I'm wondering if there is an example of a nilpotent not abelian group $G$ acting linearly on a infinite dimensional vector space (over some field $F$) $M$ such that $H^{0}(G,M)=0$ and $H^{i}(G,M)\neq 0$ for some $i\geq 0$.
It is know by a theorem due to Morris Hirsch that: if $M$ is finite dimensional , there is no such and example because if $H^{0}(G,M)=0$ then $H^{i}(G,M)=0$ for all $i\geq 0$. (Theorem C of the article "Flat Manifolds And The Cohomology Of Groups")