Rationality of almost simple algebraic groups
type: Item
instance of: scholarly article
title: Rationality of almost simple algebraic groups
zbMATH Open document ID: 0941.20051
published in: Journal of Mathematics of Kyoto University
publication date: 2000-05-16T00:00:00Z
review text: The author establishes rationality properties for a class of almost simple adjoint algebraic groups. Due to results of \textit{A. S. Merkur'ev} [K-Theory 7, No. 1, 1-3 (1993; Zbl 0811.16011) and Publ. Math., Inst. Hautes Etud. Sci. 84, 189-213 (1996; Zbl 0884.20029)] one cannot hope for stable rationality in general when the group involves division algebras of degree divisible by 4. Therefore the following main result is in some sense sharp. Let \(k\) be a field and let \(G\) be an almost simple adjoint \(k\)-group. Let \(m(G)\) be the maximal length of the black segments of its Tits index that are defined over \(k\). That is, the almost simple group with root system spanned by the segment must be defined over \(k\). If \(G\) is of classical type \(X^{(d)}_{n,r}\) with \(n-rd\leq 2\), or \(G\) is of exceptional type with \(m(G)\leq 3\) then \(G\) is either rational or stably rational over \(k\). Here, following Tits, \(d\) denotes the rank of the division algebra associated with \(G\), \(n\) is the rank and \(r\) is the \(k\)-rank. The proof is case by case, using Tits diagrams and many earlier results.
reviewed by: Wilberd van der Kallen
Mathematics Subject Classification ID: 20G15
Mathematics Subject Classification ID: 14L10
Mathematics Subject Classification ID: 19B14
zbMATH DE Number: 1374927
zbMATH Keywords: anisotropic kernels
zbMATH Keywords: Tits index
zbMATH Keywords: stable rationality
zbMATH Keywords: almost simple adjoint algebraic groups
zbMATH Keywords: root systems
author: Nguyêñ Quôć Thǎńg
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1215/kjm/1250517958
OpenAlex ID: W2170238687
DOI: 10.1215/KJM/1250517958
description: scientific article; zbMATH DE number 1374927