On the existence of embeddings into modules of finite homological dimensions

TL;DR

This paper characterizes Gorenstein rings via embeddings of modules into modules with finite projective dimension, extending previous results to higher-dimensional rings and relaxing Cohen-Macaulay assumptions.

Contribution

It generalizes a known characterization of Gorenstein rings to rings of higher Krull dimension and removes the Cohen-Macaulay restriction.

Findings

R is Gorenstein iff every finitely generated R-module embeds into a module of finite projective dimension

Extends Auslander's and Bridger's result to higher dimensions

Improves Foxby's result by removing Cohen-Macaulay assumption

Abstract

Let R be a commutative Noetherian local ring. We show that R is Gorenstein if and only if every finitely generated R-module can be embedded in a finitely generated R-module of finite projective dimension. This extends a result of Auslander and Bridger to rings of higher Krull dimension, and it also improves a result due to Foxby where the ring is assumed to be Cohen-Macaulay.