Tensor powers of modules over finitely generated abelian groups
type: Item
instance of: scholarly article
title: Tensor powers of modules over finitely generated abelian groups
zbMATH Open document ID: 0705.20050
DOI: 10.1016/0021-8693(85)90074-2
author: Robert Bieri
author: J. R. J. Groves
published in: Journal of Algebra
publication date: 1985-01-01T00:00:00Z
review text: 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 \(\nu\in V(R)\).
Mathematics Subject Classification ID: 20J05
Mathematics Subject Classification ID: 20C07
Mathematics Subject Classification ID: 20K99
Mathematics Subject Classification ID: 13A18
Mathematics Subject Classification ID: 15A72
Mathematics Subject Classification ID: 15A75
zbMATH DE Number: 4156788
zbMATH Keywords: finitely generated abelian group
zbMATH Keywords: finitely generated RG-module
zbMATH Keywords: tensor power
zbMATH Keywords: diagonal action
zbMATH Keywords: invariant
zbMATH Keywords: real valuations
MaRDI profile type: Publication
full work available at URL: 0021-8693(85)90074-2
OpenAlex ID: W2078570695
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description: scientific article; zbMATH DE number 4156788