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Classifying Orders in the Sklyanin Algebra
- Source :
- Algebra Number Theory 9 (2015) 2055-2119
- Publication Year :
- 2013
-
Abstract
- One of the major open problems in noncommutative algebraic geometry is the classification of noncommutative surfaces, and this paper resolves a significant case of this problem. Specifically, let S denote the 3-dimensional Sklyanin algebra over an algebraically closed field k and assume that S is not a finite module over its centre. (This algebra corresponds to a generic noncommutative P^2.) Let A be any connected graded k-algebra that is contained in and has the same quotient ring as a Veronese ring S^(3n). Then we give a reasonably complete description of the structure of A. This is most satisfactory when A is a maximal order, in which case we prove, subject to a minor technical condition, that A is a noncommutative blowup of S^(3n) at a (possibly non-effective) divisor on the associated elliptic curve E. It follows that A has surprisingly pleasant properties; for example it is automatically noetherian, indeed strongly noetherian, and has a dualizing complex.<br />Comment: 55 pages
- Subjects :
- Mathematics - Rings and Algebras
14A22, 14H52, 16E65, 16P40, 16S38, 16W50, 18E15
Subjects
Details
- Database :
- arXiv
- Journal :
- Algebra Number Theory 9 (2015) 2055-2119
- Publication Type :
- Report
- Accession number :
- edsarx.1308.2213
- Document Type :
- Working Paper
- Full Text :
- https://doi.org/10.2140/ant.2015.9.2055