The Golod property for powers of ideals and Koszul ideals

TL;DR

This paper investigates when powers of ideals in regular local or polynomial rings are Golod, revealing that Golodness can be detected via homology maps and extending known results to broader classes of rings.

Contribution

It generalizes the criteria for Golod ideals from graded to local rings and identifies new classes of Golod ideals based on homological properties.

Findings

Golod property characterized by vanishing homology maps

Extension of Golod criteria from graded to local rings

Identification of new classes of Golod ideals

Abstract

Let $S$ be a regular local ring or a polynomial ring over a field and $I$ be an ideal of $S$. Motivated by a recent result of Herzog and Huneke, we study the natural question of whether $I^m$ is a Golod ideal for all $m\geq 2$. We observe that the Golod property of an ideal can be detected through the vanishing of certain maps induced in homology. This observation leads us to generalize some known results from the graded case to local rings and obtain new classes of Golod ideals.