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), μn⊗2) consisting of classes which are unramified on X. It is known to be isomorphic to H0(XZar, R3π*μn⊗2), where π: Xet → XZar is the canonical map. A purely topological description of this group is given, using the Berkovich analytification Xan of X. More precisely, denote by Y a smooth and irreducible Berkovich-analytic k-curve, by Ytop the underlying topological space, by Yet the etale-analytic site and by πan the canonical map Yet → Ytop. 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(Ytop, R3πan*μn⊗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 ℤ/n. Now, if X is a smooth algebraick-curve the natural map H0(XZar, R3π*μn⊗2) → H0(Xantop, R3πan*μn⊗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.