Computation of almost split sequences with applications to relatively projective and prinjective modules. (Q1411042)

Let \(R\) be an Artin algebra. Denote by \(\text{mod}\, R \) the category of finitely generated left \(R\)-modules. Assume that \({\mathcal A}\) is a contravariantly finite and extension closed full subcategory of \(\text{mod}\, R \). Given an indecomposable non-Ext-projective module \(N\) in \({\mathcal A}\), the authors give a simple method for computing the almost split sequence \(0\to L\to M \to N\to 0\) in \({\mathcal A}\) from the almost split sequence \(0\to \tau N\to M' \to N\to 0\) in \(\text{mod}\, R \). In particular cases, the method uses the technique of orthogonal subcategories to a given full subcategory \({\mathcal C}\) of \(\text{mod}\, R\) with respect to the functors \(\text{Hom}_R\) and \(\text{Ext}^1_R\). As an application, the authors present a new method for the computation of almost split sequences in the category of relatively projective modules, in the category of linear representations of a class of bocses and in the category of prinjective modules over a bipartite algebra. The technique allows them also to get a simplified proof of some of the old results of Maurice Auslander on the computation of almost split sequences in subcategories.