Fat point modules over generalized Laurent polynomial rings

Recently a new construction of rings was introduced by Cassidy, Goetz, and Shelton. Some of these rings, called generalized Laurent polynomial rings, are quadratic Artin–Schelter regular algebras of global dimension 4. We study a family of such algebras which have finite-order point-scheme automorphisms but which are not finitely generated over their centers. Our main result is the classification of all fat point modules for each algebra in the family. We also consider the action of the shift functor τ and prove τ has infinite order on a fat point module F precisely when the center acts trivially on F. The proofs of these facts use the noncommutative geometry of some cubic Artin–Schelter regular algebras of global dimension 3.