Date of Award

6-26-2026

Date Published

August 2026

Degree Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Department

Mathematics

Advisor(s)

Dan Zacharia

Second Advisor

Graham Leushke

Keywords

Homological Algebra;Koszul Algebras;Representation Theory

Subject Categories

Mathematics | Physical Sciences and Mathematics

Abstract

We investigate questions related to the extension conjecture for finite-dimensional algebras. Suppose $\Lambda = \kk Q/I$ is a finite-dimensional algebra given by a quiver with relations. The extension conjecture asserts that if $S$ is a simple right $\Lambda$-module corresponding to a vertex with a loop, that is, $\Ext_\Lambda^1(S,S)\neq 0$, then $\Ext^n_\Lambda(S,S)\neq 0$ for infinitely many $n$. We show that one can obtain information about the non-vanishing of the extensions $\Ext^n_\Lambda(S,S)$ by looking at the graded Cartan matrix and its inverse. We use this connection to prove the extension conjecture for standardly graded $\kk$-algebras on two vertices with Loewy length at most $4$. We also use the graded Cartan matrix to provide a completely different proof of the no loop conjecture. In \cite{BHLL}, it is shown using techniques of Lenzing \cite{L}, that the extension conjecture holds for Koszul algebras whose Koszul dual is left or right Noetherian. We remark that their proof can be extended to the case where the Koszul dual is merely left or right coherent. We then give a characterization theorem for when the Koszul dual satisfies this property. We also provide a proof that monomial algebras are left and right coherent. This was proven independently by Lu and Piontkovski in \cite{LP}. Lastly we look to a specific algorithm for constructing projective resolutions due to Anick, Green, and Solberg known as the AGS resolution. We show that the AGS resolution may be used to prove the extension conjecture for Koszul algebras $\Lambda = \mathbb{k} Q/I$ such that $I$ admits a quadratic Gr{\"o}bner basis.

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Open Access

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