We introduce the notion of a quasi-connected reductive group over a field of characteristic 0 to be an almost direct product of a connected semisimple group and a quasi-torus (a group of multiplicative type). We show that a linear algebraic group is reductive and quasi-connected if and only if it is isomorphic to a normal subgroup of a connected reductive group. We compute the first Galois cohomology set H 1 (R, G) of a quasi-connected reductive group G over the field R of real numbers in terms of a certain action of a subgroup of the Weyl group on the Galois cohomology of a fundamental quasi-torus of G.
Let k be a field of characteristic zero and k an algebraic closure of k. For a geometrically integral variety X over k, we write k(X) for the function field of X = X × k k. If X has a smooth k-point, the natural embedding of multiplicative groups k * ֒→ k(X) * admits a Galois-equivariant retraction. In the first part of the paper, over local and then over global fields, equivalent conditions to the existence of such a retraction are given. They are expressed in terms of the Brauer group of X.In the second part of the paper, we restrict attention to varieties which are homogeneous spaces of connected but otherwise arbitrary algebraic groups, with connected geometric stabilizers. For k local or global, for such a variety X, in many situations but not all, the existence of a Galois-equivariant retraction to k * ֒→ k(X) * ensures the existence of a k-rational point on X. For homogeneous spaces of linear algebraic groups, the technique also handles the case where k is the function field of a complex surface. RésuméSoient k un corps de caractéristique nulle et k une clôture algébrique de k. Pour une k-variété X géométriquement intègre, on note k(X) le corps des fonctions de X = X × k k. Si X possède un k-point lisse, le plongement naturel de groupes multiplicatifs k * ֒→ k(X) * admet une rétractionéquivariante pour l'action du groupe de Galois de k sur k.Dans la première partie de l'article, sur les corps locaux puis sur les corps globaux, on donne des conditionséquivalentesà l'existence d'une telle rétractionéquivariante. Ces conditions s'expriment en terme du groupe de Brauer de la variété X.Dans la seconde partie de l'article, on considère le cas des espaces homogènes de groupes algébriques connexes, non nécessairement linéaires, avec groupes d'isotropie géométriques connexes. Pour k local ou global, pour un tel espace homogène X, 1 dans beaucoup de cas mais pas dans tous, l'existence d'une rétractionéquivariantè a k * ֒→ k(X) * implique l'existence d'un point k-rationnel sur X. Pour les espaces homogènes de groupes linéaires, la technique permet aussi de traiter le cas où k est un corps de fonctions de deux variables sur les complexes.
For a smooth geometrically integral variety X over a field k of characteristic 0, we introduce and investigate the extended Picard complex UPic(X). It is a certain complex of Galois modules of length 2, whose zeroth cohomology is k[X] × /k × and whose first cohomology is Pic(X), where k is a fixed algebraic closure of k and X is obtained from X by extension of scalars to k. When X is a k-torsor of a connected linear k-group G, we compute UPic(X) = UPic(G) (in the derived category) in terms of the algebraic fundamental group π 1 (G). As an application we compute the elementary obstruction for such X.
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