Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Department of Mathematic dành cho các bạn yêu thích môn toán học đề tài: A combinatorial proof of a formula for Betti numbers of a stacked polytope. | A combinatorial proof of a formula for Betti numbers of a stacked polytope Suyoung Choi Department of Mathematical Sciences KAIST Republic of Korea choisy@ Current Department of Mathematics Osaka City University Japan choi@ Jang Soo Kim Department of Mathematical Sciences KAIST Republic of Korea j skim@ Current LIAFA University of Paris 7 France Submitted Aug 8 2009 Accepted Dec 13 2009 Published Jan 5 2010 Mathematics Subject Classifications 05A15 05E40 05E45 52B05 Abstract For a simplicial complex A the graded Betti number flij k A of the Stanley-Reisner ring k A over a field k has a combinatorial interpretation due to Hochster. Terai and Hibi showed that if A is the boundary complex of a d-dimensional stacked polytope with n vertices for d 3 then Pk-Ỉ k k A k 1 ra-d . We prove this combinatorially. 1 Introduction A simplicial complex A on a finite set V is a collection of subsets of V satisfying 1. if v G V then v G A 2. if F G A and F c F then F G A. Each element F G A is called a face of A. The dimension of F is defined by dim F F 1. The dimension of A is defined by dim A max dim F F G A . For a subset W c V let AW denote the simplicial complex F n W F G A on W. The research of the first author was carried out with the support of the Japanese Society for the Promotion of Science JSPS grant no. P09023 and the Brain Korea 21 Project KAIST. The second author was supported by the SRC program of Korea Science and Engineering Foundation KOSEF grant funded by the Korea government MEST No. R11-2007-035-01002-0 . THE ELECTRONIC JOURNAL OF COMBINATORICS 17 2010 R9 1 Let A be a simplicial complex on V. Two elements v u G V are said to be connected if there is a sequence of vertices v u0 ui . ur u such that u u i G A for all i 0 1 . r 1. A connected component C of A is a maximal nonempty subset of V such that every two elements of C are connected. Let V x1 x2 . xn and let R be the polynomial ring k x1 . xn over a fixed field k. Then
