## Algebra & Number Theory

### Classifying orders in the Sklyanin algebra

#### Abstract

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 $A = ⊕ i≥0Ai$ 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.

#### Article information

Source
Algebra Number Theory, Volume 9, Number 9 (2015), 2055-2119.

Dates
Revised: 10 March 2015
Accepted: 3 September 2015
First available in Project Euclid: 16 November 2017

https://projecteuclid.org/euclid.ant/1510842422

Digital Object Identifier
doi:10.2140/ant.2015.9.2055

Mathematical Reviews number (MathSciNet)
MR3435812

Zentralblatt MATH identifier
1348.14006

#### Citation

Rogalski, Daniel; Sierra, Susan; Stafford, J. Classifying orders in the Sklyanin algebra. Algebra Number Theory 9 (2015), no. 9, 2055--2119. doi:10.2140/ant.2015.9.2055. https://projecteuclid.org/euclid.ant/1510842422

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