Existence of Auslander–Reiten sequences in subcategories

This paper studies the existence of Auslander–Reiten sequences in subcategories of mod(Λ), where Λ 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. Theorem – Let M be a subcategory of closed under extensions and direct summands, and let M be an indecomposable module in M such that for some in M . Then the following are equivalent.(i) has an -precover in the stable category ,(ii)There is an Auslander–Reiten sequence 0→X→Y→M→0 in M.We also have the dual result of the above theorem. Together they strengthen the results in Auslander and Smalø (1981) [3,4], and in Kleiner (1997) [7].