The Krull–Schmidt problem for modules over valuation domains

Abstract

Direct sum decompositions problems for torsion-free modules of nite rank have been the subject of recent activity in the theory of modules over valuation domains (see e.g. [11]). Indeed the nal problem (Problem 26) in the text by Fuchs and Salce [6] asks if direct decompositions of nite rank modules (over a valuation domain) into indecomposable summands are unique up to isomorphism i.e. does the Krull– Schmidt Theorem hold for this class of modules? It has been known for some time that the Krull–Schmidt Theorem fails for nite-rank torsion-free modules over the discrete valuation domain Zp, the integers localized at the prime p; see [1, Exercise 2.15] for an example due essentially to M.C.R. Butler. However, there is no immediate method of extending this result to arbitrary valuation domains; indeed it is well known that if a valuation domain is Henselian, then the Krull–Schmidt Theorem does hold for torsionfree modules of nite rank (see e.g. [11, Lemma 14] or [10, Corollary 10]. In fact for discrete rank one valuation domains the two concepts are equivalent [11, Theorem 17]. One of the principal outcomes of the present work is that the Krull–Schmidt Theorem fails for a large class of non-Henselian valuation domains. It is worth remarking that this class contains many valuation domains which are not discrete and so is a farreaching generalization of [11, Theorem 17]. (We also note that Facchini has recently shown failure of Krull–Schmidt for serial modules [5].) In order to make this comment a little more precise, let us introduce the following notation: throughout, unless speci ed to the contrary, R shall denote a valuation domain, not a eld, with completion R