Path algebras and monomial algebras of finite GK-dimension as noncommutative homogeneous coordinate rings

Abstract

This thesis sets out to understand the categories QGr A where A is either a monomial algebra or a path algebra of finite Gelfand-Kirillov dimension. The principle questions are: \begin{enumerate} \item What is the structure of the point modules up to isomorphism in QGr A? \item Given two such algebras A and A', when is QGr A equivalent to QGr A'? \end{enumerate} These two questions turn out to be intimately related. It is shown that up to isomorphism in QGr A, there are only finitely many point modules and these give all the simple objects in the category. Then, a finite quiver E_A, which can be constructed from the algebra A rather simply, is associated to the category QGr A. The vertices of E_A are in bijection with the point modules and the arrows are determined by the extensions between point modules. Lastly, it is shown that QGr A is equivalent to QGr A' if and only if E_A=E_{A'}.