An undecidability theorem for lattices over group rings
type: Item
instance of: scholarly article
title: An undecidability theorem for lattices over group rings
zbMATH Open document ID: 0893.03013
DOI: 10.1016/S0168-0072(97)00025-0
published in: Annals of Pure and Applied Logic
publication date: 1998-07-20T00:00:00Z
review text: Let \(G\) be a finite group and let \(T(\mathbb{Z} G)\) denote the theory of \(\mathbb{Z} G\)-lattices. The author proves that \(T(\mathbb{Z} G)\) is undecidable if \(G\) contains a subgroup \(S\) which is of any of the forms \(C(p^4)\), \(C(p)\times C(p)\) for \(p\neq 2\), \(C(2)\times C(4)\) where \(C(n)\) denotes the cyclic group of order \(n\) and where \(p\) is a prime. The quite lengthy and technical proof uses methods already exploited by \textit{A. Marcja} and \textit{C. Toffalori} [``Decidable representations'', J. Pure Appl. Algebra 103, 189-203 (1995; Zbl 0838.16010)] and derived in part from the methods of a paper of \textit{M. C. R. Butler} [``On the classification of local integral representations of finite Abelian \(p\)-groups'', Lect. Notes Math. 488, 54-71 (1975; Zbl 0314.20006)] which proves wildness for the category of \(\mathbb{Z} S\)-lattices for groups \(S\) as above. The main difficulty is in lifting undecidability from \(\mathbb{Z} S\)-lattices to \(\mathbb{Z} G\)-lattices. In a more recent paper [\textit{C. Toffalori}, ``Wildness implies undecidability for lattices over group rings'', J. Symb. Log. 62, 1429-1447 (1997)], the author has extended the results of this paper and shown that if \(G\) is a finite group such that for some prime \(p\) and Sylow \(p\)-subgroup \(S_p\) of \(G\) the theory of \(\mathbb{Z} S_p\)-lattices is undecidable, then so is the theory of \(\mathbb{Z} G\)-lattices.
Mathematics Subject Classification ID: 03C60
Mathematics Subject Classification ID: 16H05
Mathematics Subject Classification ID: 03B25
Mathematics Subject Classification ID: 20C10
Mathematics Subject Classification ID: 20C05
Mathematics Subject Classification ID: 16S34
Mathematics Subject Classification ID: 20A15
Mathematics Subject Classification ID: 16B70
zbMATH DE Number: 1109926
zbMATH Keywords: decidability
zbMATH Keywords: induced modules
zbMATH Keywords: representation type
zbMATH Keywords: finite group
zbMATH Keywords: theory of \(\mathbb{Z} G\)-lattices
zbMATH Keywords: wildness
zbMATH Keywords: category of \(\mathbb{Z} S\)-lattices
author: Carlo Toffalori
reviewed by: Mike Prest
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1109926