Tensor powers of modules over finitely generated abelian groups
Let R be a commutative Noetherian ring with unity, G a finitely generated abelian group, and M a finitely generated RG-module. In this paper the authors provide a necessary and sufficient condition for a tensor power \(\otimes^ k_ RM\) to be a finitely generated RG-module under the diagonal action of G. The paper uses a generalization of an invariant that was introduced earlier by \textit{R. Bieri} and \textit{R. Strebel} [Proc. Lond. Math. Soc., III. Ser. 41, 439-464 (1980; Zbl 0448.20029)]. For any commutative ring S, denote by V(S) the set of all real valuations of S (in the sense of Bourbaki). Let \(\kappa\) : RG\(\to A=RG/Ann_{RG}M\) be the natural projection. For \(\nu\in V(R)\), define \(\Delta_ M^{\nu}(G)=\{\omega \circ \kappa |\) \(G\in Hom(G,{\mathbb{R}}):\omega\in V(A)\) and \(\omega \circ \kappa | R=\nu \}\). A subset \(\Delta\subseteq Hom(G,{\mathbb{R}})\) is said to be k-tame if, for \(\chi_ 1,...,\chi_ k\in \Delta\), \(\chi_ 1+...+\chi_ k=0\) implies \(\chi_ 1=...=\chi_ k=0\). Theorem: \(\otimes^ k_ RM\) is finitely generated if and only if \(\Delta_ M^{\nu}(G)\) is k-tame for all discrete, nonnegative \(\nu\in V(R)\).
scientific article; zbMATH DE number 4156788