SEGRE PRODUCT OF ARTIN-SCHELTER REGULAR ALGEBRAS OF DIMENSION 2 AND EMBEDDINGS IN QUANTUM P-3S

摘要

The Segre embedding of P-1 x P-1 as a smooth quadric Q in P-3 corresponds to the surjection of the four-dimensional polynomial ring onto the Segre product S of two copies of the homogeneous coordinate ring of P-1. We study Segre products of noncommutative algebras. If in particular A and B are two copies of a quantum P-1 then S = +(i)(A(i) x(k) B-i) is a twisted homogeneous coordinate ring of the quadric Q. The main result of this paper is the classification of all embeddings of the Segre product of two quantum planes into so-called quantum P-3's. These are (the Proj of) Artin-Schelter regular algebras R of global dimension four with the Hilbert series of a commutative polynomial ring and which map onto S. If R is not a twist of a polynomial ring, then the point scheme of R either is the union of the quadric Q with a line or is only the quadric Q. In the first case, R is a central extension of a three-dimensional Artin-Schelter regular algebra and a twist of an algebra mapping onto the (commutative) homogeneous coordinate ring of Q; in the second case, such an algebra R is the first known example of a four-dimensional Artin-Schelter regular algebra which is not determined by its point scheme. (C) 1996 Academic Press, Inc. [References: 24]