Intersection of complete cotorsion pairs

TL;DR

This paper investigates how to combine complete cotorsion pairs in exact categories, providing new constructions and characterizations relevant to Gorenstein homological algebra.

Contribution

It proves that the intersection of certain cotorsion pairs forms a new complete cotorsion pair and applies this to Gorenstein modules and complexes.

Findings

Constructed new complete cotorsion pairs involving Gorenstein dimensions

Characterized classes of modules with finite Gorenstein homological dimensions

Provided conditions under which cotorsion pairs are hereditary

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 $\left({\rm Smd}\langle \mathcal{X}_1,\mathcal{X}_2 \rangle,\mathcal{Y}_1\cap \mathcal{Y}_2\right)$ is also a (hereditary) complete cotorsion pair, where ${\rm Smd}\langle \mathcal{X}_1,\mathcal{X}_2 \rangle$ 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 $(^\perp\mathcal{GI}^{\leqslant n},\mathcal{GI}^{\leqslant n})$, where $\mathcal{GI}^{\leqslant n}$ 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.