## 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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### References:

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