The generalized Auslander-Reiten duality on an exact category

Abstract

We introduce a notion of generalized Auslander-Reiten duality on a Hom-finite Krull-Schmidt exact category $\mathcal{C}$. This duality induces the generalized Auslander-Reiten translation functors $\tau$ and ${\tau }^{-}$. They are mutually quasi-inverse equivalences between the stable categories of two full subcategories ${\mathcal{C}}_r$ and ${\mathcal{C}}_l$ of $\mathcal{C}$. A non-projective indecomposable object lies in the domain of $\tau$ if and only if it appears as the third term of an almost split conflation; dually, a non-injective indecomposable object lies in the domain of ${\tau }^{-}$ if and only if it appears as the first term of an almost split conflation.

arXiv comment12 pages

Cited by in corpus (3)

• Almost Split Triangles and Morphisms Determined by Objects in Extriangulated Categories
• Auslander's defect formula and a commutative triangle in an exact category
• The generalized Auslander-Reiten duality on a module category