The passage among the subcategories of weakly approximable triangulated categories
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arXiv
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| Hauptverfasser: | , , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866916117581135872 |
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| author | Canonaco, Alberto Haesemeyer, Christian Neeman, Amnon Stellari, Paolo |
| author_facet | Canonaco, Alberto Haesemeyer, Christian Neeman, Amnon Stellari, Paolo |
| contents | In this article we prove that all the inclusions between the 'classical' and naturally defined full triangulated subcategories of a weakly approximable triangulated category are intrinsic (in one case under a technical condition). This extends all the existing results about subcategories of weakly approximable triangulated categories. Together with a forthcoming paper about uniqueness of enhancements, our result allows us to generalize a celebrated theorem by Rickard which asserts that if $R$ and $S$ are left coherent rings, then a derived equivalence of $R$ and $S$ is "independent of the decorations". That is, if $D^?(R\text{-}\square)$ and $D^?(S\text{-}\square)$ are equivalent as triangulated categories for some choice of decorations $?$ and $\square$, then they are equivalent for every choice of decorations. But our theorem is much more general, and applies also to quasi-compact and quasi-separated schemes -- even to the relative version, in which the derived categories consist of complexes with cohomology supported on a given closed subscheme with quasi-compact complement. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_04605 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The passage among the subcategories of weakly approximable triangulated categories Canonaco, Alberto Haesemeyer, Christian Neeman, Amnon Stellari, Paolo Algebraic Geometry Category Theory Representation Theory In this article we prove that all the inclusions between the 'classical' and naturally defined full triangulated subcategories of a weakly approximable triangulated category are intrinsic (in one case under a technical condition). This extends all the existing results about subcategories of weakly approximable triangulated categories. Together with a forthcoming paper about uniqueness of enhancements, our result allows us to generalize a celebrated theorem by Rickard which asserts that if $R$ and $S$ are left coherent rings, then a derived equivalence of $R$ and $S$ is "independent of the decorations". That is, if $D^?(R\text{-}\square)$ and $D^?(S\text{-}\square)$ are equivalent as triangulated categories for some choice of decorations $?$ and $\square$, then they are equivalent for every choice of decorations. But our theorem is much more general, and applies also to quasi-compact and quasi-separated schemes -- even to the relative version, in which the derived categories consist of complexes with cohomology supported on a given closed subscheme with quasi-compact complement. |
| title | The passage among the subcategories of weakly approximable triangulated categories |
| topic | Algebraic Geometry Category Theory Representation Theory |
| url | https://arxiv.org/abs/2402.04605 |