Products of Ideals and Golod Rings

TL;DR

This paper investigates when products of ideals in certain rings produce Golod rings, establishing conditions under which this occurs and demonstrating the prevalence of non-Golod products.

Contribution

It provides new criteria for when products of ideals in 3-dimensional regular rings are Golod and shows the widespread existence of non-Golod products.

Findings

In 3-dimensional regular local rings, I*m always defines a Golod ring.

For ideals with grade ≥ 4, some products are not Golod.

If I contains a complete intersection, then mfa*I is Golod.

Abstract

In this paper, we study conditions guaranteeing that a product of ideals defines a Golod ring. We show that for a $3$-dimensional regular local ring (or $3$-variable polynomial ring) $(R , \m)$, the ideal $I \m$ always defines a Golod ring for any proper ideal $I \subset R$. We also show that non-Golod products of ideals are ubiquitous; more precisely, we prove that for any proper ideal with grade $\geq 4$, there exists an ideal $J \subseteq I$ such that $IJ$ is not Golod. We conclude by showing that if $I$ is any proper ideal in a $3$-dimensional regular local ring and $\mfa \subseteq I$ a complete intersection, then $\mfa I$ is Golod.