The Auslander-Reiten duality via morphisms determined by objects

TL;DR

This paper explores conditions under which deflations are right determined by objects in an exact category, linking these to the intrinsic kernel and providing characterizations for the existence of Auslander-Reiten duality.

Contribution

It establishes a criterion for when a deflation is right determined by an object based on the intrinsic kernel and characterizes categories with Auslander-Reiten duality.

Findings

Deflations are right determined by an object iff their intrinsic kernel is in a specific subcategory.

Provides characterizations for categories possessing Auslander-Reiten duality.

Links the structure of deflations to the subcategory generated by injectives and certain indecomposables.

Abstract

Given an exact category $\mathcal{C}$, we denote by $\mathcal{C}_l$ the smallest additive subcategory containing injectives and indecomposable objects which appear as the first term of an almost split conflation. We prove that a deflation is right determined by some object if and only if its intrinsic kernel lies in $\mathcal{C}_l$. We give characterizations for $\mathcal{C}$ having Auslander-Reiten duality.