Group actions on monoidal triangulated categories and Balmer spectra
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866911793240080384 |
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| author | Huang, Hongdi Vashaw, Kent B. |
| author_facet | Huang, Hongdi Vashaw, Kent B. |
| contents | Let $G$ be a group acting on a left or right rigid monoidal triangulated category ${\mathbf K}$ which has a Noetherian Balmer spectrum. We prove that the Balmer spectrum of the crossed product category of ${\mathbf K}$ by $G$ is homeomorphic to the space of $G$-prime ideals of ${\mathbf K}$, give a concrete description of this space, and classify the $G$-invariant thick ideals of ${\mathbf K}$. Under some additional technical conditions, we prove that the Balmer spectrum of the equivariantization of ${\mathbf K}$ by $G$ is also homeomorphic to the space of $G$-prime ideals. Examples of stable categories of finite tensor categories and perfect derived categories of coherent sheaves on Noetherian schemes are used to illustrate the theory. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2311_18638 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Group actions on monoidal triangulated categories and Balmer spectra Huang, Hongdi Vashaw, Kent B. Category Theory Algebraic Geometry Quantum Algebra Rings and Algebras Representation Theory 18G80, 18M05 (Primary) 14L30, 16W22, 16T05 (Secondary) Let $G$ be a group acting on a left or right rigid monoidal triangulated category ${\mathbf K}$ which has a Noetherian Balmer spectrum. We prove that the Balmer spectrum of the crossed product category of ${\mathbf K}$ by $G$ is homeomorphic to the space of $G$-prime ideals of ${\mathbf K}$, give a concrete description of this space, and classify the $G$-invariant thick ideals of ${\mathbf K}$. Under some additional technical conditions, we prove that the Balmer spectrum of the equivariantization of ${\mathbf K}$ by $G$ is also homeomorphic to the space of $G$-prime ideals. Examples of stable categories of finite tensor categories and perfect derived categories of coherent sheaves on Noetherian schemes are used to illustrate the theory. |
| title | Group actions on monoidal triangulated categories and Balmer spectra |
| topic | Category Theory Algebraic Geometry Quantum Algebra Rings and Algebras Representation Theory 18G80, 18M05 (Primary) 14L30, 16W22, 16T05 (Secondary) |
| url | https://arxiv.org/abs/2311.18638 |