Detecting Golodness via Gr\"obner Degeneration

TL;DR

This paper explores how Golodness, a property in algebra, can be transferred via Gr"obner degeneration, establishing new results for monomial and rainbow monomial ideals, and proving Golodness for maximal minors of generic matrices.

Contribution

It introduces methods to transfer Golodness through Gr"obner degeneration and proves Golodness for rainbow monomial ideals and maximal minors of generic matrices.

Findings

Golodness of fiber invariant ideals is equivalent to that of their initial ideals.

Rainbow monomial ideals with linear resolution define Golod rings.

Maximal minors of sparse generic matrices are Golod rings, regardless of characteristic.

Abstract

In this paper we study the extent to which Golodness may be transferred along morphisms of DG-algebras. In particular, we show that if $I$ is a so-called fiber invariant ideal, then Golodness of $I$ is equivalent to Golodness of the initial ideal of $I$. We use this to transfer Golodness results for monomial ideals to more general classes of ideals. We also prove that any so-called rainbow monomial ideal with linear resolution defines a Golod ring; this result encompasses and generalizes many known Golodness results for classes of monomial ideals. We then combine the techniques developed to give a concise proof that maximal minors of (sparse) generic matrices define Golod rings, independent of characteristic.