## 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.

http://www.math.jussieu.fr/~keller/publ/camabs.html

Bernhard Keller, le 7 Avril 1997.

keller@math.jussieu.fr

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