Silting interval reduction and 0-Auslander extriangulated categories
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866911909690736640 |
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| author | Pan, Jixing Zhu, Bin |
| author_facet | Pan, Jixing Zhu, Bin |
| contents | We give a reduction technique for silting intervals in extriangulated categories, which we call "silting interval reduction". It provides a reduction technique for tilting subcategories when the extriangulated categories are exact categories.
In 0-Auslander extriangulated categories (a generalization of the well-known two-term category $K^{[-1,0]}(\mathsf{proj}Λ)$ for an Artin algebra $Λ$), we provide a reduction theory for silting objects as an application of silting interval reduction. It unifies two-term silting reduction and Iyama-Yoshino's 2-Calabi-Yau reduction. The mutation theory developed by Gorsky, Nakaoka and Palu recently can be deduced from it. Since there are bijections between the silting objects and the support $τ$-tilting modules over certain finite dimensional algebras, we show it is compatible with $τ$-tilting reduction. This compatibility theorem also unifies the two compatibility theorems obtained by Jasso in his work on $τ$-tilting reduction.
We give a new construction for 0-Auslander extriangulated categories using silting mutation, together with silting interval reduction, we obtain some results on silting quivers. Finally, we prove that $d$-Auslander extriangulated categories are related to a certain sequence of silting mutations. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2401_13513 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Silting interval reduction and 0-Auslander extriangulated categories Pan, Jixing Zhu, Bin Representation Theory 16G10, 18G80, 18E40, 16S90 We give a reduction technique for silting intervals in extriangulated categories, which we call "silting interval reduction". It provides a reduction technique for tilting subcategories when the extriangulated categories are exact categories. In 0-Auslander extriangulated categories (a generalization of the well-known two-term category $K^{[-1,0]}(\mathsf{proj}Λ)$ for an Artin algebra $Λ$), we provide a reduction theory for silting objects as an application of silting interval reduction. It unifies two-term silting reduction and Iyama-Yoshino's 2-Calabi-Yau reduction. The mutation theory developed by Gorsky, Nakaoka and Palu recently can be deduced from it. Since there are bijections between the silting objects and the support $τ$-tilting modules over certain finite dimensional algebras, we show it is compatible with $τ$-tilting reduction. This compatibility theorem also unifies the two compatibility theorems obtained by Jasso in his work on $τ$-tilting reduction. We give a new construction for 0-Auslander extriangulated categories using silting mutation, together with silting interval reduction, we obtain some results on silting quivers. Finally, we prove that $d$-Auslander extriangulated categories are related to a certain sequence of silting mutations. |
| title | Silting interval reduction and 0-Auslander extriangulated categories |
| topic | Representation Theory 16G10, 18G80, 18E40, 16S90 |
| url | https://arxiv.org/abs/2401.13513 |