Shellability and the strong gcd-condition
2009 (English)In: The Electronic Journal of Combinatorics, ISSN 1077-8926, Vol. 16, no 2, R1- p.Article in journal (Refereed) Published
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, Jollenbeck 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(V)] is sequentially Cohen-Macaulay, where Delta(V) 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(V) 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.
Place, publisher, year, edition, pages
2009. Vol. 16, no 2, R1- p.
monomial rings, poincare-series, golod property
IdentifiersURN: urn:nbn:se:su:diva-60364ISI: 000263259900001OAI: oai:DiVA.org:su-60364DiVA: diva2:435767
authorCount :12011-08-192011-08-162011-08-19Bibliographically approved