Nakayama functors on proper abelian subcategories

Abstract

We construct Nakayama functors on proper abelian subcategories of triangulated categories with a Serre functor using approximation theory. This, in turn, allows for the construction of Auslander-Reiten translates. As a result, we prove that suitable proper abelian subcategories are dualising $k$-varieties and have enough projectives if and only if they have enough injectives. As an application, we provide a new proof of the existence of Auslander-Reiten sequences in the category of finite dimensional modules over a finite dimensional algebra.

32 pages. Title change, added '0.2. Some useful results', included Lemma 1.2 and fixed some minor corrections. Accepted for publication in Publicacions Matemátiques