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

This article sets out to understand the categories $\text{QGr}A$ where $A$ is either a monomial algebra or a path algebra of finite Gelfand-Kirillov dimension. The principle questions are: 1) What is the structure of the point modules up to isomorphism in $\text{QGr}A$? 2) When is $\text{QGr}A\equiv \text{QGr}{A}^{\mathrm{\prime }}$? These two questions turn out to be intimately related. It is shown that up to isomorphism in $\text{QGr}A$, 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 $\text{QGr}A$. It is shown that 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 $\QGr A\equiv \QGr A’$ifandonlyif$E_A=E_{A’}$.