Some finite quantum s that are infinite modules over their centers

A result of M. Artin, J. Tate and M. Van den Bergh asserts that a regular algebra of global dimension three is a finite module over its center if and only if the automorphism encoded in the point scheme has finite order. We prove that the analogous result for a regular algebra of global dimension four is false by presenting families of quadratic, noetherian regular algebras A of global dimension four such that (i) A is an infinite module over its center, (ii) A has a finite point scheme, which is the graph of an automorphism of finite order, and (iii) A has a one-parameter family of line modules. Such algebras are candidates for generic regular algebras of global dimension four. The methods used to construct the algebras provide new techniques for creating other potential candidates.