A characterisation of higher torsion classes
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.
- Publication:
-
arXiv e-prints
- Pub Date:
- January 2023
- DOI:
- 10.48550/arXiv.2301.10463
- arXiv:
- arXiv:2301.10463
- Bibcode:
- 2023arXiv230110463A
- Keywords:
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- Mathematics - Representation Theory;
- 18E40;
- 16S90;
- 18E10;
- 18G25;
- 18G99
- E-Print:
- 34 pages. Final version, accepted for publication in Forum Math. Sigma