On the equivalence of integral representations of groups

Let $R$ be a local ring and let $R’$ be a commutative $R$-algebra faithfully flat as an $R$-module. Let $G$ be a finitely generated group and let $M,N$ be $RG$-modules, finitely presented over $R$. Let $M’ = M{ \otimes _R}R’;N’ = N{ \otimes _R}R’$, then $M’,N’$ can be considered as $R’G$-modules. We shall prove that the $R’G$-modules $M’,N’$ are isomorphic if and only if the $RG$-modules $M$ and $N$ are isomorphic. The proof depends on a theorem on noncommutative cohomology which is presented in the first part of the paper.