Triangulation et cohomologie \'{e}tale sur une courbe analytique

Abstract

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 \'{e}tale sheaf on $k$ whose torsion is prime to the residue characteristic. We denote by $|X|$ the underlying topological space and by $\pi$ the canonical map from the \'{e}tale site to $|X|$. In this text we define a triangulation of $X$, we show that it always exists and use it to compute $H^{0}(|X|,R^{q}\pi\_{*}F)$ and $H^{1}(|X|,R^{q}\pi\_{*}F)$. 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 $H^{0}(|X|,R^{q}\pi\_{*}F)$ and $H^{1}\_ {c}(|X|,R^{d+1}\pi\_{*}G)$ 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.