Universal Extensions and Ext-Orthogonal Complements of Torsion Classes

TL;DR

This paper establishes a general equivalence between certain subcategories related to torsion pairs in Krull--Schmidt abelian categories, extending previous results and providing new insights into their structure and applications.

Contribution

It generalizes known equivalences for tilting-torsion pairs and functorially finite torsion pairs, offering a more direct proof without relying on silting complexes.

Findings

Establishes a new equivalence between subcategories of torsion-free objects and ext-orthogonal complements.

Provides a simplified proof for the functorially finite case.

Illustrates the equivalence in tube categories.

Abstract

We show that torsion pairs in Krull--Schmidt abelian categories induce an equivalence between the subcategory of torsion-free objects admitting universal extensions to the torsion subcategory, and a quotient of the ext-orthogonal complement of the torsion subcategory. This generalize an equivalence described by Bauer--Botnan--Oppermann--Steen for tilting-torsion pairs and by Buan--Zhou for functorially finite torsion pairs. The result also provides a more direct proof of the functorially finite case, not relying on the machinery of two-term silting complexes. We illustrate the equivalence in the special case of tube categories. After first appearing on the arXiv, the author learnt that the equivalence has previously been described by Demonet--Iyama.