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

Abstract

We identify minimal cases in which a power $m^i\ne 0$ 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 Şega, we prove that for a generic artinian Gorenstein local ring with $m^4=0\ne m^3$, 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 $m^i\ne 0$ is Golod.

arXiv commentUpdated after peer review. Minor edits and new section with remarks on codepth 4 rings. Final version, so appear in Arch. Math. (Basel); 12 pp