#### Vol. 9, No. 9, 2015

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Classifying orders in the Sklyanin algebra

### Daniel Rogalski, Susan J. Sierra and J. Toby Stafford

Vol. 9 (2015), No. 9, 2055–2119
##### 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 ${ℙ}^{2}$.) 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.

##### Keywords
noncommutative projective geometry, noncommutative surfaces, Sklyanin algebras, noetherian graded rings, noncommutative blowing-up
##### Mathematical Subject Classification 2010
Primary: 14A22
Secondary: 16P40, 16W50, 16S38, 14H52, 16E65, 18E15