Vanishing of certain cohomology sets for \(\text{SL}_n(R_{\mathcal M})\). (English) Zbl 1003.20040

Let \((R,{\mathcal M)}\) be a complete local ring with finite residue field of odd characteristic \(p\). Let the group \(\langle g\rangle\) of order two with generator \(g\) act on \(\text{SL}_n(R)\) by \(g(A)=(A^{-1})^t\). So the group of fixed points is \(\text{SO}_n(R)\). The author shows that \(H^1(\langle g\rangle,\text{SL}_n(R))\) vanishes. This is then used to compute the order of \(\text{SO}_2(\mathbb{Z}/p^m)\).


20G10 Cohomology theory for linear algebraic groups
20G35 Linear algebraic groups over adèles and other rings and schemes
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