Gluing of cotorsion pairs via recollements of abelian categories
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866917321293955072 |
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| author | Yang, Jinrui Qin, Yongyun |
| author_facet | Yang, Jinrui Qin, Yongyun |
| contents | Let $( \mathcal{A^{'}},\mathcal{A},\mathcal{A^{''}},i^\ast,i_\ast,i^!,j_!,j^\ast,j_\ast)$ be a recollement of abelian categories. Suppose that we are given two cotorsion pairs $({\mathcal{U^{'}}},\mathcal{V{'}})$ and $({\mathcal{U}^{''}},{\mathcal{V}^{''}})$ in $\mathcal{A}^{'}$ and $\mathcal{A}^{''}$, respectively. We construct two cotorsion pairs $(^{\bot}{\mathcal{N}_{\mathcal{V^{''}}}^{\mathcal{V^{'}}}},\mathcal{N}_{\mathcal{V^{''}}}^{\mathcal{V^{'}}})$ and $(\mathcal{M}_{\mathcal{U^{''}}}^{\mathcal{U^{'}}}, ({\mathcal{M}_{\mathcal{U^{''}}}^{\mathcal{U^{'}}}})^\bot)$ in $\mathcal{A}$. Moreover, we provide a sufficient condition for these two cotorsion pairs to coincide, and we investigate the heredity and completeness of $(\mathcal{M}_{\mathcal{U^{''}}}^{\mathcal{U^{'}}},\mathcal{N}_{\mathcal{V^{''}}}^{\mathcal{V^{'}}})$. These results are applied to construct new cotorsion pairs in Morita rings. In the course of proof, we introduce a specific constraint on recollements of abelian categories, requiring $\varepsilon_P$ to be a monomorphism for any projective $P \in \mathcal{A}$, with $\varepsilon: j_!j^* \to \mathrm{id}_{\mathcal{A}}$ being the counit of $(j_!, j^*)$. Such recollements enjoy rich homological properties and hence might be of independent interest. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_06994 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Gluing of cotorsion pairs via recollements of abelian categories Yang, Jinrui Qin, Yongyun Representation Theory Rings and Algebras Let $( \mathcal{A^{'}},\mathcal{A},\mathcal{A^{''}},i^\ast,i_\ast,i^!,j_!,j^\ast,j_\ast)$ be a recollement of abelian categories. Suppose that we are given two cotorsion pairs $({\mathcal{U^{'}}},\mathcal{V{'}})$ and $({\mathcal{U}^{''}},{\mathcal{V}^{''}})$ in $\mathcal{A}^{'}$ and $\mathcal{A}^{''}$, respectively. We construct two cotorsion pairs $(^{\bot}{\mathcal{N}_{\mathcal{V^{''}}}^{\mathcal{V^{'}}}},\mathcal{N}_{\mathcal{V^{''}}}^{\mathcal{V^{'}}})$ and $(\mathcal{M}_{\mathcal{U^{''}}}^{\mathcal{U^{'}}}, ({\mathcal{M}_{\mathcal{U^{''}}}^{\mathcal{U^{'}}}})^\bot)$ in $\mathcal{A}$. Moreover, we provide a sufficient condition for these two cotorsion pairs to coincide, and we investigate the heredity and completeness of $(\mathcal{M}_{\mathcal{U^{''}}}^{\mathcal{U^{'}}},\mathcal{N}_{\mathcal{V^{''}}}^{\mathcal{V^{'}}})$. These results are applied to construct new cotorsion pairs in Morita rings. In the course of proof, we introduce a specific constraint on recollements of abelian categories, requiring $\varepsilon_P$ to be a monomorphism for any projective $P \in \mathcal{A}$, with $\varepsilon: j_!j^* \to \mathrm{id}_{\mathcal{A}}$ being the counit of $(j_!, j^*)$. Such recollements enjoy rich homological properties and hence might be of independent interest. |
| title | Gluing of cotorsion pairs via recollements of abelian categories |
| topic | Representation Theory Rings and Algebras |
| url | https://arxiv.org/abs/2603.06994 |