Ofer Gabber

Fibrés principaux sur les corps valués henséliens

Let K be the field of fractions of a henselian valuation ring A. Assume that the completion b K is a separable extension of K. Let Y be a K-variety, let G be an algebraic group over K, and let f : X! Y be a G-torsor over Y . We consider the induced map X(K)! Y (K), which is continuous for the topologies deduced from the valuation. If I denotes the image of this map, we prove that I is locally closed in Y (K); moreover, the induced surjection X(K)! I is a principal bundle with group G(K) (also topologized by the valuation).

7. Analytic geometry

7.1 Derived completion functor 7.2 Cotangent complex for formal schemes and adic spaces 7.3 Deformations of formal schemes and adic spaces 7.4 Analytic geometry over a deeply ramified base 7.5 Semicontinuity of the discriminant

6. Valuation theory

6.1 Ordered groups and valuations 6.2 Basic ramification theory 6.3 Algebraic extensions 6.4 Logarithmic differentials 6.5 Transcendental extensions 6.6 Deeply ramified extensions

5. Henselization and completion of almost algebras

5.1 Henselian pairs 5.2 Criteria for unramified morphisms 5.3 Topological algebras and modules 5.4 Henselian approximation of structures over adically complete rings 5.5 Lifting theorems for henselian pairs 5.6 Smooth locus of an affine almost scheme 5.7 Quasi-projective almost schemes 5.8 Lifting and descent of torsors

2. Homological theory

2.1 Some ring-theoretic preliminaries 2.2 Categories of almost modules and algebras 2.3 Uniform spaces of almost modules 2.4 Almost homological algebra 2.5 Almost homotopical algebra