Tuyển tập các báo cáo nghiên cứu khoa học trên tạp chí toán học quốc tế đề tài: Face numbers and nongeneric initial ideals. | Face numbers and nongeneric initial ideals Eric Babson and Isabella Novik Department of Mathematics University of Washington Seattle WA 98195-4350 USA babson novik @ Submitted Jun 30 2005 Accepted Dec 26 2005 Published Jan 3 2006 Mathematics Subject Classifications 52B05 13F55 05E25 Dedicated to Richard Stanley on the occasion of his 60th birthday. Abstract Certain necessary conditions on the face numbers and Betti numbers of sim-plicial complexes endowed with a proper action of a prime order cyclic group are established. A notion of colored algebraic shifting is defined and its properties are studied. As an application a new simple proof of the characterization of the flag face numbers of balanced Cohen-Macaulay complexes originally due to Stanley necessity and Bjorner Frankl and Stanley sufficiency is given. The necessity portion of their result is generalized to certain conditions on the face numbers and Betti numbers of balanced Buchsbaum complexes. 1 Introduction In this paper we study the face numbers of two classes of simplicial complexes complexes endowed with a group action and balanced complexes. We accomplish this by exploring the behavior of a special only partially generic initial ideal of the Stanley-Reisner ideal of a simplicial complex. The face numbers are basic invariants of simplicial complexes and their study goes back to Kruskal 14 and Katona 12 who characterized the face numbers of all finite sim-plicial complexes. Since then many powerful tools and techniques have been developed among them are the theory of Stanley-Reisner rings and the method of algebraic shifting introduced by Kalai and closely related to the notion of generic initial ideals. Both techniques have resulted in many beautiful applications including the characterization of the face numbers of all Cohen-Macaulay complexes due to Stanley 20 the characterization of the flag face numbers of all balanced Cohen-Macaulay complexes due to Stanley 21 necessity and .
