This paper details work done by the authors at the Hope College Research Experiences for Undergraduates Site in Summer 1998.
Abstract: Let A be an Artin-Schelter regular algebra of global dimension 3 having 3 generators of weights (1,1,2). All such algebras have been classified. We use these classification results to study some Artin-Schelter regular algebras of global dimension 4 having 4 generators and 6 quadratic defining relations.
To be precise, the 2-Veronese ring A(2) has 4 generators and 7 quadratic defining relations. We study certain Artin-Schelter regular algebras S of global dimension 4 which have A(2) as a graded quotient algebra. Thus, the defining relations of S are obtained by finding appropriate 6-dimensional subspaces of the space of defining relations of A(2). In this article, we focus on the case where A is an Ore extension of the algebra kq[x,y]=k{x,y : yx=qxy}.
We study the geometry of these regular algebras of dimension 4 by determining the associated varieties of point modules.
This paper appeared as Rocky Mountain Journal of Mathematics 35 (2005), 415-444. Please contact Darin Stephenson for reprints.