Introduction to Abelian and Derived Categories

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 [12]. We then characterize module categories among abelian categories. Finally we sketch a proof of Mitchell’s full embedding theorem [25]: 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 [33] and the total right derived functor of an additive functor following Deligne [6]. We treat the basics of triangulated categories including K0-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 [29]. 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 [2]. 1. Abelian categories 1.1. Definition and basic properties. A Z-category is a category C whose morphism sets HomC(X,Y ) are abelian groups such that all composition maps HomC(Y,Z)×HomC(X,Y ) → HomC(X,Z) are bilinear. For example, if R is a ring (associative, with 1) and C is the category having exactly one object, whose endomorphism set is R, then C is a Z-category. A general Z-category should be thought of as a ‘ring with several objects’ [25]. An additive category is a Z-category A which has a zero object 0 (i.e. we have HomA(0,X) = 0 = HomA(X, 0) for all X) and such that all pairs of objects X,Y ∈ C, admit a product in C, i.e. an object X ∏ Y endowed with morphisms pX : X ∏