Gluing of cotorsion pairs via recollements of abelian categories

TL;DR

This paper develops a method to construct and analyze cotorsion pairs in abelian categories using recollements, with applications to Morita rings, and introduces a special class of recollements with notable homological properties.

Contribution

It introduces a new approach to glue cotorsion pairs via recollements of abelian categories, including conditions for their coincidence and properties, and applies these to Morita rings.

Findings

Constructed new cotorsion pairs in abelian categories.

Provided conditions for cotorsion pairs to coincide.

Applied results to Morita rings.

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.