The Endocategory of a Module

Dedicated to the memory of Maurice Auslander Given a ring we introduce the endocategory E M of a-module M. It is an abelian subcategory of Mod(?) where ? = End (M) op , and E M is constructed in such a way that it is the smallest abelian subcategory of Mod(?) containing M regarded in the natural way as a ?-module and all the endomorphisms of M induced by multiplication with an element from. The rst aim of this paper is to discuss some basic properties of this category. For instance, we show that E M reeects various properties of the module M which are related to purity. Another aim is to give a functorial description of the Ziegler spectrum of which is, by deenition, a representative set of indecomposable pure-injective-modules, together with a topology introduced by Ziegler 14]. We shall also discuss the relation between certain right and left-modules which arises from the well-known duality between the categories of nitely presented functors from mod((op) and mod((), respectively, into the category of abelian groups. In fact, this duality induces a bijection M 7 ! DM between certain subsets of the Ziegler spectra of and op , respectively, which has been studied by Herzog 6] using positive primitive formulas, and by Crawley-Boevey 1] using characters. The functoriality of this bijection is expressed by dualities between the endocategories E M and E DM. A nal example illustrates these dualities as well as their limitations. This paper is dedicated to the memory of Maurice Auslander. In fact, the material presented here depends in an essential way on homological and categorical techniques and ideas that indelibly bear his mark. 1. Preliminaries Let C be a skeletally small pre-additive category. A (right) C-module is an additive functor from C op into the category Ab of abelian groups and we denote by Mod(C) the category of all C-modules. Recall that M 2 Mod(C) is said to be nitely presented provided that there