Homological properties of Sklyanin algebras
A Sklyanin algebra of dimension \(r\) is a graded algebra \(A = A(E, \sigma, \mathcal L)\) associated to an elliptic curve \(E\), a translation \(\sigma\) of \(E\) and a line bundle \(\mathcal L\) of degree \(r\) over \(E\). Such algebras were introduced in \textit{A. V. Odesskij} and \textit{B. L. Feigin} [Preprint Inst. Theor. Phys., Kiev (1989); see also Funkts. Anal. Prilozh. 23, No. 3, 45-54 (1989; Zbl 0687.17001)]; they are the algebras denoted \(Q_{1,k}(E, \tau)\) there. It was proved in loc. cit. that, for \(\sigma\) generic, \(A\) is a flat deformation of a polynomial algebra and has as consequence good homological properties. The aim of the paper under review is to extend these results on homological properties to arbitrary \(\sigma\). The authors prove: \(A\) is a Noetherian domain, is Koszul, has the Hilbert series of a polynomial ring in \(r\) variables and is regular in the sense of \textit{M. Artin} and \textit{W. Schelter} [Adv. Math. 66, 171-216 (1987; Zbl 0633.16001)]. As the case when \(\sigma\) has finite order is difficult to treat with the methods in previous papers, they introduce the notion of ``modules of \(I\)-type''. Via a general result they establish several properties of \(A\) by showing that it has modules of \(I\)-type. Then they assume that \(\sigma\) has finite order, construct a commutative subring of \(A\) over which \(A\) is finite and conclude from this the study of \(A\). The general case follows by reduction to finite characteristic via some fundamental results of \textit{J. T. Stafford} and \textit{J. J. Zhang} [J. Algebra 168, 988-1026 (1994; Zbl 0812.16046)]. The methods in this paper give also information about linear modules over \(A\). For another article concerning the topic of this paper, see \textit{A. V. Odesskij} and \textit{B. L. Feigin} [Funct. Anal. Appl. 29, No. 2, 81-90 (1995); translation from Funkts. Anal. Prilozh. 29, No. 2, 9-21 (1995); see the review Zbl 0876.17011 above].
scientific article; zbMATH DE number 866104