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Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí toán học quốc tế đề tài: Shellability and the strong gcd-condition. | Shellability and the strong gcd-condition Alexander Berglund Department of Mathematics Stockholm University Sweden alexb@math.su.se Submitted Aug 13 2008 Accepted Feb 3 2009 Published Feb 11 2009 Mathematics Subject Classification 55U10 13F55 Abstract Shellability is a well-known combinatorial criterion on a simplicial complex A for verifying that the associated Stanley-Reisner ring k A 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 Av is sequentially Cohen-Macaulay where Av is the Alexander dual of A then k A is Golod. In this paper we present a combinatorial companion of this result namely that if Av is non-pure shellable then A satisfies the strong gcd-condition. Moreover we show that all implications just mentioned are strict in general but that they are equivalences if A is a flag complex. To Anders Bjorner on his sixtieth birthday 1 Introduction Let A be a finite simplicial complex with vertex set V vi . vn and let k be a field. Recall that the Stanley-Reisner ring associated to A is the quotient k A k xi . xn I where Ia is the ideal in the polynomial ring k x1 . xn generated by the monomials xi1 . xir for which vi1 . vir ị A. The Cohen-Macaulay property of Stanley-Reisner Current affiliation Department of Mathematical Sciences University of Copenhagen Denmark. Email alexb@math.ku.dk THE ELECTRONIC JOURNAL OF COMBINATORICS 16 2 2009 R1 1 rings has been intensely studied and this has led to several important results in combinatorics. See the book 12 for an overview. The generalized .
