Gluing of cotorsion pairs via recollements of abelian categories

Abstract

Let $({\mathcal{A}}^{{}^{\prime }},\mathcal{A},{\mathcal{A}}^{{}^{\prime \prime }},i^{\ast },i_{\ast },i^{!},j_{!},j^{\ast },j_{\ast })$ be a recollement of abelian categories. Suppose that we are given two cotorsion pairs $({\mathcal{U}}^{{}^{\prime }},\mathcal{V}{}^{\prime })$ and $({\mathcal{U}}^{{}^{\prime \prime }},{\mathcal{V}}^{{}^{\prime \prime }})$ in ${\mathcal{A}}^{{}^{\prime }}$ and ${\mathcal{A}}^{{}^{\prime \prime }}$, respectively. We construct two cotorsion pairs ${(}^{\perp }{\mathcal{N}}_{{\mathcal{V}}^{{}^{\prime \prime }}}^{{\mathcal{V}}^{{}^{\prime }}},{\mathcal{N}}_{{\mathcal{V}}^{{}^{\prime \prime }}}^{{\mathcal{V}}^{{}^{\prime }}})$ and $({\mathcal{M}}_{{\mathcal{U}}^{{}^{\prime \prime }}}^{{\mathcal{U}}^{{}^{\prime }}},({\mathcal{M}}_{{\mathcal{U}}^{{}^{\prime \prime }}}^{{\mathcal{U}}^{{}^{\prime }}}{)}^{\perp })$ in $\mathcal{A}$. Moreover, we provide a sufficient condition for these two cotorsion pairs to coincide, and we investigate the heredity and completeness of $({\mathcal{M}}_{{\mathcal{U}}^{{}^{\prime \prime }}}^{{\mathcal{U}}^{{}^{\prime }}},{\mathcal{N}}_{{\mathcal{V}}^{{}^{\prime \prime }}}^{{\mathcal{V}}^{{}^{\prime }}})$. 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 ${\epsilon }_P$ to be a monomorphism for any projective $P\in \mathcal{A}$, with $\epsilon :j_{!}{j}^{\ast }\to {\mathrm{id}}_{\mathcal{A}}$ being the counit of $(j_{!},j^{\ast })$. Such recollements enjoy rich homological properties and hence might be of independent interest.