Classifying orders in the Sklyanin algebra

Rogalski, Daniel; Sierra, Susan; Stafford, J.

doi:10.2140/ant.2015.9.2055

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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

DOI: 10.2140/ant.2015.9.2055

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 $ℙ^{2}$ .) Let $A = \oplus_{i \geq 0} A_{i}$ be any connected graded $k$ -algebra that is contained in and has the same quotient ring as a Veronese ring $S^{(3 n)}$ . 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 $S^{(3 n)}$ 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

Milestones

Received: 8 April 2014

Revised: 10 March 2015

Accepted: 3 September 2015

Published: 4 November 2015

Authors

Daniel Rogalski
Department of Mathematics University of California, San Diego 9500 Gilman Drive #0112 La Jolla, CA 92093-0112 United States

Susan J. Sierra
School of Mathematics University of Edinburgh James Clerk Maxwell Building Peter Guthrie Tait Road Edinburgh EH9 3FD United Kingdom

J. Toby Stafford
School of Mathematics University of Manchester Alan Turing Building Oxford Road Manchester M13 9PL United Kingdom

Vol. 9, No. 9, 2015