Maximal Order 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 noetherian, contributing to the understanding of noncommutative projective surfaces.

Contribution

It completes the classification of maximal orders in the Sklyanin algebra, including higher Veronese subrings, and demonstrates their noetherian property.

Findings

Maximal orders in the Sklyanin algebra are classified as blowups at divisors.

Maximal orders are automatically noetherian.

The work extends and completes previous classifications by Rogalski, Sierra, and Stafford.

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 thesis 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 $S^{(3)}$ 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. This work both relies upon, and lends back to, the work of Rogalski, Sierra and Stafford. In particular, we also provide a converse for their classification of maximal orders in the higher Veronese subrings $S^{(3n)}$ of $S^{(3)}$.