Shellability and the Strong gcd-Condition

Shellability is a well-known combinatorial criterion on a simplicial complex ∆ for verifying that the associated Stanley-Reisner ring k[∆] 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[∆] is sequentially Cohen-Macaulay, where ∆ is the Alexander dual of ∆, then k[∆] is Golod. In this paper, we present a combinatorial companion of this result, namely that if ∆ is (non-pure) shellable then ∆ satisfies the strong gcd-condition. Moreover, we show that all implications just mentioned are strict in general but that they are equivalences if ∆ is a flag complex. To Anders Björner on his sixtieth birthday