Date of Award

August 2016

Degree Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Department

Mathematics

Advisor(s)

Dan Zacharia

Keywords

Homological Algebra, Representation Theory

Subject Categories

Physical Sciences and Mathematics

Abstract

In this thesis, we investigate the Ext-algebra of a basic, finite dimensional $K$-algebra $A=K\mathcal{Q}/I$, where $K$ is an algebraically closed field and $\mathcal{Q}$ is a finite quiver. We denote the Ext-algebra of $A$ by $E(A)$. We denote $\bar{A}=A/A^+$ to be the direct sum of all simple modules over $A$.

In the first part, we use the work of Green, Solberg, and Zacharia to construct a family of elements in $K\mathcal{Q}$, which we call $\{f_i^j\}$. These elements yield a minimal projective resolution of $\bar{A}$ over $A$. Consequently, $\{f_i^j\}$ form a dual basis of $E(A)$. In Chapter 2, we see that the subalgebra of $E(A)$ generated in degrees 0 and 1 is of the form $K\mathcal{Q}^*/I^!$ and prove the relations in $I^!$ can be directly computed using $\{f_i^j\}$. In the case $A$ is graded, we provide an alternate proof to the result of L{\"o}fwall and Priddy, namely that $A^!$ is quadratic. Then we proceed to compute the relations which generate $I^!$. In the case $A$ is monomial, we prove that the family $\{f_i^m\}$ is exactly the set of $m$-chains used by Green and Zacharia.

In the second part, we use a construction by Anick, Green, and Solberg to form a family $\{x_i^j\}$ which yields a projective resolution of $\bar{A}$, called the AGS resolution. If $A$ is a monomial algebra, we prove there are easily checked conditions for $E(A)$ to be generated in degrees 0,1, and 2. If $A$ is not necessarily monomial, we consider the case where the AGS resolution is minimal. In that situation, we look to the associated monomial algebra of $A$, found in \cite{G1} and \cite{G2}, which we denote $\Am$. We prove that if the AGS resolution is minimal and $E(\Am)$ is finitely generated, then $E(A)$ is finitely generated.

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