Principal bundles admitting a rational section
type: Item
instance of: scholarly article
title: Principal bundles admitting a rational section
zbMATH Open document ID: 0807.14012
DOI: 10.1007/BF01231567
author: M. S. Raghunathan
published in: Inventiones Mathematicae
publication date: 1995-03-05T00:00:00Z
full work available at URL: https://eudml.org/doc/144194
review text: The main result of the paper is the following theorem. Let \(k\) be an infinite field and \(G\) a smooth reductive group-scheme over \(k\). Let \(\mathbb{X}\) be an irreducible smooth scheme of finite type over \(k\). Let \((P, \pi, m)\) be a principal \(G\)-bundle on \(\mathbb{X}\) which is trivial over a nonempty Zariski open subset of \(\mathbb{X}\). Then for each \(x \in \mathbb{X}\) there is a Zariski open neighbourhood \(\mathbb{X} (x)\) of \(x\) in \(\mathbb{X}\) such that \((P, \pi, m)\) is trivial over \(\mathbb{X} (x)\). This result is an affirmation of the conjecture of Grothendieck and Serre in the case of an infinite field and \(\mathbb{X}\) is of finite type over \(k\). It is new only when \(k\) is not perfect and \(G\) has an anisotropic factor over \(k\); when \(k\) is infinite perfect or \(G\) has no anisotropic factors, the theorem is due to \textit{J.-L. Colliot-Thélène} and \textit{M. Ojanguren} [see Publ. Math., Inst. Hautes Étud. Sci. 75, 97-122 (1992; Zbl 0795.14029)].
Mathematics Subject Classification ID: 14F05
Mathematics Subject Classification ID: 14L30
zbMATH DE Number: 554023
zbMATH Keywords: principal \(G\)-bundle
zbMATH Keywords: group-scheme
reviewed by: Vladimir L. Popov
MaRDI profile type: Publication
cites work: Groupes reductifs
cites work: Locally trivial principal homogeneous spaces
cites work: Q3207121
cites work: Principal bundles on the affine line
OpenAlex ID: W1983633458
description: scientific article; zbMATH DE number 554023