Stability of Gorenstein objects in triangulated categories

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 $\mathcal{C}$.