Triangulation et cohomologie étale sur une courbe analytique

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Let k be a non-archimedean complete valued field and let X be a smooth Berkovich analytic k-curve. Let F be a finite locally constant étale sheaf on k whose torsion is prime to the residue characteristic. We denote by |X| the underlying topological space and by π the canonical map from the étale site to |X|. In this text we define a triangulation of X, we show that it always exists and use it to compute H⁰(|X|,Rqπ_*F) and H¹(|X|,Rqπ_*F). If X is the analytification of an algebraic curve we give sufficient conditions so that those groups are isomorphic to their algebraic counterparts; if the cohomology of k has a dualizing sheaf in some degree d (e.g k is p-adic, or k=C((t))) then we prove a duality theorem between H⁰(|X|,Rqπ_*F) and H¹_ c(|X|,Rᵈ⁺¹π_*G) where G is the tensor product of the dual sheaf of F with the dualizing sheaf and the sheaf of n-th roots of unity.