Author ORCID Identifier

https://orcid.org/0009-0008-0155-081X

Date of Award

8-2024

Degree Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Department

Mathematics and Statistics

First Advisor

Yongwei Yao

Second Advisor

Florian Enescu

Third Advisor

Mark Grinshpon

Fourth Advisor

Zhongshan Li

Abstract

This dissertation explores the use of the Frobenius endomorphism to detect the regularity of a ring within the framework of commutative algebra. In commutative local rings of prime characteristic, invariants such as the Hilbert-Kunz multiplicity, F-signature, and Frobenius Betti numbers have been shown to detect regularity. Polstra and Smirnov [PS21] showed that the Frobenius Euler characteristic can be used to determine regularity for the class of strongly F-regular rings. Here, we extend their results by relaxing the condition of strong F-regularity. We show that the Frobenius Euler characteristic can detect regularity for rings with sufficient F-splitting and for rings that are Cohen-Macaulay.

DOI

https://doi.org/10.57709/37395360

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