Fibr\'es principaux sur les corps valu\'es hens\'eliens
Abstract
Let (K,v) be a valued field, Y a K-variety, G an algebraic group over K (not necessarily smooth), and f: X->Y a G-torsor over Y. We consider the induced map X(K)-->Y(K), which is continuous for the topologies deduced from the valuation. Let I denote the image of this map. We prove that if (K,v) is henselian and its completion is a separable extension, then: - I is locally closed in Y(K); - the induced surjection X(K)-->I is a principal bundle with group G(K) (also topologized by the valuation).
Cite
@article{arxiv.1309.6430,
title = {Fibr\'es principaux sur les corps valu\'es hens\'eliens},
author = {Ofer Gabber and Philippe Gille and Laurent Moret-Bailly},
journal= {arXiv preprint arXiv:1309.6430},
year = {2014}
}
Comments
59 pages, in French. Minor improvements, some changes in presentation required by journal. Compactification theorem 4.2 improved (the compactification is now equivariant under a smooth group action). Accepted for publication in Algebraic Geometry (June 4, 2014)