A key module over pure-semisimple hereditary rings

Abstract

Let R be a hereditary, indecomposable, left pure semisimple ring. Inspired by [I. Reiten, C.M. Ringel, Infinite dimensional representations of canonical algebras, Canad. J. Math. 58 (2006) 180–224], we investigate the perfect cotorsion pair ( C , D ) in R - Mod generated by the preinjective component q. We show that there is a finitely generated product-complete tilting and cotilting left R-module W such that C = Cogen W and D = Gen W . The finite subcategory w of R - mod given by the indecomposable summands of W stores important information on R. For example, if we assume R of infinite representation type, then by [B. Zimmermann-Huisgen, Strong preinjective partitions and representation type of artinian rings, Proc. Amer. Math. Soc. 109 (1990) 309–322] there are non-preinjective indecomposable modules occurring as direct summands of products of preinjective modules, and it turns out that w is precisely the class of such modules. Moreover, we prove that R has finite representation type if and only if every module in w is source of a left almost split map in R - mod . Finally, we address the question when W is endofinite.