Classifying Orders in the Sklyanin Algebra

TL;DR

This paper classifies certain noncommutative graded algebras related to the Sklyanin algebra, revealing their structure as noncommutative blowups and establishing their favorable algebraic properties.

Contribution

It provides a detailed classification of algebras contained in Sklyanin algebra quotients, showing they are noncommutative blowups with strong noetherian properties.

Findings

A complete description of algebra A structure

A is a noncommutative blowup of S^(3n)

A is strongly noetherian with a dualizing complex

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.