Fibrés principaux sur les corps valués henséliens

Ofer Gabber (1) , Philippe Gille (2) , Laurent Moret-Bailly (3)

1 IHES - Institut des Hautes Études Scientifiques
2 DMA - Département de Mathématiques et Applications - ENS Paris
3 IRMAR - Institut de Recherche Mathématique de Rennes

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).

Mots clés: Local fields; compactifications; homogeneous spaces; torsors; algebraic groups; valuation fields

Domaines: Géométrie algébrique [math.AG]

Ofer Gabber, Philippe Gille, Laurent Moret-Bailly. Fibrés principaux sur les corps valués henséliens. Algebraic Geometry, 2014, 1 (5), pp.573-612. ⟨10.14231/AG-2014-025⟩. ⟨hal-00880469⟩