Abstract: 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) the point scheme of A
is finite and is the graph of an automorphism of finite order, and
(iii) the line scheme of A has dimension one.
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.
This paper appeared in J. Algebra 297 (2006), 208-215. Please contact
either author for reprints.