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Báo cáo toán học: " Algebraic Shifting and Sequentially Cohen-Macaulay Simplicial Complexe"

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí toán học quốc tế đề tài: Algebraic Shifting and Sequentially Cohen-Macaulay Simplicial Complexes. | Algebraic Shifting and Sequentially Cohen-Macaulay Simplicial Complexes Art M. Duval University of Texas at El Paso Department of Mathematical Sciences El Paso TX 79968-0514 artduval@math.utep.edu Submitted February 2 1996 Accepted July 23 1996. Abstract Bjorner and Wachs generalized the dehnition of shellability by dropping the assumption of purity they also introduced the h-triangle a doubly-indexed generalization of the h-vector which is combinatorially signihcant for nonpure shellable complexes. Stanley subsequently dehned a nonpure simplicial complex to be sequentially Cohen-Macaulay if it satishes algebraic conditions that generalize the Cohen-Macaulay conditions for pure complexes so that a nonpure shellable complex is sequentially Cohen-Macaulay. We show that algebraic shifting preserves the h-triangle of a simplicial complex K if and only if K is sequentially Cohen-Macaulay. This generalizes a result of Kalai s for the pure case. Immediate consequences include that nonpure shellable complexes and sequentially Cohen-Macaulay complexes have the same set of possible h-triangles. 1991 Mathematics Subject Classification Primary 06A08 Secondary 52B05. 1 Introduction A simplicial complex is pure if all of its facets maximal faces ordered by inclusion have the same dimension. Cohen-Macaulayness algebraic shifting shellability and the h-vector are significantly interrelated for pure simplicial complexes. We will be concerned with extending some of these relations to nonpure complexes but first we briefly review the pure case. More detailed definitions are in later sections. A simplicial complex is Cohen-Macaulay if its face-ring is a Cohen-Macaulay ring an algebraic property or equivalently if the complex satisfies certain topological conditions see e.g. St3 St6 . In particular the complex must be pure. A pure simplicial complex 1 THE ELECTRONIC .JOURNAL OF COMBINATORICS 3 1996 R21 2 is shellable if it can be constructed one facet at a time subject to certain .