The Weil-étale fundamental group of a number field II

We define the fundamental group underlying to Lichtenbaum's Weil-étale cohomology for number rings. To this aim, we define the Weil-étale topos as a refinement of the Weil-étale sites introduced in \cite{Lichtenbaum}. We show that the (small) Weil-étale topos of a smooth projective curve defined in this paper is equivalent to the natural definition given in \cite{Lichtenbaum-finite-field}. Then we compute the Weil-étale fundamental group of an open subscheme of the spectrum of a number ring. Our fundamental group is a projective system of locally compact topological groups, which represents first degree cohomology with coefficients in locally compact abelian groups. We apply this result to compute the Weil-étale cohomology in low degrees and to prove that the Weil-étale topos of a number ring satisfies the expected properties of the conjectural Lichtenbaum topos.

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