The extension of the reduced Clifford algebra and its Brauer class.
type: Item
instance of: scholarly article
title: The extension of the reduced Clifford algebra and its Brauer class.
zbMATH Open document ID: 1037.11025
author: Rajesh S. Kulkarni
published in: Manuscripta Mathematica
publication date: 2003-12-16T00:00:00Z
review text: For \(f\) a form of degree \(d\) in two variables over a field \(k\), the Clifford algebra \(C_f\) is the quotient of the free associative \(k\)-algebra \(k\{ x_1, x_2 \}\) modulo the ideal \(I\) generated by \[ \{ (\alpha_1x_1 + \alpha _2x_2)^d - f(\alpha _1, \alpha _2) \mid \alpha _1, \alpha _2 \in k \}. \] In [Trans. Am. Math. Soc. 355, 3181--3208 (2003; Zbl 1035.11015)] the author defined the reduced Clifford algebra \(A_f\), and showed that \(A_f\) is Azumaya over its center, hence represents a class in \(\text{Br} (U)\), the Brauer group of the affine ring of the complement \(U\) of a certain divisor in \(\text{Pic}_{C/k}^{d+g-1}\), where \(C\) is the curve defined by \(w^d = f(u, v)\) of genus \(g\). Here the author shows that the Brauer class of \(A_f\) extends to a class in \(\text{Br} (\text{Pic}_{C/k}^{d+g-1})\). If \(d\) is odd, so that \(\text{Pic}_{C/k}^{d+g-1}\) and the Jacobian \(J(C)\) are isomorphic over \(k\), then \(A_f\) in \(\text{Br}(U)\) is split iff \(J(C)\) has a \(k\)-rational point. This generalizes a result for \(d=3\) of \textit{D. E. Haile} [J. Algebra 146, 514--520 (1992; Zbl 0758.11022)].
Mathematics Subject Classification ID: 11E76
Mathematics Subject Classification ID: 16K50
Mathematics Subject Classification ID: 14H40
Mathematics Subject Classification ID: 15A66
zbMATH DE Number: 2018252
zbMATH Keywords: Clifford algebra
zbMATH Keywords: generalized Clifford algebra
zbMATH Keywords: Azumaya algebra
zbMATH Keywords: Jacobian
zbMATH Keywords: Brauer group
reviewed by: Lindsay N. Childs
MaRDI profile type: Publication
full work available at URL: s00229-003-0392-2
OpenAlex ID: W2074794346
DOI: 10.1007/S00229-003-0392-2
description: scientific article; zbMATH DE number 2018252