Tensor powers of modules over finitely generated abelian groups

Abstract

In an earlier paper [ 11, the authors investigated, as a major component of their final result, a special case of the following question. Let R be a commutative ring with unity, G a finitely generated abelian group and A4 a finitely generated RG-module. Consider the tensor power @“, M as an RG-module via the diagonal action of G. Under what conditions is @“, M a finitely generated RG-module? The answer obtained in [ 1 ] was for the case when R is a field and used an invariant Z;M introduced by the first author and Strebel in [3]. We briefly describe CL. Firstly, if n is the torsion-free rank of G, we can identify Hom(G, [w) with [w”. Defining two elements of Hom(G, Iw) to be equivalent if they differ by a positive real multiple, we obtain a natural projection of Hom(G, [w)\(O) onto the sphere s” I. Denote the equivalence class of XE Hom(G, 1w) by [x] and define GcXI = G,= { ge G: x(g) 30). Then Z:, is the subset of S”’ consisting of all those [x] for which M is a finitely generated RG,-module and CL is the complement of C, in S”‘. We say that M is k-tame if there do not exist [x, I,..., [xk ] in Zh such that x1 + “. + xk = 0. Then the partial answer referred to above (Theorem 3.4 of [l]) is