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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2023
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2306.09832 |
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| _version_ | 1866929531791605760 |
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| author | Zhang, Xiaolei Zhao, Tiwei Wang, Dingguo |
| author_facet | Zhang, Xiaolei Zhao, Tiwei Wang, Dingguo |
| contents | For each $n\in\mathbb{N}\cup\{\infty\}$, we introduce the notion of $n$-singularity category $\mathbf{D}_{n{\rm-}sg}(R)$ of a given ring $R$, which can be seen as a generalization of the classical singularity category. Moreover, the $n$-global dimension $n$-gldim$(R)$ of $R$ is investigated. We show that $\mathbf{D}_{n{\rm-}sg}(R)=0$ if and only if $n$-gldim$(R)$ is finite. Furthermore, we characterize the vanishing property of $n$-singularity categories in terms of recollements. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2306_09832 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On the generalizations of global dimensions and singularity categories Zhang, Xiaolei Zhao, Tiwei Wang, Dingguo Rings and Algebras For each $n\in\mathbb{N}\cup\{\infty\}$, we introduce the notion of $n$-singularity category $\mathbf{D}_{n{\rm-}sg}(R)$ of a given ring $R$, which can be seen as a generalization of the classical singularity category. Moreover, the $n$-global dimension $n$-gldim$(R)$ of $R$ is investigated. We show that $\mathbf{D}_{n{\rm-}sg}(R)=0$ if and only if $n$-gldim$(R)$ is finite. Furthermore, we characterize the vanishing property of $n$-singularity categories in terms of recollements. |
| title | On the generalizations of global dimensions and singularity categories |
| topic | Rings and Algebras |
| url | https://arxiv.org/abs/2306.09832 |