Algebra & Number Theory

Classifying orders in the Sklyanin algebra

Daniel Rogalski, Susan Sierra, and J. Stafford

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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 2.) Let A = i0Ai 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

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

Received: 8 April 2014
Revised: 10 March 2015
Accepted: 3 September 2015
First available in Project Euclid: 16 November 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 14A22: Noncommutative algebraic geometry [See also 16S38]
Secondary: 16P40: Noetherian rings and modules 16W50: Graded rings and modules 16S38: Rings arising from non-commutative algebraic geometry [See also 14A22] 14H52: Elliptic curves [See also 11G05, 11G07, 14Kxx] 16E65: Homological conditions on rings (generalizations of regular, Gorenstein, Cohen-Macaulay rings, etc.) 18E15: Grothendieck categories

noncommutative projective geometry noncommutative surfaces Sklyanin algebras noetherian graded rings noncommutative blowing-up


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.

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