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Comptes Rendus Mathématique
Volume 339, n° 5
pages 331-334 (septembre 2004)
Doi : 10.1016/j.crma.2004.06.012
Received : 1 June 2004 ;  accepted : 8 June 2004
Résolutions flasques des groupes réductifs connexes
Flasque resolutions for connected reductive algebraic groups.
 

Jean-Louis Colliot-Thélène
CNRS UMR 8628, mathématiques, bâtiment 425, université Paris-Sud, 91405 Orsay, France 

Résumé

Tout groupe réductif connexe G sur un corps k (de caractéristique nulle) peut sʼécrire comme un quotient  , où S est un k -tore flasque central dans un k -groupe réductif H extension dʼun k -tore quasitrivial par un k -groupe semisimple simplement connexe. De telles présentations de G permettent de simplifier lʼétude du groupe   des points rationnels de G , de la cohomologie galoisienne de G et du groupe de Brauer dʼune compactification lisse de G . Pour citer cet article : J.-L. Colliot-Thélène, C. R. Acad. Sci. Paris, Ser. I 339 (2004).

The full text of this article is available in PDF format.
Abstract

A connected reductive group G over a (characteristic zero) field k may be written as a quotient  , where the k -group H is an extension of a quasitrivial torus by a simply connected group, and S is a flasque k -torus, central in H . Such presentations   lead to a simplified approach to the Galois cohomology of G and related objects, such as the Brauer group of a smooth compactification of G . When k is a number field, one also recovers known formulas, in terms of S , for the quotient of the group   of rational points by R -equivalence, and for the Abelian groups which measure the lack of weak approximation and the failure of the Hasse principle for principal homogeneous spaces. To cite this article: J.-L. Colliot-Thélène, C. R. Acad. Sci. Paris, Ser. I 339 (2004).

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