Maximum Betti Numbers for a Given Hilbert Function

Hulett, Heather Ann

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Description

Title

Maximum Betti Numbers for a Given Hilbert Function

Author(s)

Hulett, Heather Ann

Issue Date

1993

Doctoral Committee Chair(s)

Evans, E.G.,

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

Abstract

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.

Graduation Semester

1993

Type of Resource

text

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