Let $G$ be a finite group and $k$ a field whose characteristic does not divide the order of $G$. Then if $M,N$ are $k[G]$-modules which are finitely generated as $k$-module, then $$\text{Ext}^i(M,N)=\text{H}^i(k,\text{Hom}_k(M,N))=H^i(G,\text{Hom}_k(M,N)).$$ Since $\text{Hom}_k(M,N)$ is finitely generated as a $k$-module, we can conclude that $\text{Ext}^i(M,N)$ is finitely generated and vanishes for sufficiently large $i$ (depending on $M,N$).
My question is, how can we extend this?
The particular case I have in mind are algebraic groups (well besides the case of algebraic groups with finite set of rational points).