Date of Graduation

8-2018

Document Type

Dissertation

Degree Name

Doctor of Philosophy in Mathematics (PhD)

Degree Level

Graduate

Department

Mathematical Sciences

Advisor

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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