2010/06/02 by Morin, Baptiste
#11R42 #14F20 #14F35 #Algebraic Geometry (math.AG) #FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.1006.0525
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 \citeLichtenbaum. 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 \citeLichtenbaum-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.