The Auslander bijections and universal extensions

TL;DR

This paper explores the interplay between Auslander bijections, universal extensions, and Auslander-Reiten duality in abelian categories, leading to new insights and verification of a conjecture by Ringel.

Contribution

It introduces a bijection triangle connecting these concepts and applies it to derive new results, including confirming Ringel's conjecture.

Findings

Established a bijection triangle involving Auslander bijections, universal extensions, and Auslander-Reiten duality.

Verified Ringel's conjecture using the developed framework.

Provided new consequences and insights in the structure of abelian categories with Auslander-Reiten duality.

Abstract

Universal extensions arise naturally in the Auslander bijections. For an abelian category having Auslander-Reiten duality, we exploit a bijection triangle, which involves the Auslander bijections, universal extensions and the Auslander-Reiten duality. Some consequences are given, in particular, a conjecture by Ringel is verified.