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

Nisnevich, Yevsey A.

C. R. Acad. Sci., Paris, Sér. I 299, 5-8 (1984).

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: Hiroshi Yanagihara (Yashiro)

MSC:

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

Keywords:

rationally trivial principal homogeneous space; henselian local ring; G-torseur; locally trivial for the Zariski topology