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

TL;DR

This paper investigates the structure of categories of graded modules over monomial and path algebras with finite Gelfand-Kirillov dimension, revealing a finite classification of point modules and a quiver-based criterion for category equivalence.

Contribution

It introduces a finite quiver construction from such algebras that classifies their noncommutative projective geometries and characterizes when two categories are equivalent.

Findings

Finitely many point modules up to isomorphism in $ ext{QGr} A$

Vertices of the quiver correspond to point modules

Category equivalence characterized by equality of associated quivers

Abstract

This article 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: 1) What is the structure of the point modules up to isomorphism in $\QGr A$? 2) When is $\QGr A \equiv \QGr A'$? 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 $\QGr A$. 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 $\QGr A\equiv \QGr A'$ if and only if $E_A=E_{A'}$.