Noetherian rings of low global dimension and syzygetic prime ideals

TL;DR

This paper characterizes Noetherian rings with low global dimension by properties of their prime ideals, linking algebraic homological conditions to geometric ideal properties, and uses André-Quillen homology for these characterizations.

Contribution

It establishes new equivalences between global dimension bounds and prime ideal properties in Noetherian rings, utilizing homological tools.

Findings

R has global dimension ≤ 2 iff all prime ideals are of linear type

R has global dimension ≤ 3 iff all prime ideals are syzygetic

Characterization via André-Quillen homology

Abstract

Let $R$ be a Noetherian ring. We prove that $R$ has global dimension at most two if, and only if, every prime ideal of $R$ is of linear type. Similarly, we show that $R$ has global dimension at most three if, and only if, every prime ideal of $R$ is syzygetic. As a consequence, one derives a characterization of these rings using the Andr\'e-Quillen homology.