Date of Graduation

8-2018

Document Type

Dissertation

Degree Name

Doctor of Philosophy in Mathematics (PhD)

Degree Level

Graduate

Department

Mathematical Sciences

Advisor/Mentor

Mark Johnson

Committee Member

Lance Miller

Second Committee Member

Wenbo Niu

Keywords

Closures, Commutative Algebra, Hilbert-Kunz, Hilbert-Samuel, Multiplicity

Abstract

We define a family of functions, called s-multiplicity for each s>0, that interpolates between Hilbert-Samuel multiplicity and Hilbert-Kunz multiplicity by comparing powers of ideals to the Frobenius powers of ideals. The function is continuous in s, and its value is equal to Hilbert-Samuel multiplicity for small values of s and is equal to Hilbert-Kunz multiplicity for large values of s. We prove that it has an associativity formula generalizing the associativity formulas for Hilbert-Samuel and Hilbert-Kunz multiplicity. We also define a family of closures, called s-closures, such that if two ideals have the same s-closure then they have the same s-multiplicity, and the converse holds under mild conditions. We describe methods for computing the $F$-threshold, the $s$-multiplicity, and the $s$-closure of monomial ideals in toric rings using the geometry of the cone defining the ring.

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