Stability of Gorenstein objects in triangulated categories

Abstract

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