The generalized Auslander-Reiten duality on an exact category

TL;DR

This paper generalizes Auslander-Reiten duality within exact categories, establishing new translation functors that relate stable subcategories and characterize indecomposable objects via almost split conflations.

Contribution

It introduces a generalized Auslander-Reiten duality and translation functors on exact categories, expanding classical theory to a broader categorical context.

Findings

Defines generalized Auslander-Reiten duality on exact categories

Establishes mutually quasi-inverse translation functors between stable subcategories

Characterizes indecomposable objects via almost split conflations

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.