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Published January 2011 | public
Journal Article

The Weil-étale fundamental group of a number field II

Abstract

We define the fundamental group underlying the Weil-étale cohomology of number rings. To this aim, we define the Weil-étale topos as a refinement of the Weil-étale sites introduced by Lichtenbaum (Ann Math 170(2):657–683, 2009). We show that the (small)Weil-étale topos of a smooth projective curve defined in this paper is equivalent to the natural definition. 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 theWeil-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.

Additional Information

© 2010 Springer Basel AG. Published online: 13 October 2010. I am very grateful to Matthias Flach for his comments and for many stimulating discussions related to Weil-étale cohomology.

Additional details

Created:
August 22, 2023
Modified:
October 23, 2023