Shellability and the Strong gcd-Condition

Abstract

Shellability is a well-known combinatorial criterion on a simplicial complex $\Delta$ for verifying that the associated Stanley-Reisner ring $k[\Delta]$ is Cohen-Macaulay. A 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, Jöllenbeck introduced a criterion on simplicial complexes reminiscent of shellability, called the strong gcd-condition, and he together with the author proved that it implies Golodness of the associated Stanley-Reisner ring. The two algebraic notions were earlier tied together by Herzog, Reiner and Welker, who showed that if $k[\Delta^\vee]$ is sequentially Cohen-Macaulay, where $\Delta^\vee$ is the Alexander dual of $\Delta$, then $k[\Delta]$ is Golod. In this paper, we present a combinatorial companion of this result, namely that if $\Delta^\vee$ is (non-pure) shellable then $\Delta$ satisfies the strong gcd-condition. Moreover, we show that all implications just mentioned are strict in general but that they are equivalences if $\Delta$ is a flag complex.