An undecidability theorem for lattices over group rings
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.
scientific article; zbMATH DE number 1109926
scientific article