A characterisation of higher torsion classes

Abstract

Let A be an abelian length category containing a d-cluster tilting subcategory M . We prove that a subcategory of 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 M form a complete lattice. Moreover, we use the characterisation to classify the d-torsion classes associated to higher Auslander algebras of type A , and give an algorithm to compute them explicitly. The classification is furthermore extended to the setup of higher Nakayama algebras.