On higher torsion classes

TL;DR

This paper explores the relationship between $n$-torsion classes in $n$-abelian categories and torsion classes in abelian categories, establishing their correspondence and implications for filtrations and coverings.

Contribution

It demonstrates that $n$-torsion classes correspond to intersections with torsion classes in abelian categories and shows their preservation under Galois coverings.

Findings

Every $n$-torsion class is an intersection with a torsion class in the abelian category.

Chains of $n$-torsion classes induce Harder-Narasimhan filtrations.

$n$-torsion classes are preserved by Galois covering functors.

Abstract

Building on the embedding of an $n$-abelian category $\mathscr{M}$ into an abelian category $\mathcal{A}$ as an $n$-cluster-tilting subcategory of $\mathcal{A}$, in this paper we relate the $n$-torsion classes of $\mathscr{M}$ with the torsion classes of $\mathcal{A}$. Indeed, we show that every $n$-torsion class in $\mathscr{M}$ is given by the intersection of a torsion class in $\mathcal{A}$ with $\mathscr{M}$. Moreover, we show that every chain of $n$-torsion classes in the $n$-abelian category $\mathscr{M}$ induces a Harder-Narasimhan filtration for every object of $\mathscr{M}$. We use the relation between $\mathscr{M}$ and $\mathcal{A}$ to show that every Harder-Narasimhan filtration induced by a chain of $n$-torsion classes in $\mathscr{M}$ can be induced by a chain of torsion classes in $\mathcal{A}$. Furthermore, we show that $n$-torsion classes are preserved by Galois covering functors, thus we provide a way to systematically construct new (chains of) $n$-torsion classes.