Gorenstein projective complexes with respect to cotorsion pairs

TL;DR

This paper introduces Gorenstein projective complexes relative to cotorsion pairs in module categories, establishing their properties and stability, thus extending Gorenstein homological algebra.

Contribution

It defines Gorenstein projective complexes with respect to cotorsion pairs and proves their equivalence to complexes of Gorenstein projective modules, highlighting their stability.

Findings

Gorenstein projective complexes are characterized as complexes of Gorenstein projective modules.

Both Gorenstein projective modules and complexes with respect to cotorsion pairs are stable.

The paper extends Gorenstein homological algebra to complexes relative to cotorsion pairs.

Abstract

Let $(\mathcal{A,B})$ be a complete and hereditary cotorsion pair in the category of left $R$-modules. In this paper, the so-called Gorenstein projective complexes respect to the cotorsion pair $(\mathcal{A}, \mathcal{B})$ are introduced. We show that these complexes are just the complexes of Gorenstein projective modules respect to the cotorsion pair $(\mathcal{A}, \mathcal{B})$. As applications, we prove that both Gorenstein projective modules with respect to cotorsion pairs and Gorenstein projective complexes with respect to cotorsion pairs possess of stability.