Ph.D. (2004) Cornell University
Abstract: This thesis discusses the connection between Hilbert functions and graded free resolutions. One can compute the Hilbert function of a module from its graded free resolution. However, there may be modules with the same Hilbert function but different graded Betti numbers. We investigate what graded free resolutions actually occur for a given Hilbert function. We provide a quick review of background material in the first chapter.
In Chapter 2, we consider the simplest case of this question: For what Hilbert functions is there only one possible set of graded Betti numbers? We characterize these Hilbert functions for ideals in k [x, y]. In more variables, the question is more difficult, and we give examples of some families of Hilbert functions for which there is a unique set of attainable graded Betti numbers.
The third chapter focuses on a stable ideals. Fix a Hilbert function H. We ask if there exists a stable ideal I with Hilbert function H such that the graded Betti numbers of any other stable ideal J with the same Hilbert function are at least as large as those of I. We answer the question affirmatively for ideals in k [x_1, x_2, x_3] but provide a counterexample in four or more variables.
In the fourth and fifth chapters, we investigate the Lex-Plus-Powers Conjecture on graded Betti numbers and the EGH Conjecture on Hilbert functions. We devote Chapter 4 to introducing versions of the conjectures and the relationships among them. In Chapter 5, we prove that the graded Betti numbers of a lex-plus-powers almost complete intersection are an upper bound for the graded Betti numbers of any ideal with the same Hilbert function and regular sequence in the same degrees. In addition, we give some related results on Hilbert functions.
We conclude in Chapter 6 by investigating a conjecture of Huneke and Srinivasan on the multiplicity of an ideal. They conjecture bounds for the multiplicity in terms of the graded free resolution. We prove some special cases of the conjecture for ideals with generators in high degrees relative to their regularity, survey some computational work, and discuss some reductions of the conjecture.