Existence of Auslander-Reiten sequences in subcategories

TL;DR

This paper establishes necessary and sufficient conditions for the existence of Auslander-Reiten sequences within subcategories of module categories over finite-dimensional algebras, advancing the understanding of their structure.

Contribution

It provides new criteria characterizing when Auslander-Reiten sequences exist in subcategories, including necessary and sufficient conditions involving precover existence.

Findings

Characterization of Auslander-Reiten sequences in subcategories

Equivalence between precover existence and AR sequence existence

Dual results complement the main theorems

Abstract

This paper studies the existence of Auslander-Reiten sequences in subcategories of mod A, where A is a finite dimensional algebra over a field. The two main theorems give necessary and sufficient conditions for the existence of Auslander-Reiten sequences in subcategories. Let M be a subcategory of mod A closed under extensions and direct summands, and let X be an indecomposable module in M such that Ext^{1}(X, X') is not zero for some X' in M. Then the following are equivalent: (i) DTrX has a precover in the stable category of mod A, (ii) There exists an Auslander-Reiten sequence 0 --> U --> W --> X --> 0 in M. We also have the dual result of the above theorem.