Upper bounds for dimensions of singularity categories and their annihilators
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866908398090452992 |
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| author | Dey, Souvik Mifune, Yuki |
| author_facet | Dey, Souvik Mifune, Yuki |
| contents | Let $R$ be a commutative noetherian ring. Denote by $\operatorname{mod} R$ the category of finitely generated $R$-modules and by $\operatorname{D^b}(R)$ the bounded derived category of $\operatorname{mod} R$. In this paper, we first investigate localizations and annihilators of Verdier quotients of $\operatorname{D^b}(R)$. After that, we explore upper bounds for the dimension of the singularity category of $R$ and its (strong) generators. We extend a theorem of Liu to the case where $R$ is neither an isolated singularity nor even a local ring. Some of our results are more generally stated in terms of Spanier--Whitehead category of a resolving subcategory. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_12206 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Upper bounds for dimensions of singularity categories and their annihilators Dey, Souvik Mifune, Yuki Commutative Algebra Representation Theory 13C60, 13D09, 18G80 Let $R$ be a commutative noetherian ring. Denote by $\operatorname{mod} R$ the category of finitely generated $R$-modules and by $\operatorname{D^b}(R)$ the bounded derived category of $\operatorname{mod} R$. In this paper, we first investigate localizations and annihilators of Verdier quotients of $\operatorname{D^b}(R)$. After that, we explore upper bounds for the dimension of the singularity category of $R$ and its (strong) generators. We extend a theorem of Liu to the case where $R$ is neither an isolated singularity nor even a local ring. Some of our results are more generally stated in terms of Spanier--Whitehead category of a resolving subcategory. |
| title | Upper bounds for dimensions of singularity categories and their annihilators |
| topic | Commutative Algebra Representation Theory 13C60, 13D09, 18G80 |
| url | https://arxiv.org/abs/2408.12206 |