A characterisation of higher torsion classes
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arXiv
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| Main Authors: | , , , , , |
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| Format: | Preprint |
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2023
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| _version_ | 1866917918016536576 |
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| author | August, Jenny Haugland, Johanne Jacobsen, Karin M. Kvamme, Sondre Palu, Yann Treffinger, Hipolito |
| author_facet | August, Jenny Haugland, Johanne Jacobsen, Karin M. Kvamme, Sondre Palu, Yann Treffinger, Hipolito |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2301_10463 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | A characterisation of higher torsion classes August, Jenny Haugland, Johanne Jacobsen, Karin M. Kvamme, Sondre Palu, Yann Treffinger, Hipolito Representation Theory 18E40, 16S90, 18E10, 18G25, 18G99 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. |
| title | A characterisation of higher torsion classes |
| topic | Representation Theory 18E40, 16S90, 18E10, 18G25, 18G99 |
| url | https://arxiv.org/abs/2301.10463 |