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.
Bernhard Keller, le 7 Avril 1997.

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