Curves are algebraic $K(\ddot{I}€,1)$: theoretical and practical aspects

Abstract

We prove that any geometrically connected curve $X$ over a field $k$ is an algebraic $K(\ddot{I}€,1)$, as soon as its geometric irreducible components have nonzero genus. This means that the cohomology of any locally constant constructible étale sheaf of $\mathbb{Z}/n\mathbb{Z}$-modules, with $n$ invertible in $k$, is canonically isomorphic to the cohomology of its corresponding $\ddot{I}{€}_1(X)$-module. To this end, we explicitly construct some Galois coverings of $X$ corresponding to Galois coverings of the normalisation of its irreducible components. When $k$ is finite or separably closed, we explicitly describe finite quotients of $\ddot{I}{€}_1(X)$ that allow to compute the cohomology groups of the sheaf, and give explicit descriptions of the cup products $H^1\times H^1\to H^2$ and $H^1\times H^2\to H^3$ in terms of finite group cohomology.

Final text, to be published in Bulletin of the LMS