Maximum Betti Numbers for a Given Hilbert Function

In his 1927 paper, Macaulay gave a necessary and sufficient condition for a function H to be the Hilbert function of a cyclic module over a polynomial ring $R = k\lbrack x\sb1,\..., x\sb{n}\rbrack$ where k is a field of characteristic 0. He constructed the lex-segment ideal which has a given Hilbert function through degree d, and he showed that this ideal gives a lower bound for H(d + 1). From this, Macaulay showed that the lex-segment ideal has the most generators (that is, the largest first Betti number) of a homogeneous ideal with that Hilbert function. This thesis proves that the lex-segment ideal having a given Hilbert function always has the largest graded Betti numbers of any homogeneous ideal with that Hilbert function. We also construct a submodule of a given free module which has graded Betti numbers at least as large as the graded Betti numbers of any other submodule of the given free module with the same Hilbert function. Moreover, this submodule gives a lower bound on the Hilbert function in degree d + 1 for any submodule with the given Hilbert function through degree d, thus generalizing Macaulay's theorem above.

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

2014

Author dc:creator

• Hulett, Heather Ann

Contributors dc:contributor

• Evans, E.G.,

• Mathematics

Identifier

(UMI)AAI9411657

OAI identifier oai:identifier

oai:www.ideals.illinois.edu:2142/72543