INTRODUCTION TO ABELIAN AND DERIVED CATEGORIES 

Bernhard Keller

April 6, 1997

Abstract. This is an account of three 1-hour lectures given at the
Instructional Conference on Representation Theory of Algebraic Groups
and Related Finite Groups, Isaac Newton Institute, Cambridge, 6-11
January 1997. 

In section 1, we define abelian categories following Grothendieck. We then
characterize module categories among abelian categories. Finally we
sketch a proof of Mitchell's full embedding theorem: each small abelian
category embeds fully and exactly into a module category. 

We come to our main topic in section 2, where we define the derived
category of an abelian category following Verdier and the total right
derived functor of an additive functor following Deligne. 

We treat the basics of triangulated categories including K_0-groups and
the example of perfect complexes over a ring in section 3. 

Section 4 is devoted to Rickard's Morita theory for derived categories. We
give his characterization of derived equivalences, list the most important
invariants under derived equivalence, and conclude by stating the simplest
version of Broué's conjecture.