Ordinary and symbolic powers are Golod

TL;DR

This paper establishes a criterion involving derivatives for when a quotient of a polynomial ring is Golod, and applies it to identify broad classes of Golod ideals such as powers and symbolic powers.

Contribution

It introduces a new condition based on partial derivatives that guarantees Golodness for quotients, expanding the understanding of Golod ideals.

Findings

S/I is Golod if the square of the ideal generated by partial derivatives is contained in I.

Identifies large classes of Golod ideals including powers, symbolic powers, and saturations.

Provides a new criterion for Golodness involving derivatives.

Abstract

Let $S$ be a positively graded polynomial ring over a field of characteristic 0, and $I\subset S$ a proper graded ideal. In this note it is shown that $S/I$ is Golod if $\partial(I)^2\subset I$. Here $\partial(I)$ denotes the ideal generated by all the partial derivatives of elements of $I$. We apply this result to find large classes of Golod ideals, including powers, symbolic powers, and saturations of ideals.