Extriangulated length categories: torsion classes and $τ$-tilting theory
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| Format: | Preprint |
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2025
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| _version_ | 1866918019809148928 |
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| author | Wang, Li Wei, Jiaqun Zhang, Haicheng Zhou, Panyue |
| author_facet | Wang, Li Wei, Jiaqun Zhang, Haicheng Zhou, Panyue |
| contents | This paper introduces the notion of extriangulated length categories, whose prototypical examples include abelian length categories and bounded derived categories of finite dimensional algebras with finite global dimension. We prove that an extriangulated category $\mathcal{A}$ is a length category if and only if $\mathcal{A}$ admits a simple-minded system. Subsequently, we study the partially ordered set ${\rm tor}_Θ(\mathcal{A})$ of torsion classes in an extriangulated length category $(\mathcal{A},Θ)$ from the perspective of lattice theory. It is shown that ${\rm tor}_Θ(\mathcal{A})$ forms a complete lattice, which is further proved to be completely semidistributive and algebraic. Moreover, we describe the arrows in the Hasse quiver of ${\rm tor}_Θ(\mathcal{A})$ using brick labeling. Finally, we introduce the concepts of support torsion classes and support $τ$-tilting subcategories in extriangulated length categories and establish a bijection between these two notions, thereby generalizing the Adachi-Iyama-Reiten bijection for functorially finite torsion classes. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2502_07367 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Extriangulated length categories: torsion classes and $τ$-tilting theory Wang, Li Wei, Jiaqun Zhang, Haicheng Zhou, Panyue Representation Theory Category Theory This paper introduces the notion of extriangulated length categories, whose prototypical examples include abelian length categories and bounded derived categories of finite dimensional algebras with finite global dimension. We prove that an extriangulated category $\mathcal{A}$ is a length category if and only if $\mathcal{A}$ admits a simple-minded system. Subsequently, we study the partially ordered set ${\rm tor}_Θ(\mathcal{A})$ of torsion classes in an extriangulated length category $(\mathcal{A},Θ)$ from the perspective of lattice theory. It is shown that ${\rm tor}_Θ(\mathcal{A})$ forms a complete lattice, which is further proved to be completely semidistributive and algebraic. Moreover, we describe the arrows in the Hasse quiver of ${\rm tor}_Θ(\mathcal{A})$ using brick labeling. Finally, we introduce the concepts of support torsion classes and support $τ$-tilting subcategories in extriangulated length categories and establish a bijection between these two notions, thereby generalizing the Adachi-Iyama-Reiten bijection for functorially finite torsion classes. |
| title | Extriangulated length categories: torsion classes and $τ$-tilting theory |
| topic | Representation Theory Category Theory |
| url | https://arxiv.org/abs/2502.07367 |