On the Ziegler Spectrum of a Group Ring

Carlo Toffalori

This is an abstract of joint work with Mike Prest (Manchester) and Annalisa Marcja (Florence) [MPT].

Let R be a Dedekind domain, G be a finite group. We consider R-torsionfree modules over the group ring RG and we study their Ziegler topology. First we show that, for every R-torsionfree N in the Ziegler space ZG_{RG}, either N is injective and simple, or there is some maximal ideal P of R such that N is a reduced point in Zg_{R_{P}} G where R_{P} denotes the completion of the localization of R at P. In the latter case, the induced topology over RG or over R_{P} G is the same. So we examine R_{P} G for a given P. We see that a reduced torsionfree N in Zg_{R_{P} G} is in the closure of finitely generated points (i. e. R_{P} G-lattices) and is ”almost” isolated and closed. We analyse the case when R_{P} G-lattices are a classe of finite representation type, in particular when the order of G is not in P. Finally we discuss the role of RG-lattices within Zg_{RG}.

[MPT] A. Marcja-M. Prest-C. Toffalori, The torsionfree part of the Ziegler spectrum of RG when R is a Dedekind domain and G is a finite group, preprint

Address: Dipartimento di Matematica e Fisica, Università di Camerino, Via Madonna delle Carceri, 62032 Camerino (Italy)