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. LetMbe a subcategory ofmod(Λ)closed under extensions and direct summands, and letMbe an indecomposable module inMsuch thatfor someinM. Then the following are equivalent.

(i)

has an-precover in the stable category,

(ii)

There is an Auslander-Reiten sequence0→X→Y→M→0inM.