Minimally non-Golod face rings and Massey products

TL;DR

This paper corrects a previous criterion for when a face ring is Golod, and constructs an example of a minimally non-Golod complex with specific cohomological properties, advancing understanding of face rings and Massey products.

Contribution

It provides a corrected criterion for Golodness of face rings and constructs a novel example illustrating complex cohomological phenomena.

Findings

Corrected the Golod criterion for face rings.

Constructed a minimally non-Golod complex with trivial cup product.

Found a complex with non-trivial triple Massey product.

Abstract

We give a correct statement and a complete proof of the criterion obtained by Grbi\'c, Panov, Theriault and Wu for the face ring $\Bbbk[K]$ of a simplicial complex $K$ to be Golod over a field $\Bbbk$. (The original argument depended on the main result of a paper by Berglund and J\"ollenbeck, which was shown to be false by Katth\"an.) We also construct an example of a minimally non-Golod complex $K$ such that the cohomology of the corresponding moment-angle complex $\mathcal Z_K$ has trivial cup product and a non-trivial triple Massey product.