Homological dimensions relative to preresolving subcategories II

Abstract

Abstract Let 𝒜 {\mathscr{A}} be an abelian category having enough projective and injective objects, and let 𝒯 {\mathscr{T}} be an additive subcategory of 𝒜 {\mathscr{A}} closed under direct summands. A known assertion is that in a short exact sequence in 𝒜 {\mathscr{A}} , the 𝒯 {\mathscr{T}} -projective (resp. 𝒯 {\mathscr{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 {\mathscr{T}} contains all projective (resp. injective) objects of 𝒜 {\mathscr{A}} , then the above assertion holds true if and only if 𝒯 {\mathscr{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 {\mathscr{C}} of 𝒯 {\mathscr{T}} , we show that the finitistic 𝒞 {\mathscr{C}} -projective and 𝒯 {\mathscr{T}} -projective dimensions of 𝒜 {\mathscr{A}} are identical.