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Zbl 0823.17020
Levasseur, Thierry; Smith, S.Paul
Modules over the 4-dimensional Sklyanin algebra. (English)
[J] Bull. Soc. Math. Fr. 121, No.1, 35-90 (1993). ISSN 0037-9484

Let $E$ be an elliptic curve, $j$ an embedding of $E$ in $\bbfP\sp 3$ and $\tau \in E$ not of order 4. The authors show that many algebraic properties of the Sklyanin algebra $A$ associated to this data, first defined in [{\it E. K. Sklyanin}, Funct. Anal. Appl. 16, 263-270 (1983); translation from Funkts. Anal. Prilozh. 16, 27-34 (1982; Zbl 0513.58028)], have geometric interpretations in terms of $E,j, \tau$. They first prove the equivalence between the original definition and a geometric one given in the present paper. A point, resp. line, plane, module is a graded module whose Hilbert series coincides with the Hilbert series of a polynomial ring in 1, resp. 2, 3, variables. One of the main results of this article is the characterization of the Cohen-Macaulay (graded) modules over $A$ of multiplicity 1 and Gelfand-Kirillov dimension 1,2,3: they are precisely shifts of point, resp. line, plane, modules. It is also proved that line modules are in bijective correspondence with lines in $\bbfP\sp 3$ secant to $E$. (Analogously, it was proved in [the second author and {\it J. T. Stafford} [Compos. Math. 83, 259-289 (1992; Zbl 0758.16001)] that point modules are in bijective correspondence with points of $E$ plus 4 more points; these 4 points also play a role in the new definition of $A$ and are, in fact, the only points which lie on infinitely many secants of $E)$.
[N.Andruskiewitsch (Córdoba)]

MSC 2000:: *17B37 Quantum groups and related deformations
16W50 Graded associative rings and modules
16E10 Homological dimensions (assoc. rings and algebras)
16P40 Associative Noetherian rings and modules
14H52 Elliptic curves

Keywords: Cohen-Macaulay modules; Sklyanin algebra; graded module; Hilbert series; Gelfand-Kirillov dimension

Citations: Zbl 0513.58029; Zbl 0758.16001; Zbl 0513.58028

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