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

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 -