$\tau$-tilting theory in abelian categories

TL;DR

This paper explores $ au$-tilting theory within Hom-finite abelian categories, establishing relationships between covariantly finite $ au$-rigid subcategories, support $ au$-tilting subcategories, and torsion classes, with applications.

Contribution

It demonstrates that covariantly finite $ au$-rigid subcategories are contained in support $ au$-tilting subcategories and establishes a bijection with certain torsion classes, extending $ au$-tilting theory.

Findings

Covariantly finite $ au$-rigid subcategories are contained in support $ au$-tilting subcategories.

Support $ au$-tilting subcategories correspond bijectively to finitely generated torsion classes.

Main results have several applications in the structure theory of abelian categories.

Abstract

Let $\mathcal A$ be a Hom-finite abelian category with enough projectives. In this note, we show that any covariantly finite $\tau$-rigid subcategory is contained in a support $\tau$-tilting subcategory. We also show that support $\tau$-tilting subcategories are in bijection with certain finitely generated torsion classes. Some applications of our main results are also given.