On an algebra associated to a ternary cubic curve.
type: Item
instance of: scholarly article
title: On an algebra associated to a ternary cubic curve.
zbMATH Open document ID: 1229.16013
author: Jung-Miao Kuo
published in: Journal of Algebra
publication date: 2011-07-25T00:00:00Z
review text: Let \(f(x_1,\dots,x_n)\) be an \(n\)-ary form of degree \(d\) over a field \(F\). The `Clifford algebra' of \(f\) is defined to be the \(F\)-algebra \(C_f=F\{x_1,\dots,x_n\}/I\) where \(I\) is the two-sided ideal generated by elements of the form \((\alpha_1x_1+\cdots+\alpha_nx_n)^d-f(\alpha_1,\dots,\alpha_n)\) for \(\alpha_1,\dots,\alpha_n\in F\). When \(d=2\), this is the classical Clifford algebra. When \(d>2\), the Clifford algebra \(C_f\) has been studied by various authors. For \(d>3\) or \(d=3\) and \(n=2\), \(C_f\) contains a free \(F\)-algebra on two generators and hence it is not finitely generated over its center and therefore Azumaya. \textit{D.~E.~Haile} proved [in Am. J. Math. 106, 1269-1280 (1984; Zbl 0585.13001)] that the Clifford algebra of a binary cubic form is Azumaya over its center of rank 9. He also described this center as the affine coordinate ring of an elliptic curve related to the given form. In the current paper, the author generalizes those results. Starting with a binary cubic form \(f(x,y)=ax^3+3bx^2y+3cxy^2+dy^3\), she defines a cubic plane curve \(C\) by the equation \(z^3-exyz-f(x,y)=0\) where \(e\) is a parameter. She then defines a (non-usual) Clifford algebra \(A_C\) associated to this curve as \(A_C=F\{x,y\}/I\) where \(I\) is the two-sided ideal generated by elements of the form \((\alpha x+\beta y)^3-e\alpha\beta(\alpha x+\beta y)-f(\alpha,\beta)\). She proves that the center of \(A_C\) is isomorphic to the affine coordinate ring of the Jacobian of the curve \(C\). She defines a group homomorphism \(\Phi\colon E(F)\to\text{Br}(F)\) in the obvious manner, and she proves that the image is the relative Brauer group \(\text{Br}(F(C)/F)\). Finally, she describes \(\Phi\) explicitly in the diagonal case, i.e. when \(b=c=0\). It must be noted that for \(e=0\), these results give Haile's results mentioned above.
reviewed by: Emre Coskun
Mathematics Subject Classification ID: 16H05
Mathematics Subject Classification ID: 11E76
Mathematics Subject Classification ID: 14H05
Mathematics Subject Classification ID: 15A66
Mathematics Subject Classification ID: 16K50
zbMATH DE Number: 5931637
zbMATH Keywords: Clifford algebras
zbMATH Keywords: Azumaya algebras
zbMATH Keywords: Brauer groups
zbMATH Keywords: cubic curves
describes a project that uses: RelBrauer
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2010.12.021
OpenAlex ID: W2068552449
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DOI: 10.1016/J.JALGEBRA.2010.12.021
description: scientific article; zbMATH DE number 5931637