Subcategories of the derived category and cotilting complexes

Abstract

We show that there is a one-to-one correspondence between basic cotilting complexes and certain contravariantly finite subcategories of the bounded derived category of an artin algebra. This is a triangulated version of a result by Auslander and Reiten. We use this to find an existence criterion for complements to exceptional complexes. Introduction. Homologically finite subcategories were introduced by Auslander and Smalo (3), and they have proved to be important in the study of artin algebras. Homologically finite subcategories of the category of finitely generated modules have been studied by several authors. In (1), Auslander and Reiten showed that there is a correspondence between cer- tain contravariantly finite subcategories and basic cotilting modules. In this paper we consider some subcategories of the bounded derived category of an artin algebra that are associated with cotilting complexes. In the first sec- tion we give definitions and basic results that we use in the second section, where we show that there is a correspondence between cotilting complexes and certain contravariantly finite subcategories of the derived category. The third section is devoted to examples. In the fourth section we use the cor- respondence to prove an existence criterion for complements of exceptional complexes. 1. Subcategories of the derived category. Let Λ be an artin algebra. Let mod Λ be the category of finitely generated left Λ-modules, and let D = D b (mod Λ) be the bounded derived category. This is a triangulated category. We denote the shift functor by (1), and its inverse by (−1). We let I(Λ) be the full subcategory of mod Λ formed by the injective objects, and, similarly, P(Λ) stands for the projectives. Then D b (mod Λ) is equivalent to K +,b (I(Λ)). We consider this an identification, and let K b (I(Λ)) denote the coperfect complexes. By a subcategory, we will always mean a full additive