Massey products and the Golod property for simplicially resolvable rings

TL;DR

This paper uses algebraic Morse theory to analyze Massey products and the Golod property in monomial rings, providing new criteria and combinatorial characterizations for when such rings are Golod.

Contribution

It introduces a novel application of algebraic Morse theory to describe Massey products and characterizes Golod rings via combinatorial conditions in the simplicial case.

Findings

Higher Massey products vanish for simplicial resolutions.

A ring is Golod iff the product on Tor vanishes in this setting.

Provides two combinatorial criteria for Golodness.

Abstract

We apply algebraic Morse theory to the Taylor resolution of a monomial ring $R = S/I$ to obtain an $A_{\infty}$-structure on the minimal free resolution of $R$. Using this structure we describe the vanishing of higher Massey products in case the minimal free resolution is simplicial. Under this assumption, we show that R is Golod if and only if the product on $\text{Tor}^S(R, k)$ vanishes. Lastly, we give two combinatorial characterizations of the Golod property in this case.