The Golod property of powers of the maximal ideal of a local ring

TL;DR

This paper characterizes when powers of the maximal ideal in a local ring are Golod, showing that for certain generic artinian Gorenstein rings with specific nilpotency, the quotient is not Golod, except when the ideal is generated by two elements.

Contribution

It identifies minimal cases where powers of the maximal ideal are not Golod, extending previous results and providing new conditions based on the number of generators.

Findings

For 2-generated maximal ideals, all powers are Golod.

In generic artinian Gorenstein rings with m^4=0 and m^3≠0, the quotient R/m^3 is not Golod.

The result complements and extends earlier work by Rossi and Šega.

Abstract

We identify minimal cases in which a power $m^i\not=0$ of the maximal ideal of a local ring $R$ is not Golod, i.e.\ the quotient ring $R/m^i$ is not Golod. Complementary to a 2014 result by Rossi and \c{S}ega, we prove that for a generic artinian Gorenstein local ring with $m^4=0\not= m^3$, the quotient $R/m^3$ is not Golod. This is provided that $m$ is minimally generated by at least $3$ elements. Indeed, we show that if $m$ is $2$-generated, then every power $m^i\not= 0$ is Golod.