Homological Dimensions Relative to Preresolving Subcategories II

Abstract

Let $\mathscr{A}$ be an abelian category having enough projective and injective objects, and let $\mathscr{T}$ be an additive subcategory of $\mathscr{A}$ closed under direct summands. A known assertion is that in a short exact sequence in $\mathscr{A}$, the $\mathscr{T}$-projective (respectively, $\mathscr{T}$-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 $\mathscr{T}$ contains all projective (respectively, injective) objects of $\mathscr{A}$, then the above assertion holds true if and only if $\mathscr{T}$ is resolving (respectively, coresolving). As applications, we get that a left and right Noetherian ring $R$ is $n$-Gorenstein if and only if the Gorenstein projective (respectively, injective, flat) dimension of any left $R$-module is at most $n$. In addition, in several cases, for a subcategory $\mathscr{C}$ of $\mathscr{T}$, we show that the finitistic $\mathscr{C}$-projective and $\mathscr{T}$-projective dimensions of $\mathscr{A}$ are identical.