The theory of \({\mathbf Z}C(2)^2\)-lattices is decidable
The authors prove the decidability of the theory of \({\mathbf Z}C(2)^2\)-lattices, where \(C(2)^2\) is Klein's four group. For an arbitrary finite group \(G\) and a countable Dedekind domain \(R\) such that the residue field \(R/P\) is finite for every maximal \(R\)-ideal \(P\), they show that the genus of an \(RG\)-lattice \(M\) equals the elementary equivalence class of \(M\), i.e. the localizations at every maximal ideal of two \(RG\)-lattices are isomorphic if and only if the two lattices satisfy the same first-order sentences. Then the authors investigate generalizations of the above results to arbitrary \(R\)-torsion-free \(RG\)-modules and they apply the previous results to show the decidability of the theory of \({\mathbf Z}C(2)^2\)-lattices. This proof is obtained on the ground of the proof of Marcja and Toffalori that \({\mathbf Z}_2C(2)^2\)-lattices have a decidable theory, where \({\mathbf Z}_2\) is the localization of \({\mathbf Z}\) at the prime 2. Eventually, the authors show that \({\mathbf Z}[i]C(2)^2\)-lattices have an undecidable theory.
scientific article; zbMATH DE number 1187515