Vanishing of certain cohomology sets for $SL_n(R_{\mathcal {M}})$

Abstract

Let $\mathcal{M}$ be a maximal ideal of a commutative ring $R$ such that $\sharp(R / \mathcal{M}) < \infty$ and $\operatorname{char} R / \mathcal{M} \ne 2$. Denoting the $\mathcal{M}$-adic completion of $R$ by $R_{\mathcal{M}}$, we will show $H^1(g, SL_n(R_{\mathcal{M}}))$ vanishes for $g = \langle s \rangle$ acting on $SL_n(R_{\mathcal{M}})$ via $A^s = (A^{-1})^t$ where $t$ is the transpose operator.