International Journal of Mathematics and Mathematical Sciences
Volume 17 (1994), Issue 3, Pages 545-552
doi:10.1155/S0161171294000797
Pascal type properties of Betti numbers
Department of Mathematics, Southeastern Louisiana University, Hammond 70402, Louisiana, USA
Received 13 January 1993
Copyright © 1994 Tilak de Alwis. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
In this paper, we will describe the Pascal Type properties of Betti numbers of ideals associated to n-gons. These are quite similar to the properties enjoyed by the Pascal's Triangle, concerning the binomial coefficients. By definition, the Betti numbers βt(n) of an ideal I associated to an n-gon are the ranks of the modules in a free minimal resolution of the R-module R/I, where R is the polynomial ring k[x1,x2,…,xn]. Here k is any field and x1,x2,…,xn are indeterminates. We will prove those properties using a specific formula for the Betti numbers.