A characterisation of higher torsion classes

TL;DR

This paper characterizes higher torsion classes within a $d$-cluster tilting subcategory of an abelian length category, establishing their lattice structure and providing explicit classifications and algorithms for higher Auslander and Nakayama algebras.

Contribution

It generalizes classical torsion class results to higher torsion classes, offering a new characterization, lattice structure proof, and explicit classification algorithms.

Findings

Higher torsion classes are characterized by closure under $d$-extensions and $d$-quotients.

The set of $d$-torsion classes forms a complete lattice.

Explicit classification and algorithms are provided for higher Auslander and Nakayama algebras.

Abstract

Let $\mathcal{A}$ be an abelian length category containing a $d$-cluster tilting subcategory $\mathcal{M}$. We prove that a subcategory of $\mathcal{M}$ is a $d$-torsion class if and only if it is closed under $d$-extensions and $d$-quotients. This generalises an important result for classical torsion classes. As an application, we prove that the $d$-torsion classes in $\mathcal{M}$ form a complete lattice. Moreover, we use the characterisation to classify the $d$-torsion classes associated to higher Auslander algebras of type $\mathbb{A}$, and give an algorithm to compute them explicitly. The classification is furthermore extended to the setup of higher Nakayama algebras.