Homological dimensions relative to preresolving subcategories II

Let A be an abelian category having enough projective and injective objects, and let T be an additive subcategory of A closed under direct summands. A known assertion is that in a short exact sequence in A, the T-projective (resp. T-injective) dimensions of any two terms can sometimes induce an upper bound of that of the third term by using the same comparison expressions. We show that if T contains all projective (resp. injective) objects of A, then the above assertion holds true if and only if T is resolving (resp. coresolving). As applications, we get that a left and right Noetherian ring R is n-Gorenstein if and only if the Gorenstein projective (resp. injective, flat) dimension of any left R-module is at most n. In addition, in several cases, for a subcategory C of T, we show that the finitistic C-projective and T-projective dimensions of A are identical.