Stably rational algebraic tori
type: Item
instance of: scholarly article
title: Stably rational algebraic tori
zbMATH Open document ID: 0946.14030
author: V. E. Voskresenskiĭ
published in: Journal de Théorie des Nombres de Bordeaux
publication date: 2000-04-27T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/223753
full work available at URL: jtnb11-1.html
review text: Suppose that \(T\) is a torus over a field \(k\) of characteristic zero. If \(T\) is stably rational, that is, \(T\times_k{\mathbb A}^m_k\) is rational over~\(k\) for some~\(m\), then it is conjectured that \(T\) is rational over~\(k\). Here this conjecture is proved under the additional assumption that the splitting field of \(T\) is a cyclic extension of~\(k\), by reformulating it in terms of linear representations of~\(T\). The method is first used to reprove a weaker result of \textit{A. A. Klyachko} [in: Arithmetic and geometry of varieties, Interuniv. Collect. Sci. Works, Kujbyshev, 73-78 (1988; Zbl 0751.14031)] and then extended to give the main result of the paper.
Mathematics Subject Classification ID: 14M20
Mathematics Subject Classification ID: 14G25
Mathematics Subject Classification ID: 14M25
Mathematics Subject Classification ID: 14L99
zbMATH DE Number: 1370116
zbMATH Keywords: stably rational torus
zbMATH Keywords: cyclic splitting field
zbMATH Keywords: rationality of torus
reviewed by: G. K. Sankaran
MaRDI profile type: Publication
OpenAlex ID: W2313812678
cites work: Stably rational non rational varieties
cites work: The geometry of linear algebraic groups
cites work: FIELDS OF INVARIANTS FOR ABELIAN GROUPS
cites work: Q3972613
cites work: Q4133741
DOI: 10.5802/JTNB.250
description: scientific article; zbMATH DE number 1370116