A characterisation of higher torsion classes
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...
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| Main Authors: | , , , , , |
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| Format: | Article |
| Language: | English |
| Published: |
Cambridge University Press
2025-01-01
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| Series: | Forum of Mathematics, Sigma |
| Subjects: | |
| Online Access: | https://www.cambridge.org/core/product/identifier/S2050509424000732/type/journal_article |
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| Summary: | 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. |
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| ISSN: | 2050-5094 |