On Auslander-Reiten translates in functorially finite subcategories and applications

Functorially finite subcategories in module categories over Artin algebras have been introduced by Auslander and Smalø [2] to provide a convenient setting for existence of relative Auslander-Reiten sequences in subcategories. Given a functorially finite subcategory, it is generally not well understood how to compute approximations, and the end terms of relative Auslander-Reiten sequences. In this paper, we present a general method, in the setting of bounded derived categories. This was partly inspired by [3], and it generalizes [6, Theorem 3]. Furthermore, we prove an Auslander-Reiten formula for the setting of functorially finite subcategories.