Derived Categories , Derived Equivalences and Representation Theory

Deenition: A derived category ... is when you take complexes seriously! (L.L. Scott Sc]) The aim of this chapter is to give a fairly elementary introduction to the (not very elementary) subject of derived categories and equivalences. Especially, we emphasize the applications of derived equivalences in representation theory of groups and algebras in order to illustrate the importance and usefulness of the concept. We try to keep the necessary prerequisites as low as possible. Ideally, these notes should be accessible for an audience with a good background in general algebra and some basic knowledge in representation theory, category theory and homological algebra. Of course, this means that these notes cannot be a comprehensive treatment. Most theorems have to be stated without proof but in any case we point to the relevant literature and wherever possible we provide examples in order to illustrate the results. In Section 1 abelian categories are introduced and we give a proof of Morita's theorem. Section 2 deals with triangulated categories; in particular we study the homotopy category of complexes. Section 3 contains the deenition of derived categories. In Section 4 we discuss tilting theory and indicate the development towards the important results of J. Rickard. As an illustration we discuss derived equivalences for Brauer tree algebras. In Section 5 we sketch some of the main applications of derived equivalences in representation theory. 1. Abelian Categories Let R be a ring (always associative and with a unit element). Denote by R-Mod the category of (left-) R-modules and by R-mod the category of nitely generated (left-) R-modules. These module categories are the main object of study in representation theory and they carry a lot of important additional structure.