email: stephenson@hope.edu
Abstract: The purpose of this article and the companion article [St1] is to complete the classification of Artin-Schelter regular algebras of global dimension three. For algebras generated by elements of degree one, this has been achieved by M. Artin, W. Schelter, J. Tate and M. Van den Bergh. Therefore, this paper deals with algebras which are not generated in degree one. As in the work of Artin, Tate and Van den Bergh, there exist some exceptional algebras, each of which has geometric data consisting of an elliptic curve together with an automorphism.
In particular, we study the elliptic algebras A(+), A(-) and
A(a) (where a is a point in the projective
plane) which were first defined in
[St1]. We omit a set S consisting of eleven points of the
projective plane where the algebras A(a) become
too degenerate to be regular.
Theorem: Let A represent A(+), A(-) or
A(a), where a is not in S. Then A is an
Artin-Schelter regular algebra of global dimension three.
Moreover, A is a Noetherian domain with the same Hilbert series
as the (appropriately graded) commutative polynomial ring in
three variables.
This, combined with the results of [St1], completes the classification.
This article has appeared in Transactions of the AMS 349 (1997), pp. 2317-2340. .