Minimal graded Betti numbers and stable ideals

摘要

Let k be a field, and let R = k[x(1), x(2), x(3)]. Given a Hilbert function H for a cyclic module over R, we give an algorithm to produce a stable ideal I such that R/I has Hilbert function H and uniquely minimal graded Betti numbers among all R/J with the same Hilbert function, where J is another stable ideal in R. We also show that such ail algorithm is impossible in more variables and disprove a related conjecture made by Decry (Deery, T. (1995). Rev-Lex segment ideals and minimal Betti numbers. Queen's Papers Pure Appl. Math. 102:193-219). [References: 12]