A characterisation of higher torsion classes

August, Jenny  ; Haugland, Johanne  ; Jacobsen, Karin M.  ; Kvamme, Sondre  ; Palu, Yann  ; Treffinger, Hipolito 

Forum of Mathematics, Sigma, Tome 13 (2025) no. 1, p. e33

Résumé

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.

Détail

DOI : 10.1017/fms.2024.73

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