The non-commutative algebraic geometry of some skew polynomial algebras

The purpose of this dissertation is to apply the techniques of non-commutative algebraic geometry, developed by Artin, Tate and Van den Bergh to certain algebras arising in the study of three dimensional skew polynomial rings.; There are five chapters to this dissertation. In Chapter 1, we define the class of three dimensional skew polynomial rings and their homogenizations. The homogenizations are quadratic algebras of four generators and six defining relations.; Not all of these homogenizations are Artin-Schelter regular. We study in detail the Type I and Type II algebras, whose homogenizations are Artin-Schelter regular. These families of algebras include Gurevich's example and Odesskii's example: we define these examples and analyze their finite dimensional simple modules in Chapters 3 and 4.; In Chapter 2, we describe the geometric data associated with Type I and Type II algebras, especially the point modules and line modules. The point modules of an algebra are parameterized by its point variety, a subvariety of the 3-dimensional projective space. The point varieties of the Type I algebras contain three lines, a quadric and an isolated point and those of Type II algebras contain three lines and five isolated points. The line modules are parameterized by the lines in the hyperplane at infinity and by certain extra lines. A quadric {dollar}Qsb{lcub}p{rcub}{dollar} is associated to each point p in the part of the point variety at infinity, and the extra lines are precisely those that pass through such a point p and lie on {dollar}Qsb{lcub}p{rcub}.{dollar}; We develop methods to compute all finite dimensional simple modules that are quotients of line modules. If {dollar}sigma{dollar} is of infinite order, any finite dimensional simple module is a quotient of a line module. We show that this can be the case even if the order of {dollar}sigma{dollar} is finite; one instance of this is Gurevich's example.; In chapter 5, we consider a Type II algebra that is finitely generated over its center. We show that it can be embedded into a matrix ring over a polynomial ring in 3 variables. We show that there is a bound on the dimension of the simple modules and that not all of the fat points lie on extra lines.