Curves are algebraic K (π, 1) K (, 1): Theoretical and practical aspects

Abstract We prove that any geometrically connected curve over a field is an algebraic , as soon as its geometric irreducible components have nonzero genus. This means that the cohomology of any locally constant constructible étale sheaf of ‐modules, with invertible in , is canonically isomorphic to the cohomology of its corresponding ‐module. To this end, we explicitly construct some Galois coverings of corresponding to Galois coverings of the normalisation of its irreducible components. When is finite or algebraically closed, we precisely describe finite quotients of that allow to compute the cohomology groups of the sheaf, and give explicit descriptions of the cup products and in étale cohomology in terms of finite group cohomology.