Models for chain homotopy category of relative acyclic complexes
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866911218420154368 |
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| author | Hu, Jiangsheng Ren, Wei Yang, Xiaoyan You, Hanyang |
| author_facet | Hu, Jiangsheng Ren, Wei Yang, Xiaoyan You, Hanyang |
| contents | Let $(\mathcal{X}, \mathcal{Y})$ be a balanced pair in an abelian category $\mathcal{A}$. Denote by ${\bf K}_{\mathcal{E}\text{-}{\rm ac}}(\mathcal{X})$ the chain homotopy category of right $\mathcal{X}$-acyclic complexes with all items in $\mathcal{X}$, and dually by ${\bf K}_{\mathcal{E}\text{-}{\rm ac}}(\mathcal{Y})$ the chain homotopy category of left $\mathcal{Y}$-acyclic complexes with all items in $\mathcal{Y}$. We establish realizations of ${\bf K}_{\mathcal{E}\text{-}{\rm ac}}(\mathcal{X})$ and ${\bf K}_{\mathcal{E}\text{-}{\rm ac}}(\mathcal{Y})$ as homotopy categories of model categories under mild conditions. Consequently, we obtain relative versions of recollements of Krause and Neeman-Murfet. We further give applications to Gorenstein projective and Gorenstein injective modules. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2510_16294 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Models for chain homotopy category of relative acyclic complexes Hu, Jiangsheng Ren, Wei Yang, Xiaoyan You, Hanyang Representation Theory Category Theory Rings and Algebras 18G25, 18N40, 18E10, 18G35, 18G80 Let $(\mathcal{X}, \mathcal{Y})$ be a balanced pair in an abelian category $\mathcal{A}$. Denote by ${\bf K}_{\mathcal{E}\text{-}{\rm ac}}(\mathcal{X})$ the chain homotopy category of right $\mathcal{X}$-acyclic complexes with all items in $\mathcal{X}$, and dually by ${\bf K}_{\mathcal{E}\text{-}{\rm ac}}(\mathcal{Y})$ the chain homotopy category of left $\mathcal{Y}$-acyclic complexes with all items in $\mathcal{Y}$. We establish realizations of ${\bf K}_{\mathcal{E}\text{-}{\rm ac}}(\mathcal{X})$ and ${\bf K}_{\mathcal{E}\text{-}{\rm ac}}(\mathcal{Y})$ as homotopy categories of model categories under mild conditions. Consequently, we obtain relative versions of recollements of Krause and Neeman-Murfet. We further give applications to Gorenstein projective and Gorenstein injective modules. |
| title | Models for chain homotopy category of relative acyclic complexes |
| topic | Representation Theory Category Theory Rings and Algebras 18G25, 18N40, 18E10, 18G35, 18G80 |
| url | https://arxiv.org/abs/2510.16294 |