Four dimensional regular algebras with point scheme, anonsingular quadric in P³

Abstract

In [22], a class of four-dimensional, quadratic, Artin-Schelter regular algebras was introduced, whose point scheme is the graph of an automorphism of a nonsingular quadric in P³. These algebras are the first examples of quadratic Artin-Schelter regular algebras whose defining relations are not determined by the point scheme and, hence, not determined by the algebraic data obtained from the point modules. In this paper, we study these algebras via their line modules. In particular, the set of lines in P³ that correspond to left line modules is not the set of lines in P³ that correspond to right line modules. Our analysis focuses on a distinguished member R _λ of this class of algebras, where R _λ is a twist by a twisting system of the other algebras. We prove that R _λ is a finite module over its center and that its central Proj is a smooth quadric inP⁴.

Keywords:

• Regular algebra
• quadratic algebra
• line module
• center

^∗The first author was supported in part by NSF grant DMS-9996056

^∗The first author was supported in part by NSF grant DMS-9996056

Notes

^∗The first author was supported in part by NSF grant DMS-9996056