Localization of triangulated categories with respect to extension-closed subcategories
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866916748468420608 |
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| author | Ogawa, Yasuaki |
| author_facet | Ogawa, Yasuaki |
| contents | The aim of this paper is to develop a framework for localization theory of triangulated categories $\mathcal{C}$, that is, from a given extension-closed subcategory $\mathcal{N}$ of $\mathcal{C}$, we construct a natural extriangulated structure on $\mathcal{C}$ together with an exact functor $Q:\mathcal{C}\to\widetilde{\mathcal{C}}_\mathcal{N}$ satisfying a suitable universality, which unifies several phenomena. Precisely, a given subcategory $\mathcal{N}$ is thick if and only if the localization $\widetilde{\mathcal{C}}_\mathcal{N}$ corresponds to a triangulated category. In this case, $Q$ is nothing other than the usual Verdier quotient. Furthermore, it is revealed that $\widetilde{\mathcal{C}}_\mathcal{N}$ is an exact category if and only if $\mathcal{N}$ satisfies a generating condition $\mathsf{cone}(\mathcal{N},\mathcal{N})=\mathcal{C}$. Such an (abelian) exact localization $\widetilde{\mathcal{C}}_\mathcal{N}$ provides a good understanding of some cohomological functors $\mathcal{C}\to\mathsf{Ab}$, e.g., the heart of $t$-structures on $\mathcal{C}$ and the abelian quotient of $\mathcal{C}$ by a cluster-tilting subcategory $\mathcal{N}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2205_12116 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Localization of triangulated categories with respect to extension-closed subcategories Ogawa, Yasuaki Category Theory 18G80, 18E35, 18E10 The aim of this paper is to develop a framework for localization theory of triangulated categories $\mathcal{C}$, that is, from a given extension-closed subcategory $\mathcal{N}$ of $\mathcal{C}$, we construct a natural extriangulated structure on $\mathcal{C}$ together with an exact functor $Q:\mathcal{C}\to\widetilde{\mathcal{C}}_\mathcal{N}$ satisfying a suitable universality, which unifies several phenomena. Precisely, a given subcategory $\mathcal{N}$ is thick if and only if the localization $\widetilde{\mathcal{C}}_\mathcal{N}$ corresponds to a triangulated category. In this case, $Q$ is nothing other than the usual Verdier quotient. Furthermore, it is revealed that $\widetilde{\mathcal{C}}_\mathcal{N}$ is an exact category if and only if $\mathcal{N}$ satisfies a generating condition $\mathsf{cone}(\mathcal{N},\mathcal{N})=\mathcal{C}$. Such an (abelian) exact localization $\widetilde{\mathcal{C}}_\mathcal{N}$ provides a good understanding of some cohomological functors $\mathcal{C}\to\mathsf{Ab}$, e.g., the heart of $t$-structures on $\mathcal{C}$ and the abelian quotient of $\mathcal{C}$ by a cluster-tilting subcategory $\mathcal{N}$. |
| title | Localization of triangulated categories with respect to extension-closed subcategories |
| topic | Category Theory 18G80, 18E35, 18E10 |
| url | https://arxiv.org/abs/2205.12116 |