Cotorsion pairs in extensions of abelian categories

TL;DR

This paper explores how cotorsion pairs in an abelian category extend to certain categories constructed from it, analyzing their properties and providing applications in algebraic structures.

Contribution

It constructs and studies cotorsion pairs in extension categories derived from an abelian category, including their heredity and completeness, with applications to algebraic structures.

Findings

Constructed cotorsion pairs in extension categories

Analyzed heredity and completeness of these cotorsion pairs

Provided applications to comma categories and Morita context rings

Abstract

Let $\mathcal{B}$ be an abelian category with enough projective objects and enough injective objects and let $\mathcal{A}=\mathcal{B}\ltimes_\eta\mathsf{F}$ be an $\eta$-extension of $\mathcal{B}$. Given a cotorsion pair $(\mathcal{X},\;\mathcal{Y})$ in $\mathcal{B}$, we construct a cotorsion pair $(^{\perp}{\mathsf{U}^{-1}(\mathcal{Y})}, \mathsf{U}^{-1}(\mathcal{Y}))$ in $\mathcal{A}$ and a cotorsion pair $(\Delta(\mathcal{X}),\;\Delta(\mathcal{X})^\perp)$ in $\mathcal{A}$ for $\mathsf{F}^2=0$. In addition, the heredity and completeness of these cotorsion pairs are studied. Finally, we give some applications and examples in comma categories, some Morita context rings and trivial extensions of rings.