Support $\tau_2$-tilting and 2-torsion pairs

TL;DR

This paper generalizes the bijection between support $ au$-tilting modules and torsion classes to the context of higher Auslander--Reiten theory, specifically for 2-cluster-tilting subcategories.

Contribution

It extends the support $ au$-tilting theory to 2-cluster-tilting subcategories, establishing a correspondence with torsion pairs under certain conditions.

Findings

Established a bijection between support $ au_2$-tilting modules and torsion pairs.

Generalized classical $ au$-tilting results to higher Auslander--Reiten theory.

Provided conditions for functorial finiteness in the correspondence.

Abstract

The theory of $\tau$-tilting was introduced by Adachi--Iyama--Reiten; one of the main results is a bijection between support $\tau$-tilting modules and torsion classes. We are able to generalise this result in the context of the higher Auslander--Reiten theory of Iyama. For a finite-dimensional algebra $A$ with 2-cluster-tilting subcategory $\mathcal{C}\subseteq\mathrm{mod}A$, we are able to find a correspondence between support $\tau_2$-tilting $A$-modules and torsion pairs in $\mathcal{C}$ satisfying an additional functorial finiteness condition.