Homological properties of Sklyanin algebras
type: Item
instance of: scholarly article
title: Homological properties of Sklyanin algebras
zbMATH Open document ID: 0876.17010
author: Michel Van den Bergh
author: John Tate
published in: Inventiones Mathematicae
publication date: 1997-09-16T00:00:00Z
review text: 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].
Mathematics Subject Classification ID: 17B37
Mathematics Subject Classification ID: 16E10
Mathematics Subject Classification ID: 16P40
Mathematics Subject Classification ID: 16W50
zbMATH DE Number: 866104
zbMATH Keywords: Sklyanin algebras
zbMATH Keywords: elliptic curves
reviewed by: Nicolás Andruskiewitsch
MaRDI profile type: Publication
full work available at URL: s002220050065
OpenAlex ID: W2039726285
DOI: 10.1007/S002220050065
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description: scientific article; zbMATH DE number 866104