\(R\)-equivalence and special unitary groups
type: Item
instance of: scholarly article
title: \(R\)-equivalence and special unitary groups
zbMATH Open document ID: 0947.14023
published in: Journal of Algebra
publication date: 2000-11-05T00:00:00Z
review text: The notion of \(R\)-equivalence, introduced by \textit{Yu. I. Manin} [``Cubic forms: Algebra, geometry, arithmetic'' (1972; Zbl 0255.14002; English translation 1974; Zbl 0277.14014)], is an important birational invariant of an algebraic variety defined over an arbitrary field \(F\). Let \(G\) be an algebraic group over an arbitrary field \(F\). The set of \(R\)-equivalence classes \(G(F)/R\), has a natural group structure. If for instance the group \(G\) is connected and a direct factor of a stably rational group, then the group of \(R\)-equivalence classes of \(G\) is trivial. But the converse is known only for tori. However, the group of \(R\)-equivalence classes carries some information about the birational geometry of \(G\). For any simply connected semisimple classical algebraic group \(G\), a norm homomorphism is constructed and for all the special unitary groups the group of \(R\)-equivalence classes is calculated.
Mathematics Subject Classification ID: 14L30
Mathematics Subject Classification ID: 14M20
Mathematics Subject Classification ID: 20G15
zbMATH DE Number: 1234247
zbMATH Keywords: group of \(R\)-equivalence classes
zbMATH Keywords: norm homomorphism
zbMATH Keywords: algebraic groups
zbMATH Keywords: birational geometry
author: V. Chernousov
author: Alexander Merkurjev
reviewed by: Stefan Helmke
MaRDI profile type: Publication
full work available at URL: jabr.1998.7534
OpenAlex ID: W2020632254
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DOI: 10.1006/JABR.1998.7534
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description: scientific article; zbMATH DE number 1234247