A graph-theoretical characterization of the order complexes on the 2-sphere

✍ Scribed by Takayuki Hibi; Hiroshi Narushima; Morimasa Tsuchiya; Kei-ichi Watanabe

• Publisher: Elsevier Science
• Year: 1988
• Tongue: English
• Weight: 376 KB
• Volume: 72
• Category: Article
• ISSN: 0012-365X
• DOI: 10.1016/0012-365x(88)90206-3

✦ Synopsis

Background

Recently, some remarkable connections between commutative algebra and combinatorics have been discovered. Among the main topics in this area are the concepts of Cohen-Macaulay and Gorenstein complexes. Recall some basic definitions. Let V = {x1, x2, . . . , x,} be a finite set and A a simplicial complex on V, i.e. A is a collection of subsets of V satisfying (1) {v} E A for all IJ E V and (2) o E A, t c o imply t E A. The dimension of A is ( maxoEA #(u)) -1, where #(a) is the cardinality of u as a set. If #(a) = i + 1, then u is called an i-face. Also, let A = k[x,, x2, . . . , x,] be the polynomial ring in n-variables over a field k. We regard the elements xi as indeterminates over k. Let Z, be the ideal of A generated by all square-free monomials not contained in A, namely 1~ = (Xi, Xi2 . * . Xir 1 il < i2 < ' . 'i,,