Gorenstein homological dimensions for triangulated categories

Let CC be a triangulated category with a proper class EE of triangles. Asadollahi and Salarian introduced and studied EE-GGprojective and EE-GGinjective objects, and developed a relative homological algebra in CC.In this paper, we further study Gorenstein homological dimensions for triangulated categories. First, we discuss the finiteness of EE-GGprojective and EE-GGinjective dimensions. Then we define EE-Gorenstein derived functors GExtEi(−,−), and characterize EE-GGprojective and EE-GGinjective dimensions of objects in CC by vanishing of such functors. Consequently we can prove the equality sup{E-GpdM|for any M∈C}=sup{E-GidM|for any M∈C}sup{E-GpdM|for any M∈C}=sup{E-GidM|for any M∈C}, which is used to define the global EE-Gorenstein dimension of CC. Finally, EE-GGphantom tower and EE-GGcellular tower for objects in CC with finite EE-GGprojective dimension are constructed.