Homological dimensions via silting and tilting objects in triangulated categories

This paper investigates how classical homological invariants extend from module categories to a triangulated category T with a compact silting generator M .We construct sup-projective, inf-injective and sup-flat resolutions for objects in T , and use them to establish basic properties of the corresponding homological dimensions.We then introduce the global dimension gldimT and prove that it coincides with the global dimension of the heart H M = M ⊥ ̸ =0 and with the Add(M )-dimension of T b when M is tilting.These results provide a unified framework for homological dimension theory in triangulated categories and reveal how silting and tilting objects encode essential homological information.As applications, we study homological properties such as Gorensteinness and regularity for T with a focus on the derived category over rings and dg rings.