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Classifying orders in the Sklyanin algebra

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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 ℙ2.) Let A=⊕i≥0 Ai 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 noneffective) 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 dualising complex.

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