Products of ideals and Golod rings

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) $(R, \mathfrak {m})$, the ideal $I \mathfrak {m}$ 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 $\geqslant 4$, 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 $\mathfrak {a} \subseteq I$ a complete intersection, then $\mathfrak {a} I$ is Golod.