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arXiv · 1308.2213

Classifying Orders in the Sklyanin Algebra

D. Rogalski·S. J. Sierra·J. T. Stafford
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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.

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D. Rogalski, S. J. Sierra, J. T. Stafford. 2013-08-09. Classifying Orders in the Sklyanin Algebra. https://doi.org/10.2140/ant.2015.9.2055

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