2005/01/28 by Antoine Ducros, Ducros, Antoine
Mathematics · #14F20 #14G20 #14G22 #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #math.AG #msc:14F20 #msc:14G20 #msc:14G22
paper · pdf · doi:10.48550/arxiv.math/0501508
arxiv created 2005/01/28 · openalex publication_date 2005/01/28 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.