Maximal Orders in the Sklyanin Algebra

TL;DR

This paper classifies all maximal orders within the 3-dimensional Sklyanin algebra, extending previous work, and shows these orders are inherently noetherian, advancing the understanding of noncommutative projective surfaces.

Contribution

It completes and extends the classification of maximal orders inside the Sklyanin algebra, providing a comprehensive understanding of their structure and properties.

Findings

Maximal orders are classified inside the Sklyanin algebra.

Maximal orders are automatically noetherian.

Classification relates to blowups at divisors.

Abstract

A major current goal of noncommutative geometry is the classification of noncommutative projective surfaces. The generic case is to understand algebras birational to the Sklyanin algebra. In this work we complete a considerable component of this problem. Let S denote the 3-dimensional Sklyanin algebra over an algebraically closed field, and assume that S is not a finite module over its centre. In earlier work Rogalski, Sierra and Stafford classified the maximal orders inside the 3-Veronese of S. We complete and extend their work and classify all maximal orders inside S. As in Rogalski, Sierra and Stafford's work, these can be viewed as blowups at (possibly non-effective) divisors. A consequence of this classification is that maximal orders are automatically noetherian among other desirable properties.