2-Cohomology of semi-simple simply connected group-schemes over curves defined over p -adic fields

Jean-Claude Douai[1]

  • [1] UFR de Mathématiques Laboratoire Paul Painlevé CNRS UMR 8524 Université de Lille 1 59665 Villeneuve d’Ascq Cedex

Journal de Théorie des Nombres de Bordeaux (2013)

  • Volume: 25, Issue: 2, page 307-316
  • ISSN: 1246-7405

Abstract

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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 ) / 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 ( k P ) where P X and k P denotes the residual field of the point P . The aim of this paper is to prove that if G = G ˜ is an X e t - scheme of semi-simple simply connected groups (s.s.s.c groups), then we can deduce from Lichtenbaum’s results the neutrality of every X e t -gerb which is locally tied by G ˜ . 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 𝔛 e t -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 dimension greater than 1, then every class of H 2 ( X e t , ) H 2 ( 𝔛 e t , ) is neutral whenever 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.

How to cite

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Douai, Jean-Claude. "2-Cohomology of semi-simple simply connected group-schemes over curves defined over $p$-adic fields." Journal de Théorie des Nombres de Bordeaux 25.2 (2013): 307-316. <http://eudml.org/doc/275725>.

@article{Douai2013,
abstract = {Let $X$ be a proper, smooth, geometrically connected curve over a $p$-adic field $k$. Lichtenbaum proved that there exists a perfect duality:\[\mathop \{\hbox\{\rm Br\}\}\nolimits (X)\times \mathop \{\hbox\{\rm Pic\}\}\nolimits (X)\rightarrow \mathbb\{Q\}/\mathbb\{Z\}\]between the Brauer and the Picard group of $X$, from which he deduced the existence of an injection of $\mathop \{\hbox\{\rm Br\}\}\nolimits (X)$ in $\displaystyle \{\prod _\{P\in X\} \mathop \{\hbox\{\rm Br\}\}\nolimits (k_P)\}$ where $P\in X$ and $k_P$ denotes the residual field of the point $P$. The aim of this paper is to prove that if $G=\widetilde\{G\}$ is an $X_\{et\}$- scheme of semi-simple simply connected groups (s.s.s.c groups), then we can deduce from Lichtenbaum’s results the neutrality of every $X_\{et\}$-gerb which is locally tied by $\widetilde\{G\}$. In particular, if $\mathfrak\{X\}$ is a model of $X$ over the ring of integers $\mathcal\{O\}$ in $k$, i.e $X=\mathfrak\{X\}\times _\mathcal\{O\}k$, then every $\mathfrak\{X\}_\{et\}$-gerb which is locally tied by a s.s.s.c $\mathfrak\{X\}$-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 dimension greater than 1, then every class of $H^2(X_\{et\},\mathcal\{L\}) \diagup H^2(\mathfrak\{X\}_\{et\},\mathcal\{L\})$ is neutral whenever $\mathcal\{L\}$ is a $\mathfrak\{X\}$-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.},
affiliation = {UFR de Mathématiques Laboratoire Paul Painlevé CNRS UMR 8524 Université de Lille 1 59665 Villeneuve d’Ascq Cedex},
author = {Douai, Jean-Claude},
journal = {Journal de Théorie des Nombres de Bordeaux},
keywords = {semi-simple simply connected group schemes},
language = {eng},
month = {9},
number = {2},
pages = {307-316},
publisher = {Société Arithmétique de Bordeaux},
title = {2-Cohomology of semi-simple simply connected group-schemes over curves defined over $p$-adic fields},
url = {http://eudml.org/doc/275725},
volume = {25},
year = {2013},
}

TY - JOUR
AU - Douai, Jean-Claude
TI - 2-Cohomology of semi-simple simply connected group-schemes over curves defined over $p$-adic fields
JO - Journal de Théorie des Nombres de Bordeaux
DA - 2013/9//
PB - Société Arithmétique de Bordeaux
VL - 25
IS - 2
SP - 307
EP - 316
AB - Let $X$ be a proper, smooth, geometrically connected curve over a $p$-adic field $k$. Lichtenbaum proved that there exists a perfect duality:\[\mathop {\hbox{\rm Br}}\nolimits (X)\times \mathop {\hbox{\rm Pic}}\nolimits (X)\rightarrow \mathbb{Q}/\mathbb{Z}\]between the Brauer and the Picard group of $X$, from which he deduced the existence of an injection of $\mathop {\hbox{\rm Br}}\nolimits (X)$ in $\displaystyle {\prod _{P\in X} \mathop {\hbox{\rm Br}}\nolimits (k_P)}$ where $P\in X$ and $k_P$ denotes the residual field of the point $P$. The aim of this paper is to prove that if $G=\widetilde{G}$ is an $X_{et}$- scheme of semi-simple simply connected groups (s.s.s.c groups), then we can deduce from Lichtenbaum’s results the neutrality of every $X_{et}$-gerb which is locally tied by $\widetilde{G}$. In particular, if $\mathfrak{X}$ is a model of $X$ over the ring of integers $\mathcal{O}$ in $k$, i.e $X=\mathfrak{X}\times _\mathcal{O}k$, then every $\mathfrak{X}_{et}$-gerb which is locally tied by a s.s.s.c $\mathfrak{X}$-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 dimension greater than 1, then every class of $H^2(X_{et},\mathcal{L}) \diagup H^2(\mathfrak{X}_{et},\mathcal{L})$ is neutral whenever $\mathcal{L}$ is a $\mathfrak{X}$-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.
LA - eng
KW - semi-simple simply connected group schemes
UR - http://eudml.org/doc/275725
ER -

References

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  9. G. Wiesend, Local-Global Prinzipien für die Brauergruppe. Manuscripta math. 86 (1995), 455–466. Zbl0846.12007MR1324682
  10. S.G.A.D., Séminaire de géometrie algébrique 1963-1964. Lectures Notes in Math., 151–153, Springer, 1970. 
  11. 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. Zbl0587.14033MR756297

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