Finite Generation of Ext-Algebras of Finite Dimensional Algebras and Associated Monomial Algebras

<p>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 $A$.</p> <p> 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. </p> <p> 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.</p>

Name thesis:degree_name

Doctor of Philosophy (PhD)

Level thesis:degree_level

Dissertation

Discipline thesis:degree_discipline

Mathematics

Year

2016

Author dc:creator

• DiMarco, Melissa Margaret

Contributors dc:contributor

• Dan Zacharia

• Homological Algebra
• Representation Theory
• Physical Sciences and Mathematics

Repository record dc:identifier

https://surface.syr.edu/etd/639

OAI identifier oai:identifier

oai:surface.syr.edu:etd-1639