On the equivalence of integral representations of groups

Abstract

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