Classifying orders in the Sklyanin algebra

Abstract

Let $S$ denote the 3-dimensional Sklyanin algebra over an algebraically closed field $\mathbb{k}$ and assume that $S$ is not a finite module over its centre. (This algebra corresponds to a generic noncommutative ${\mathbb{P}}^2$.) Let $A={\oplus }_{i\ge 0}{A}_i$ be any connected graded $\mathbb{k}$-algebra that is contained in and has the same quotient ring as a Veronese ring $S^{\left(3n\right)}$. 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 $S^{\left(3n\right)}$ 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.