Gluing of cotorsion pairs via recollements of abelian categories

Abstract

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.

Theorems & Definitions (48)

• Theorem 1.1
• Theorem 1.2
• Definition 2.1
• Remark 2.2
• Definition 2.3
• Proposition 2.4
• Proposition 2.5
• Lemma 3.1
• proof
• Lemma 3.2