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Galois 2-cohomology of the residually neutral component of connected reductive groups defined over local fields. (Sur la 2-cohomologie galoisienne de la composante résiduellement neutre des groupes réductifs connexes définis sur les corps locaux.) (French) Zbl 1101.11014

Summary: Let \(K\) be a complete discrete valuation field, \(k\) its residue field, \(\mathcal O\) its ring of integers, and \(G\) a connected reductive \(K\)-group. Bruhat and Tits have defined the residually neutral component \(G^{00}\) of \(G\) and have calculated \(H^{1}(K,G^{00})\). The aim of this Note is to calculate the Galois 2-cohomology of \(G^{00}\). We extend our results where the case \(G = \widetilde G\) simply connected and \(k\) of cohomological dimension \(\leqslant 1\) is treated. We show that each class of Galois 2-cohomology in \(G^{00}\) reduces to a class into a maximal \(k\)-torus of the special fiber of an \(\mathcal O\)-model of \(G\). We deduce, in particular, that, if \(G\) is a connected reductive \(K\)-group and if \(cd \cdot(k) \leqslant 1\), then each class of Galois 2-cohomology into the residually neutral component is neutral.

MSC:

11E72 Galois cohomology of linear algebraic groups
12G05 Galois cohomology
20G10 Cohomology theory for linear algebraic groups
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