Stable Module Categories and Their Representation Type

Given a nite dimensional algebra over an algebraically closed eld one frequently disregards the projective objects in the category mod of nite dimensional-modules and focuses on the stable category mod. The objects of mod are those of mod but the Hom-groups are the ordinary Hom-groups modulo the subgroup of those morphisms which factor through a projective-module. In this paper we show that mod determines the representation type of. Recall that the algebra is either tame, i.e. all nite dimensional indecomposable-modules belong to one-parameter families, or is wild, i.e. there are two-parameter families of nite dimensional indecomposable-modules 8]. Of course, one feels that this dichotomy should not depend on the deletion of nitely many objects in the category mod , and this is precisely one of the main results of this paper. More precisely, given another algebra ? and an equivalence mod ! mod ?, then ? is tame if is tame. Moreover, we show that the equivalence sends the one-parameter families in mod to one-parameter families in mod ?. The fact that mod determines the representation type of also follows, for some classes of symmetric algebras, from recent work of Assem, de la Pe~ na and Erdmann 2, 9]; however their methods are completely diierent. Equivalences between stable module categories have been studied by many authors. They naturally occur for instance in representation theory of nite groups. Another source of examples , which includes every algebra of Loewy length 2, is the class of algebras stably equivalent to a hereditary algebra. Usually the analysis concentrates on homological properties of the category mod which are preserved by an equivalence mod ! mod ?. In this paper we follow a diierent approach. We investigate pure-injective modules which are not necessarily nitely presented. Among them the endoonite modules are of particular interest. Recall that a module is endoonite if it is of nite length when regarded in the natural way as a module over its endomorphism ring. In order to study the non-nitely presented-modules we introduce a new category ?! lim mod which is essentially determined by the following three properties: (i) ?! lim mod contains, up to equivalence, mod as a full subcategory. (ii) ?! lim mod is an additive category with direct limits. (iii) Every object in ?! lim mod is a direct limit of objects in mod. The canonical functor mod ! mod induces a functor Mod ! ?! lim …