Cohomologie non ramifiée sur une courbe p-adique lisse (Unramified Cohomology on a Smooth p-Adic Curve)

Abstract

Let k be a local field with residue characteristic p. Let n be a prime-to-p integer. Consider a smooth, irreducible k-curve X. In this article, we study the subgroup of H³(k(X), μ_n^[otimes]2) consisting of classes which are unramified on X. It is known to be isomorphic to H⁰(X_Zar, R³π_*μ_n^[otimes]2), where π: X_ét → X_Zar is the canonical map. A purely topological description of this group is given, using the Berkovich analytification X^an of X. More precisely, denote by Y a smooth and irreducible Berkovich-analytic k-curve, by Y_top the underlying topological space, by Y_ét the étale-analytic site and by π^an the canonical map Y_ét → Y_top. Let Δ be the skeleton of Y (it is a closed subset defined by Berkovich which is locally a finite graph). Then we show (th. 42) that H⁰(Y_top, R³π^an_*μ_n^[otimes]2) is naturally isomorphic (through sort of a pointwise evaluation of cohomology classes) to the group of harmonic cochains defined on Δ with values in $Bbb Z$/n. Now, if X is a smooth algebraic k-curve the natural map H⁰(X_Zar, R³π_*μ_n^[otimes]2) → H⁰(X^an_top, R³π^an_*μ_n^[otimes]2) is shown to be an isomorphism (th. 5.2). It is a new formulation (for the case where X is proper) and a generalization (to open curves) of a result which was due to Kato. Moreover we use here some steps of Kato's proof, but not his whole result. So our method gives, for projective curves, a (partially) new proof of Kato's theorem.