Monomial Ideals, N-Lists, and Smallest Graded Betti Numbers

The second set of results in this thesis concerns the existence of smallest graded betti numbers. Given an Artinian Hilbert function H , consider all ideals I ⊂ R such that H(R/I) = H . Then the graded betti numbers which correspond to these ideals form a finite set which we partially order component-wise. It is known that this set has a largest element, but may fail to have a smallest element. This thesis extends, to an infinite family, the previous examples of Hilbert functions which fail to have a smallest element. Then we prove a conjecture of Geramita, Harima, and Shin which states that a smallest element need not exist when we restrict our inquiry to the graded betti numbers of Gorenstein ideals attaining H . Finally, we demonstrate with an infinite family that a smallest element may fail to exist even if H is the Hilbert function of an R-sequence.

Name thesis:degree_name

Ph.D.

Level thesis:degree_level

Dissertation

Discipline thesis:degree_discipline

Mathematics

Grantor

University of Illinois at Urbana-Champaign

Year dc:date

2015

Author dc:creator

• Richert, Benjamin P.

Contributors dc:contributor

• Evans, E. Graham, Jr.

• Mathematics

Language dc:language

eng

Identifier

(MiAaPQ)AAI9971173

OAI identifier oai:identifier

oai:www.ideals.illinois.edu:2142/86997