Auslander's defects over extriangulated categories: an application for the General Heart Construction
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arXiv
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| Format: | Preprint |
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2019
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| _version_ | 1866918028393840640 |
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| author | Ogawa, Yasuaki |
| author_facet | Ogawa, Yasuaki |
| contents | The notion of extriangulated category was introduced by Nakaoka and Palu giving a simultaneous generalization of exact categories and triangulated categories. Our first aim is to provide an extension to extriangulated categories of Auslander's formula: for some extriangulated category $\mathcal{C}$, there exists a localization sequence $\operatorname{\mathsf{def}}\mathcal{C}\to\operatorname{\mathsf{mod}}\mathcal{C}\to\operatorname{\mathsf{lex}}\mathcal{C}$, where $\operatorname{\mathsf{lex}}\mathcal{C}$ denotes the full subcategory of finitely presented left exact functors and $\operatorname{\mathsf{def}}\mathcal{C}$ the full subcategory of Auslander's defects. Moreover we provide a connection between the above localization sequence and the Gabriel-Quillen embedding theorem. As an application, we show that the general heart construction of a cotorsion pair $(\mathcal{U},\mathcal{V})$ in a triangulated category, which was provided by Abe and Nakaoka, is same as the construction of a localization sequence $\operatorname{\mathsf{def}}\mathcal{U}\to\operatorname{\mathsf{mod}}\mathcal{U}\to\operatorname{\mathsf{lex}}\mathcal{U}$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_1911_00259 |
| institution | arXiv |
| publishDate | 2019 |
| record_format | arxiv |
| spellingShingle | Auslander's defects over extriangulated categories: an application for the General Heart Construction Ogawa, Yasuaki Representation Theory Category Theory 18E10, 18E30, 18E35 The notion of extriangulated category was introduced by Nakaoka and Palu giving a simultaneous generalization of exact categories and triangulated categories. Our first aim is to provide an extension to extriangulated categories of Auslander's formula: for some extriangulated category $\mathcal{C}$, there exists a localization sequence $\operatorname{\mathsf{def}}\mathcal{C}\to\operatorname{\mathsf{mod}}\mathcal{C}\to\operatorname{\mathsf{lex}}\mathcal{C}$, where $\operatorname{\mathsf{lex}}\mathcal{C}$ denotes the full subcategory of finitely presented left exact functors and $\operatorname{\mathsf{def}}\mathcal{C}$ the full subcategory of Auslander's defects. Moreover we provide a connection between the above localization sequence and the Gabriel-Quillen embedding theorem. As an application, we show that the general heart construction of a cotorsion pair $(\mathcal{U},\mathcal{V})$ in a triangulated category, which was provided by Abe and Nakaoka, is same as the construction of a localization sequence $\operatorname{\mathsf{def}}\mathcal{U}\to\operatorname{\mathsf{mod}}\mathcal{U}\to\operatorname{\mathsf{lex}}\mathcal{U}$. |
| title | Auslander's defects over extriangulated categories: an application for the General Heart Construction |
| topic | Representation Theory Category Theory 18E10, 18E30, 18E35 |
| url | https://arxiv.org/abs/1911.00259 |