Classifying Orders in the Sklyanin Algebra

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 no...

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Bibliographic Details
Published in:arXiv.org 2013-08
Main Authors: Rogalski, D, Sierra, S J, Stafford, J T
Format: Article
Language:English
Subjects:
Algebra
Classification
Curves
Online Access:Get full text
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Summary: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.
ISSN:2331-8422
DOI:10.48550/arxiv.1308.2213