Homological Dimensions Relative to Preresolving Subcategories II

TL;DR

This paper explores the conditions under which the $ ext{T}$-projective and $ ext{T}$-injective dimensions in an abelian category relate across short exact sequences, with applications to Gorenstein rings and projective dimensions.

Contribution

It establishes that the assertion about dimensions holds if and only if the subcategory is resolving or coresolving, and applies this to characterize Gorenstein rings and compare projective dimensions.

Findings

The assertion holds iff $ ext{T}$ is resolving or coresolving.

A ring is $n$-Gorenstein iff certain module dimensions are bounded by $n$.

Finitistic $ ext{C}$-projective and $ ext{T}$-projective dimensions coincide in several cases.

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.