$τ$-perpendicular wide subcategories
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arXiv
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| Format: | Preprint |
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2021
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| _version_ | 1866914919559987200 |
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| author | Buan, Aslak Bakke Hanson, Eric J. |
| author_facet | Buan, Aslak Bakke Hanson, Eric J. |
| contents | Let $Λ$ be a finite-dimensional algebra. A wide subcategory of $\mathsf{mod}Λ$ is called left finite if the smallest torsion class containing it is functorially finite. In this paper, we prove that the wide subcategories of $\mathsf{mod}Λ$ arising from $τ$-tilting reduction are precisely the Serre subcategories of left finite wide subcategories. As a consequence, we show that the class of such subcategories is closed under further $τ$-tilting reduction. This leads to a natural way to extend the definition of the "$τ$-cluster morphism category" of $Λ$ to arbitrary finite-dimensional algebras. This category was recently constructed by Buan-Marsh in the $τ$-tilting finite case and by Igusa-Todorov in the hereditary case. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2107_01141 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | $τ$-perpendicular wide subcategories Buan, Aslak Bakke Hanson, Eric J. Representation Theory 16G10, 16G20 Let $Λ$ be a finite-dimensional algebra. A wide subcategory of $\mathsf{mod}Λ$ is called left finite if the smallest torsion class containing it is functorially finite. In this paper, we prove that the wide subcategories of $\mathsf{mod}Λ$ arising from $τ$-tilting reduction are precisely the Serre subcategories of left finite wide subcategories. As a consequence, we show that the class of such subcategories is closed under further $τ$-tilting reduction. This leads to a natural way to extend the definition of the "$τ$-cluster morphism category" of $Λ$ to arbitrary finite-dimensional algebras. This category was recently constructed by Buan-Marsh in the $τ$-tilting finite case and by Igusa-Todorov in the hereditary case. |
| title | $τ$-perpendicular wide subcategories |
| topic | Representation Theory 16G10, 16G20 |
| url | https://arxiv.org/abs/2107.01141 |