Shellability and the strong gcd-condition

TL;DR

This paper explores the relationship between shellability, the strong gcd-condition, and Golodness in simplicial complexes, establishing new combinatorial criteria and equivalences, especially for flag complexes.

Contribution

It introduces a combinatorial criterion linking shellability of Alexander duals to the strong gcd-condition, expanding understanding of Golodness in simplicial complexes.

Findings

Shellability of Alexander dual implies the strong gcd-condition.

Equivalences between these properties hold for flag complexes.

Implications are strict in general but become equivalences for flag complexes.

Abstract

Shellability is a well-known combinatorial criterion for verifying that a simplicial complex is Cohen-Macaulay. Another notion familiar to commutative algebraists, but which has not received as much attention from combinatorialists as the Cohen-Macaulay property, is the notion of a Golod ring. Recently, a criterion on simplicial complexes reminiscent of shellability, called the strong gcd-condition, was shown to imply Golodness of the associated Stanley-Reisner ring. The two algebraic notions were tied together by Herzog, Reiner and Welker who showed that if the Alexander dual of a complex is sequentially Cohen-Macaulay then the complex itself is Golod. In this paper, we present a combinatorial companion of this result, namely that if the Alexander dual of a complex is (non-pure) shellable then the complex itself satisfies the strong gcd-condition. Moreover, we show that all implications just mentioned are strict in general but that they are equivalences for flag complexes.