trivial modules of group rings

Question

Let $R=\mathbb{F}_p[D]$ where $D$ is a finite group of order prime to $p$. Let $M$ be any simple $R$-module. If one knows that $H^0(D,M)=0$, is $M=0$? If not, under what further conditions can one show $M=0$?

Answer 1

By Maschke's theorem any $\mathbb{F}_p(D)$-module $M$ is a direct sum of simple modules. The condition $H^0(D,M)=0$ only says that $M^D=0$, that is, none of these simple summands is trivial. In your case $M$ has no nontrivial submodules so the submodule $H^0(D,M)=M^D$ is either $M^D=M$ or $M^D=0$. In the former case $M$ is the trivial module or zero. In the latter case (your case) $M$ could be zero, but it could also be any nontrivial simple module, so one cannot deduce that $M=0$.