Date of Graduation

7-2020

Document Type

Dissertation

Degree Name

Doctor of Philosophy in Mathematics (PhD)

Degree Level

Graduate

Department

Mathematical Sciences

Advisor/Mentor

Mantero, Paolo

Committee Member

Johnson, Mark

Second Committee Member

Miller, Lance E.

Keywords

Commutative Algebra; Homogeneous Ideals; Liaison; Linkage

Abstract

This thesis is concered with the graded structure of homogeneous CI-liaison. Given two homogeneous ideals in the same linkage class, we want to understand the ways in which you can link from one ideal to the other. We also use homogeneous linkage to study the socles and Hilbert functions of Artinian monomial ideals.

First, we build off the work of C. Huneke and B. Ulrich on monomial liaison. They provided an algorithm to check the licci property of Artinian monomial ideals and we use their method to characterize when two Artinian monomial ideals can be linked by monomial regular sequences. Furthermore, we use linkage to describe the socle generators of Artinian monomial ideals. This socle structure, along with techniques of B. Boyle, resulted in a partial answer to a question about unimodality of pure O-sequences; namely, we prove that in three variables, the Hilbert function of level Artinian monomial ideals linked within two steps to a CI is peaked strictly unimodal.

Our main result of this thesis was motivated by the work of C. Huneke and B. Ulrich in [35], and E. Chong in [14]. Huneke and Ulrich asked if, for any Artinian licci ideal, there was a coordinate change and monomial order for which the initial ideal would be licci. We provide a negative answer to this question by addressing a question by Chong about a weaker property. He introduced a family of homogeneous licci ideals with the property that the regular sequences linking to a CI yield descending degree sequences, and called them sequentially bounded licci (SBL) ideals. Chong used this property to provide a large class of examples satisfying the Eisenbud-Green-Harris conjecture (among them, grade 3 Gorenstein ideals), and asked if all homogeneous licci ideals were SBL. We construct an infinite family of homogeneous ideals, characterized by the graded Betti numbers, that are licci but not SBL.

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