Regular local rings of dimension four and Gorenstein syzygetic prime ideals

TL;DR

This paper characterizes four-dimensional regular local rings through the property that all Gorenstein quotient prime ideals are syzygetic, using André-Quillen homology for the characterization.

Contribution

It provides a new criterion for regularity of local rings of dimension four based on Gorenstein prime ideals and syzygetic properties, linked via André-Quillen homology.

Findings

Regular local rings of dimension at most four are characterized by syzygetic Gorenstein prime ideals.

A new homological criterion for regularity involving André-Quillen homology.

Equivalence between regularity and properties of Gorenstein quotient ideals.

Abstract

Let $R$ be a Noetherian local ring. We prove that $R$ is regular of dimension at most four if, and only if, every prime ideal, defining a Gorenstein quotient ring, is syzygetic. We deduce a characterization of these rings in terms of the Andr\'e-Quillen homology.