Criterion for surjectivity of localization in Galois cohomology of a reductive group over a number field

Borovoi, Mikhail

Comptes Rendus. Mathématique, Tome 361 (2023) no. G9, pp. 1401-1414

Let $G$ be a connected reductive group over a number field $F$, and let $S$ be a set (finite or infinite) of places of $F$. We give a necessary and sufficient condition for the surjectivity of the localization map from $H^1(F,G)$ to the “direct sum” of the sets $H^1(F_v,G)$ where $v$ runs over $S$. In the appendices, we give a new construction of the abelian Galois cohomology of a reductive group over a field of arbitrary characteristic.

Soit $G$ un groupe réductif connexe sur un corps de nombres $F$, et soit $S$ un ensemble (fini ou infini) de places de $F$. On donne une condition nécessaire et suffisante pour la surjectivité de l’application de localisation de $H^1(F,G)$ vers la « somme directe » des ensembles $H^1(F_v,G)$, où $v$ parcourt $S$. Dans les appendices on donne une nouvelle construction de la cohomologie galoisienne abélienne d’un groupe réductif sur un corps de caractéristique quelconque.

Classification : 11E72, 20G10, 20G20, 20G25, 20G30

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