Date of Award

5-10-2026

Date Published

June 2026

Degree Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Department

Mathematics

Advisor(s)

Claudia Miller

Second Advisor

Simon Catterall

Keywords

Algebras;Ideals;Resolutions

Subject Categories

Mathematics | Physical Sciences and Mathematics

Abstract

This thesis is comprised of two projects. The first is individual work where I use semifree Gamma-extensions to show, for an ideal I with infinite projective dimension over a Noetherian local ring R, there exists a splitting morphism from a Tor algebra Tor(R/I,R/I) to the exterior algebra of the R/I-free summands of the conormal module. In particular, when the conormal module is free over R/I, we see that the exterior algebra of the conormal module splits completely from Tor(R/I,R/I) as algebras. For the second project my collaborators, Hugh Geller and Henry Potts-Rubin, and I completely classify all trees and cycle graphs whose edge ideals have minimal free resolutions with differential graded (dg) algebra structures. To do this we rely on Morse Theory from Algebraic Combinatorics and also homological algebra to construct or describe the dg structures of the resolutions. We also use a pruning technique by Boocher to give an obstruction to the existence of dg algebra structures in graphs that "contain" this obstruction.

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Open Access

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Mathematics Commons

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