Endomorphism algebras

Abstract

One of the constant lines of thinking in representation theory is the comparison between the module categories of a given algebra and the endomorphism algebra of some “well-chosen” module. For instance, the classical Morita theorem asserts that, given a progenerator P of the module category of an algebra A, that is, a projective module P that is also a generator of $\operatorname {\mathrm {{mod}}} A$, the categories $\operatorname {\mathrm {{mod}}} A$ and $\operatorname {\mathrm {{mod}}} ( \operatorname {\mathrm {End}} P)$ are equivalent. This implies that, from the point of view of representation theory, we may assume that the algebras we deal with are basic, something we have done consistently. If one takes the endomorphism algebra of a module that is not a progenerator, then these modules categories are not equivalent, but nevertheless several features from one may pass to the other. This approach, initiated with the projectivisation procedure, much used by Auslander and his school, culminated in the now very important tilting theory. The aim of this chapter is to present these topics.