Author ORCID Identifier

https://orcid.org/0009-0007-3274-5911

Semester

Spring

Date of Graduation

2026

Document Type

Dissertation

Degree Type

PhD

College

Eberly College of Arts and Sciences

Department

Mathematics

Committee Chair

Olgur Celikbas

Committee Co-Chair

Ela Celikbas

Committee Member

Charis Tsikkou

Committee Member

Guangming Jing

Committee Member

Yongwei Yao

Abstract

This dissertation presents the author’s recent research, conducted under the supervision of Professor Olgur Celikbas, and based on two articles—one published and one in progress. These works develop two closely related research directions in commutative algebra. Together, they contribute to the subject by addressing aspects of existing conjectures, establishing new results, and introducing methods for studying homological invariants.

The first research direction concerns the depth formula, namely the equality \[ \depth_R(M)+\depth_R(N)=\depth(R)+\depth_R(M\otimes_RN) \] where $M$ and $N$ are finitely generated $R$-modules. A classical result of Auslander \cite{Aus} shows that the depth formula holds provided that either $M$ or $N$ has finite projective dimension.

The second research direction focuses on the notion of reducing projective dimension. This invariant was introduced in \cite{ArCe} and was motivated by the work of Bergh \cite{Ber} which focuses on reducing complexity. In particular, this generalization allows for the existence of modules with infinite complexity that nevertheless have finite reducing projective dimension.

A central theme of this dissertation is the interaction between these two aforementioned topics. Finite reducing projective dimension, together with suitable vanishing conditions on Tor, provides a framework that extends the role of finite projective dimension in the depth formula. In this way, the depth formula can be established in settings beyond the classical case. At the same time, reducing projective dimension is of independent interest, possessing its own structure and properties that are developed throughout this work.

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