Auslander-reiten Theory via Brown Representability

We develop an Auslander-Reiten theory for triangulated categories which is based on Brown’s representability theorem. In a fundamental article [3], Auslander and Reiten introduced almost split sequences for the category of finitely generated modules over an artin algebra. These are short exact sequences which look almost like split exact sequences, but many authors prefer to call them Auslander-Reiten sequences. This concept is one of the most successful in modern algebra representation theory (cf. [4] for a good introduction). In fact, the existence theorem for almost split sequences has been generalized in various directions and became the starting point of what is now called Auslander-Reiten theory. Let us mention some of the main ingredients of classical Auslander-Reiten theory: • the Auslander-Reiten formula, • almost split sequences, • morphisms determined by objects, • Auslander’s defect formula. In this paper we discuss analogous concepts and results for compactly generated triangulated categories. This includes, for example, the stable homotopy category of CWspectra, or the derived category of modules over some fixed ring. For each of the above concepts from classical Auslander-Reiten theory there is a corresponding section in this paper. In addition, we have included an appendix which provides a brief introduction into classical Auslander-Reiten theory and sketches the parallel between Auslander-Reiten theory for module categories and the new AuslanderReiten theory for triangulated categories. In this paper, no attempt has been made to present a unified approach towards a general Auslander-Reiten theory which covers module categories and triangulated categories at the same time. For this we refer to recent work of Beligiannis [5]. 1. Brown representability Throughout this paper we fix a triangulated category S and make the following additional assumptions: • S has arbitrary coproducts; • the isomorphism classes of compact objects in S form a set; • Hom(C,X) = 0 for all compact C implies X = 0 for every object X in S. Recall that an object X in S is compact if the representable functor Hom(X,−) preserves arbitrary coproducts. A functor S → Ab into the category of abelian groups is exact if it sends every triangle to an exact sequence. The following characterization of representable functors is our main tool for proving the existence of maps and triangles. Theorem (Brown). A contravariant functor F : S → Ab is isomorphic to a representable functor Hom(−,X) for some X ∈ S if and only if F is exact and sends arbitrary coproducts to products.