The torsion-free part of the Ziegler spectrum of the Klein four group
The authors study the Ziegler spectrum of the integral group ring \(\mathbb{Z}V_4\) of the Klein four group \(V_4\). If \(R\) is a ring, the Ziegler spectrum \(\mathrm{Zsp}(R)\) is a compact topological space whose points are given by the isomorphism classes of indecomposable algebraically compact \(R\)-modules and whose basic open sets are of the form \((\varphi / \psi)\), where \(\varphi\) and \(\psi\) are positive primitive formulas and \((\varphi / \psi)\) consists of those \(M\in\mathrm{Zsp}(R)\) for which there exists \(m\in M\) which satisfies \(\varphi\) but not \(\psi\). If \(S\) is a Dedekind domain, \(G\) a finite group and \(R=SG\) is the group ring of \(G\) over \(S\), it was shown in [\textit{A. Marcja, M. Prest} and \textit{C. Toffalori}, ``The torsionfree part of the Ziegler spectrum of \(RG\) when \(R\) is a Dedekind domain and \(G\) is a finite group'', J. Symb. Log. 67, No. 3, 1126--1140 (2002; Zbl 1024.03035)] that every \(S\)-torsionfree indecomposable algebraically compact \(SG\)-module is either divisible or reduced and therefore carries a natural structure of an indecomposable algebraically compact \(\widehat{S}_PG\)-module for some maximal ideal \(P\) of \(S\). Moreover, it was shown that the topology of the \(S\)-torsionfree part \(\mathrm{Zsp}_{\mathrm{tf}}(SG)\) of \(\mathrm{Zsp}(SG)\) is obtained by gluing together so-called \(P\)-patches, which are topological spaces associated to maximal ideals \(P\) of \(S\). The authors give a rather explicit description of \(\mathrm{Zsp}_{\mathrm{tf}}(\mathbb{Z}V_4)\). In particular, they show that the Cantor-Bendixson rank of this space is 3 and that the only points of maximal rank are the modules \(\mathbb{Q}V_4e_i\), where the \(e_i\) are the indecomposable idempotents of \(\mathbb{Q}V_4\). Using the existence of almost split sequences, they moreover prove that the isolated points in \(\mathrm{Zsp}_{\mathrm{tf}}(\mathbb{Z}V_4)\) are exactly the indecomposable lattices over \(\widehat{\mathbb{Z}}_pV_4\).
scientific article; zbMATH DE number 5886340
scientific article