Locally trivial principal homogeneous spaces
type: Item
instance of: scholarly article
title: Locally trivial principal homogeneous spaces
zbMATH Open document ID: 0795.14029
DOI: 10.1007/BF02699492
author: Manuel Ojanguren
author: Jean-Louis Colliot-Thélène
published in: Publications Mathématiques
publication date: 1993-05-16T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/104081
review text: The following conjecture was made by \textit{J.-P. Serre} and \textit{A. Grothendieck} in 1958: Let \(X\) be a smooth, irreducible variety over an algebraically closed field \(k\); let \(G\) be a connected reductive \(k\)-group. Then every principal homogeneous space over \(G\), with base \(X\) and having a rational section, is locally trivial in the Zariski topology. This conjecture is solved in the paper under review, with weaker assumptions on \(k\) and \(G\). Namely, it holds when (i) \(k\) is infinite and perfect, and \(G\) is linear and smooth (theorem 3.2); or when (ii) \(k\) is infinite, \(G\) is reductive, and every \(k\)-simple component of the derived group of \(G\) is \(k\)-isotropic (theorem 2.1). More generally, the authors consider the following conjecture of \textit{A. Grothendieck} [in Dix Exposés Cohomologie Schémas, Adv. Stud. Pure Math. 3, 67--87 (1968; Zbl 0198.25803)]: Let \(A\) be a regular local ring, with the field of fractions \(K\). Let \(G\) be a reductive group scheme over \(A\). Then the map of pointed sets \(H^ 1(A,G)\to H^ 1(K,G_ K)\) has a trivial kernel. This conjecture is established in various cases; in theorem 1.1, the authors reduce it to checking that the functor \(X\to H^ 1(X,G)\) commutes with inductive limits (Property P1), satisfies a weak form of homotopic invariance (P2), and satisfies a glueing property (P3). The most difficult point is (P2), for which results of \textit{M. S. Raghunathan} [Math. Ann. 285, No. 2, 309--332 (1989; Zbl 0672.14007)] are used. This approach can be followed for other functors, e.g. \(K\)- functors; see \S4. In \S5, an anlogue of Gersten's conjecture is formulated and studied, by expanding ideas of \textit{D. Quillen} [in Algebraic \(K\)-theory I, Proc. Conf. Battelle Inst. 1972, Lect. Notes Math. 341, 85--147 (1973; Zbl 0292.18004)].
Mathematics Subject Classification ID: 14M17
Mathematics Subject Classification ID: 14L15
Mathematics Subject Classification ID: 20G15
zbMATH DE Number: 165867
zbMATH Keywords: local triviality of principal homogeneous space
zbMATH Keywords: reductive group scheme
MaRDI profile type: Publication
cites work: Q5562621
cites work: Analytic isomorphism and category of finitely generated modules
cites work: Gersten's conjecture and the homology of schemes
cites work: Q5607044
cites work: Fibres quadratiques et composantes connexes réelles
cites work: Principal homogeneous spaces under flasque tori; applications
cites work: Projective Modules Over Central Separable Algebras
cites work: Q5602064
cites work: Q3506242
cites work: On the Bass-Quillen conjecture concerning projective modules over polynomial rings
cites work: Stone Duality for Nominal Boolean Algebras with И
cites work: Q4205555
cites work: Q4179114
cites work: Q3757970
cites work: Failure of a Quadratic Analogue of Serre's Conjecture
cites work: Principal bundles on the affine line
OpenAlex ID: W27919915
description: scientific article; zbMATH DE number 165867