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.
@article{1148392980,
author = {Gajcowski, Nicholas Hine},
title = {Vanishing of certain cohomology sets for $SL\_n(R\_{\mathcal {M}})$},
journal = {Proc. Japan Acad. Ser. A Math. Sci.},
volume = {77},
number = {10},
year = {2001},
pages = { 157-160},
language = {en},
url = {http://dml.mathdoc.fr/item/1148392980}
}
Gajcowski, Nicholas Hine. Vanishing of certain cohomology sets for $SL_n(R_{\mathcal {M}})$. Proc. Japan Acad. Ser. A Math. Sci., Tome 77 (2001) no. 10, pp. 157-160. http://gdmltest.u-ga.fr/item/1148392980/