Algebra & Number Theory
- Algebra Number Theory
- Volume 9, Number 9 (2015), 2055-2119.
Classifying orders in the Sklyanin algebra
Let denote the 3-dimensional Sklyanin algebra over an algebraically closed field and assume that is not a finite module over its centre. (This algebra corresponds to a generic noncommutative .) Let be any connected graded -algebra that is contained in and has the same quotient ring as a Veronese ring . Then we give a reasonably complete description of the structure of . This is most satisfactory when is a maximal order, in which case we prove, subject to a minor technical condition, that is a noncommutative blowup of at a (possibly noneffective) divisor on the associated elliptic curve . It follows that has surprisingly pleasant properties; for example, it is automatically noetherian, indeed strongly noetherian, and has a dualising complex.
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
Permanent link to this document
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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
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. https://projecteuclid.org/euclid.ant/1510842422