ON HIGHER TORSION CLASSES

Abstract

AbstractBuilding 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.