Cotorsion pairs in extensions of abelian categories

Abstract

Let $\mathcal{B}$ be an abelian category with enough projective objects and enough injective objects and let $\mathcal{A}=\mathcal{B}{⋉}_{\hat{I}}\cdot \mathsf{F}$ be an $\hat{I}\cdot$-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 $(\hat{I}”(\mathcal{X}),\hat{I}”(\mathcal{X}{)}^{\perp })$ 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.

21 pages