On the $1$-cohomology of $\mathrm{SL}(n,\mathbb{K})$ on the dual of its adjoint module

Abstract

Given a field $\mathbb{K}$, for any $n\ge 3$ the first cohomology group $H^1(G_n,A_n^{\ast })$ of the special linear group $G_n=\mathrm{SL}(n,\mathbb{K})$ over the dual $A_n^{\ast }$ of its adjoint module $A_n$ is isomorphic to the space $\mathrm{Der}(\mathbb{K})$ of the derivations of $\mathbb{K}$, except possibly when $|\mathbb{K}|\in \{2,4\}$ and $n$ is even. This fact is stated by S. Smith and H. Völklein in their paper "A geometric presentation for the adjont module of ${\mathrm{SL}}_3(k)$" (J. Algebra 127 (1989), 127--138). They claim that when $|\mathbb{K}|>9$ this fact follows from the main result of Völklein's paper "The 1-cohomology of the adjoint module of a Chevalley group" (Forum Math. 1 (1989), 1--13), but say nothing that can help the reader to deduce it from that result. When $|\mathbb{K}|\le 9$ they obtain the isomorphism $H^1(G_n,A_n^{\ast })\cong \mathrm{Der}(\mathbb{K})$ by means of other results from homological algebra, which however miss the case $|\mathbb{K}|\in \{2,4\}$ with $n$ even. In the present paper we shall provide a straightforward proof of the isomorphism $H^1(G_n,A_n^{\ast })\cong \mathrm{Der}(\mathbb{K})$ under the hypothesis $n>3$. Our proof also covers the above mentioned missing case.

24 pages/revised version