Graduate Studies, UNL

 

Dissertations and Doctoral Documents, University of Nebraska-Lincoln, 2023–

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First Advisor

Jack Jeffries

Degree Name

Doctor of Philosophy (Ph.D.)

Committee Members

Alexandra Seceleanu, Eloísa Grifo, Qiuming Yao

Department

Mathematics

Date of this Version

4-21-2026

Document Type

Dissertation

Citation

A dissertation presented to the faculty of the Graduate College at the University of Nebraska in partial fulfillment of requirements for the degree Doctor of Philosophy

Major: Mathematics

Under the supervision of Professor Jack Jeffries

Lincoln, Nebraska, May 2026

Comments

Copyright 2026, Taylor Jeffrey Murray. Used by permission

Abstract

The first part of this thesis is inspired by the works of G. Lyubeznik, C. Hunkeke, R. Sharp, and L. Núñez-Betancourt on Bass numbers, associated primes, and injective dimension of local cohomology modules. In particular, we study the question: how do Bass numbers behave under the Veronese functors? We show for a positive integer n, a reasonably nice, graded, finitely generated algebra over field R, and a graded module M, that if the Bass numbers of M are finite over R, then so are the Bass numbers of Mn/Rn; this recovers and extends a theorem of L. Núñez-Betancourt on splittings of rings for this setting. Moreover, we prove a formula for most Bass numbers of Mn in terms of the Bass numbers M. In addition, we provide an application to local cohomology modules.

The second part of this thesis is motivated by the work of J. Jeffries and A. Singh on the liftability of differential operators in charactersitic p > 0. Here we study the Frobenius trace on symmetric determinantal rings over \Z/2\Z. Employing the D-module structure on local cohomology, we show that the Frobenius trace on such a ring does not lift to a differential operator on the analogous symmetric determinantal ring over \Z, extending and recovering a theorem of J. Jeffries and A. Singh. This part is based on joint work with Jack Jeffries and Victor Daniel Mendoza Rubio.

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