Monomial Ideals, N-Lists, and Smallest Graded Betti Numbers
Richert, Benjamin P.
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https://hdl.handle.net/2142/86997
Description
Title
Monomial Ideals, N-Lists, and Smallest Graded Betti Numbers
Author(s)
Richert, Benjamin P.
Issue Date
2000
Doctoral Committee Chair(s)
Evans, E. Graham, Jr.
Department of Study
Mathematics
Discipline
Mathematics
Degree Granting Institution
University of Illinois at Urbana-Champaign
Degree Name
Ph.D.
Degree Level
Dissertation
Keyword(s)
Mathematics
Language
eng
Abstract
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.
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