Segre Product of Artin–Schelter Regular Algebras of Dimension 2 and Embeddings in Quantum P3's

✍ Scribed by Kristel Van Rompay

• Publisher: Elsevier Science
• Year: 1996
• Tongue: English
• Weight: 305 KB
• Volume: 180
• Category: Article
• ISSN: 0021-8693
• DOI: 10.1006/jabr.1996.0078

✦ Synopsis

The Segre embedding of ‫ސ‬ 1 = ‫ސ‬ 1 as a smooth quadric Q in ‫ސ‬ 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 ‫ސ‬ 1 . We study Segre products of noncommutative algebras. If in particular A and B are two copies of a quantum ‫ސ‬ 1 Ž .

then S s [ A m B is a twisted homogeneous coordinate ring of the quadric

The main result of this paper is the classification of all embeddings of the Segre 3 Ž product of two quantum planes into so-called quantum ‫ސ‬ '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 Ž . 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.