Exercise 9, § 6

a) Let $G$ be a group, $M$ a $G$-module, $g$ an element of the center of $G$ such that the endomorphism $m\mapsto gm-m$ of $M$ is bijective. Prove that the groups $H^p(G,M)$ and $H_p(G,M)$ are zero for $p\geq 1$.

b) Let $V$ be a vector space over a field $k$, with $\operatorname{Card}(k)\geq 3$. Prove that the groups $H^p(\operatorname{GL}(V),V)$ and $H_p(\operatorname{GL}(V),V)$ are zero for $p\geq 1$.