A characterisation of higher torsion classes

TL;DR

This work provides a precise higher-dimensional analogue of Dickson’s torsion-class characterisation: in a finite-length abelian category $\mathcal{A}$ with a $d$-cluster tilting subcategory $\mathcal{M}$, a subcategory of $\mathcal{M}$ is a $d$-torsion class iff it is closed under $d$-extensions and $d$-quotients. The authors establish a robust lattice structure for $d$-torsion classes and derive a combinatorial description for those arising from higher Auslander algebras of type $\mathbb{A}$ and higher Nakayama algebras, including explicit algorithms to compute and count them. The results bridge higher homological algebra with concrete combinatorics and yield efficient methods to classify and manipulate higher torsion classes in key families of algebras. This advances the understanding of the interplay between higher torsion theory and representation-theoretic structures, with potential applications to related areas such as $(d+1)$-term silting theory and higher lattice theory.

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.

Figures (4)

• Figure 1: The Auslander--Reiten quiver of the module category considered in \ref{['ex:running']}, with the generators of the $2$-cluster tilting subcategory $\mathcal{M}$ marked.
• Figure 2: The Hasse diagram of the lattice of $2$-torsion classes in \ref{['ex:running4']}.
• Figure 3: The Hasse diagram of the $3$-torsion classes in the $3$-cluster tilting subcategory $\mathcal{M}_3^3$ of the higher Auslander Algebra $A_{3}^{3}$.
• Figure 4: The quiver of the $2$-cluster tilting subcategory $\mathcal{M}_{\underline{\ell}}^{2}$ in \ref{['ex:higher nakayama']}

Theorems & Definitions (100)

• Theorem 1.1: \ref{['cor: characterisation']}
• Theorem 1.2: \ref{['theorem:lattice']}
• Theorem 1.3: \ref{['prop: extension closure indec']}
• Definition 2.1
• Definition 2.2
• Theorem 2.3
• Example 2.4
• Definition 2.5
• Proposition 2.6
• Remark 2.7