Espaces homogènes principaux rationnellement triviaux et arithmétique des schémas en groupe réductifs sur les anneaux de Dedekind. (Rationally trivial principal homogeneous spaces and arithmetic of reductive group schemes over Dedekind rings). (French) Zbl 0587.14033

Let X be a noetherian integral regular scheme, \(K=k(X)\) the field of rational functions on X, G a reduced group scheme over X and E a principal homogeneous space of G over X locally trivial for the étale topology of X (i.e. a G-torseur). We say that E is rationally trivial if it has a section over K. The aim of this article is to prove the following conjecture, formulated by Serre and Grothendieck, for the special case that X is one-dimensional or henselian local. Conjecture: In the above hypothesis any rationally trivial G-torseur E is locally trivial for the Zariski topology of X.
Reviewer: H.Yanagihara


14M17 Homogeneous spaces and generalizations
14F20 Étale and other Grothendieck topologies and (co)homologies
14B15 Local cohomology and algebraic geometry