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.
","abstract_html":"<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 \\{fij\\}. These elements yield a minimal projective resolution of $\\bar{A}$ over $A$. Consequently, \\{fij\\} 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 \\{fij\\}. 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 \\{fim\\} 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 \\{xij\\} 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>","abstract_has_math":true,"creators":["DiMarco, Melissa Margaret"],"institution":null,"degree_name":"Doctor of Philosophy (PhD)","degree_level":"Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["Dan Zacharia"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2016,"date_issued":"2016-08-01T07:00:00Z","date_published":"2016-08-01T07:00:00Z","updated_at":"2026-07-24T04:55:13Z","subjects":["Homological Algebra","Representation Theory","Physical Sciences and Mathematics"],"languages":[],"rights":[],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"https://surface.syr.edu/etd/639","outbound_label":"Repository record","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Dan Zacharia"]},{"key":"dc:creator","label":"Author","values":["DiMarco, Melissa Margaret"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Dissertation"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Doctor of Philosophy (PhD)"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Homological Algebra","Representation Theory","Physical Sciences and Mathematics"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["https://surface.syr.edu/etd/639"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["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.
"]},{"key":"dc:title","label":"Title","values":["Finite Generation of Ext-Algebras of Finite Dimensional Algebras and Associated Monomial Algebras"]}]}],"canonical_facts":{"dc:contributor":["Dan Zacharia"],"dc:creator":["DiMarco, Melissa Margaret"],"dc:description.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.
"],"dc:identifier":["https://surface.syr.edu/etd/639"],"dc:subject":["Homological Algebra","Representation Theory","Physical Sciences and Mathematics"],"dc:title":["Finite Generation of Ext-Algebras of Finite Dimensional Algebras and Associated Monomial Algebras"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["Dissertation"],"thesis:degree_name":["Doctor of Philosophy (PhD)"]},"updated_at":"2026-07-24T04:55:13Z"}