Hyperbolicity of certain involutions over the function field of a quadric
type: Item
instance of: scholarly article
title: Hyperbolicity of certain involutions over the function field of a quadric
zbMATH Open document ID: 1005.16018
DOI: 10.1016/S0019-3577(01)80015-X
author: Ahmed Laghribi
published in: Indagationes Mathematicae. New Series
publication date: 2002-07-11T00:00:00Z
review text: Let \(F\) be a field of characteristic \(\neq 2\) and let \((A,\sigma)\) be a central simple algebra with involution \(\sigma\) over \(F\). The involution is called hyperbolic if there exists an idempotent \(e\in A\) with \(\sigma(e)=1-e\). Let \(\varphi\) be a quadratic form over \(F\) and denote by \(F(\varphi)\) the function field of the projective quadric defined by the equation \(\varphi=0\). Well known results imply that if there exists a homomorphism of algebras with involution \((C_0(\varphi),\sigma_0)\to(A,\sigma)\), where \(C_0(\varphi)\) denotes the even part of the Clifford algebra of \(\varphi\) and \(\sigma_0\) the standard involution on \(C_0(\varphi)\), then \((A\otimes_FF(\varphi),\sigma_{F(\varphi)})\) is hyperbolic. The author constructs an example which shows that the converse generally fails, but he gives an affirmative answer in the case of orthogonal \(\sigma\) under the assumption that either the degree of \(A\) is \(2\) or \(4\), or \(A\) is split and \(\dim\varphi\leq 4\) or \(\dim\varphi=5\) and \(\varphi\) is a Pfister neighbor. The second main result characterizes those forms \(\psi\) such that \((A,\sigma)\) is hyberbolic over \(F(\psi)\) in the case where \(A\) is a division algebra of degree \(8\), \(\sigma\) is orthogonal, and \(A\) decomposes completely, i.e. \(A\) can be written as a product of three quaternion algebras each of them being stable under the action of \(\sigma\). In the case where \(A\) does not decompose completely, a partial result is given.
Mathematics Subject Classification ID: 16K20
Mathematics Subject Classification ID: 11E04
Mathematics Subject Classification ID: 16W10
Mathematics Subject Classification ID: 11E88
zbMATH DE Number: 1766475
zbMATH Keywords: central simple algebras
zbMATH Keywords: algebras with involutions
zbMATH Keywords: Clifford algebras
zbMATH Keywords: hyperbolic involutions
zbMATH Keywords: function fields of quadratic forms
zbMATH Keywords: orthogonal involutions
reviewed by: Detlev Hoffmann
MaRDI profile type: Publication
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full work available at URL: s0019-3577(01)80015-x
OpenAlex ID: W2010982018
description: scientific article; zbMATH DE number 1766475