R'esolutions flasques des groupes lin'eaires connexes

Abstract

A connected reductive group G over a field k may be written as a quotient H/S, where the k-group H is an extension of a quasitrivial torus by a simply connected semisimple group, and S is a flasque k-torus, central in H (a flasque torus is a torus whose cocharacter group is an H^1-trivial Galois lattice). The flasque torus S is well-defined up to multiplication by a quasitrivial torus. Such presentations G=H/S lead to a simplified approach of the Galois cohomology of G and of 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.

49 pages, in French

Cited by in corpus (5)

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• Approximation forte pour les variétés avec une action d'un groupe linéaire
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• Degree 2 cohomological invariants of linear algebraic groups