Stability of Gorenstein objects in triangulated categories

TL;DR

This paper investigates the stability and characterizations of Gorenstein projective objects within triangulated categories, extending Gorenstein homological algebra by proving strong stability results and providing new equivalent definitions.

Contribution

It establishes the strong stability of $\xi$-Gorenstein projective objects and offers new characterizations of their Gorenstein projective dimension in triangulated categories.

Findings

The subcategory of $\xi$-Gorenstein projective objects is strongly stable.

Iteration of the defining procedure for $\xi$-Gorenstein projectives yields the same objects.

Provides equivalent characterizations for $\xi$-Gorenstein projective dimension.

Abstract

Let $\mathcal{C}$ be a triangulated category with a proper class $\xi$ of triangles. Asadollahi and Salarian introduced and studied $\xi$-Gorenstein projective and $\xi$-Gorenstein injective objects, and developed Gorenstein homological algebra in $\mathcal{C}$. In this paper, we further study Gorenstein homological properties for a triangulated category. First, we discuss the stability of $\xi$-Gorenstein projective objects, and show that the subcategory $\mathcal{GP}(\xi)$ of all $\xi$-Gorenstein projective objects has a strong stability. That is, an iteration of the procedure used to define the $\xi$-Gorenstein projective objects yields exactly the $\xi$-Gorenstein projective objects. Second, we give some equivalent characterizations for $\xi$-Gorenstein projective dimension of object in $\mathcal{C}$.