Intersection of complete cotorsion pairs

Abstract

Given two (hereditary) complete cotorsion pairs $({\mathcal{X}}_1,{\mathcal{Y}}_1)$ and $({\mathcal{X}}_2,{\mathcal{Y}}_2)$ in an exact category with ${\mathcal{X}}_1\subseteq {\mathcal{Y}}_2$, we prove that $\left(\mathrm{Smd}⟨{\mathcal{X}}_1,{\mathcal{X}}_2⟩,{\mathcal{Y}}_1\cap {\mathcal{Y}}_2\right)$ is also a (hereditary) complete cotorsion pair, where $\mathrm{Smd}⟨{\mathcal{X}}_1,{\mathcal{X}}_2⟩$ is the class of direct summands of extension of ${\mathcal{X}}_1$ and ${\mathcal{X}}_2$. As an application, we construct complete cotorsion pairs, such as ${(}^{\perp }{\mathcal{G}\mathcal{I}}^{⩽n},{\mathcal{G}\mathcal{I}}^{⩽n})$, where ${\mathcal{G}\mathcal{I}}^{⩽n}$ is the class of modules of Gorenstein injective dimension at most $n$. And we also characterize the left orthogonal class of exact complexes of injective modules and the classes of modules with finite Gorenstein projective, Gorenstein flat, and PGF dimensions.