δ1 is also neutral.
Remarks. a) IfX is a projective, smooth, geometrically irreducible curve
over a p-adic field k, Br(X) = Br(X′) = 0: the cohomology of X comes
from the cohomology of the special fiber by " proper base change " the-
orem and each class of H2(X,L) is neutral (cf. Theor. 1.3 of [3′′]). By
Lichtenbaum’s theorem, Br(X ′) ↪→
∏
P ′∈X′
Br(k′P ) and the restriction "
mod. p-torsion " is no more necessary.
b) Br(X) is the kernel of the pairing (3.1) and H2(X, G̃) (resp. H2(X,L))
is the kernel of the application
H2(X, G̃)× CH0(X) −→ H2(k, G̃)
(resp. H2(X,L)×CH0(X) −→ H2(k,L)) in the non abelian sense, i.e
H2(X, G̃) (resp. H2(X,L) ) is sent to the unit class of H2(k, G̃) pointed
by Tors G̃ (resp.of H2(k,L) pointed by Tors G̃L where G̃L represents
L).
4. Application
From Theorem 1.1 and Theorem 2.1, we obtain the following diagram
(D4.1) where X is a proper, smooth, geometrically connected curve over a
p-adic field k, and where G is a semi-simple group and G̃ its universal
covering, µ : G̃ −→ G:
314 Jean-Claude Douai
1
=
H
1
(k
,G̃
)

// H
1
(X
,G̃
)

'
th
eo
r.
1
// H
0
(k
,H
1 Z
a
r
(X̄
,
¯̃ G
))

// ∗
H
1
(k
,G
)
' 
// H
1
(X
,G
)
////
 
H
0
(k
,H
1 Z
a
r
(X̄
,Ḡ
))
 
// ∗
0
&&MMMMMMMMMMM
H
1
(k
,H
1
(X̄
,µ̄
))
))SSSSSSSSSSSSSSSSS

:
:
:
:
:
:
:
:
:
:
:
:
:
:
H
2
(k
,µ
)
	
 
// H
2
(X
,µ
)
  
// E
))SSSSSSSSSSSSSSSS
//
  
∗
H
0
(k
,H
2
(X̄
,µ̄
))
%%KKKKKKKKKKKK
H
2
(k
,G̃
)
//
[3
]
=
H
2
(X
,G̃
)
//
th
eo
r.
2
.1
=
H
1
(k
,H
1
(X̄
,
¯̃ G
))
//
=
∗
0
∗
∗
?
G
//
/o
/o
/o
/o
/o
/o
/o
/o
/o
/o
/o
/o
/o
/o
/o
/o
/o
/o
/o
/o
/o
/o
/o
/o
/o
/o
G
er
(G
)
(D
4.
1
)
2-cohomology of schemes. . . 315
Here Ger(G) is the sheaf of maximal subgerbs of the direct image of G
by X −→ Spec(k) (cf. [4], p.131 and p.327, Ex 3.1.9.2).
E is calculated by the following diagram (D4.2)
0

0

0

H2(k, µ)
=
// H2(X,µ)tr _

// H1(k,H1(X̄, µ̄))

//

""""
0
H2(k, µ)

// H2(X,µ)

//

&&&&
E //

&&&&

0
0 //
	


H0(k,H2(X̄, µ̄)) = H0(k,H2(X̄, µ̄))

%%%%
// 0
H2(k, G̃) //
[3]=
H2(X, G̃) //
=
H1(k,H1(X̄, ¯̃G)) = ?
∗
H2(X, G̃)tr //
=
H0(k,H2(X̄, ¯̃G)) = ∗
∗
(D4.2)
Note : H2(X, ·)tr := Ker
{
H2(X, ·) −→ H0(k,H2(X̄, ·))
}
(cf.chap V,
no 3.1.9.3 de [4]).
References
[1] M. V. Borovoi, Abelianization of the second non abelian Galois cohomology. Duke Math-
ematical journal Vol. 72 (1993), N° 1, 217–239.
[2] J.-L. Colliot-Thélène et S. Saito, Zero-cycles sur les variétés p-adiques et groupe de
Brauer. IMRN 4 (1996).
316 Jean-Claude Douai
[3] J.C. Douai, 2-cohomologie galoisienne des groupes semi-simples definis sur les corps locaux.
C.R Acad. Sci. Paris Série A 280 (1975), 321–323.
[3′] J.C. Douai, Sur la 2-cohomologie galoisienne de la composante résiduellement neutre des
groupes réductifs connexes définis sur les corps locaux. C.R Acad. Sci. Paris Série I 342
(2006), 813–818.
[3′′] J.C. Douai, Sur la 2-cohomologie non abélienne des modèles réguliers des anneaux locaux
henseliens. Journal de Théorie des Nombres de Bordeaux 21 (2009), 119–129.
[4] J.Giraud, Cohomologie non abélienne. Grundlehren Math. Wiss. 179, Springer Verlag,
Berlin, 1971.
[5] M.Kneser,Galois-Kohomologie halbeinfacher Gruppen über p-adishe Körpen II. Math. Zeit.
89 (1965), 250–272.
[6] S. Lichtenbaum, Duality theorems for curves over p-adic fields. Inv. Math. 7 (1969), 120–
136.
[7] G. Wiesend, Local-Global Prinzipien für die Brauergruppe. Manuscripta math. 86 (1995),
455–466.
[8] S.G.A.D., Séminaire de géometrie algébrique 1963-1964. Lectures Notes in Math., 151–153,
Springer, 1970.
[9] Y. A. Nisnevich, Espaces principaux rationnellemnt triviaux et arithmétique des schémas
en groupes réductifs sur les anneaux de Dedekind. C. R. Acad. Sc. Paris Série I 299 (1984),
No 1, 5–8.
Jean-Claude Douai
UFR de Mathématiques
Laboratoire Paul Painlevé CNRS UMR 8524
Université de Lille 1
59665 Villeneuve d’Ascq Cedex
E-mail: douai@math.univ-lille1.fr


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