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 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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References
Bunge, M.: Classifying topoi and fundamental localic groupoids. Category theory 1991 (Montreal, PQ, 1991). In: CMS Conference Proceedings, vol. 13, pp. 75–96. American Mathematical Society, Providence (1992)
Bunge M., Moerdijk I.: On the construction of the Grothendieck fundamental group of a topos by paths. J. Pure Appl. Algebra 116(1-3), 99–113 (1997)
Flach M.: Cohomology of topological groups with applications to the Weil group. Compos. Math. 144(3), 633–656 (2008)
Flach, M., Morin, B.: On the Weil-étale topos of regular arithmetic schemes. Preprint (2010)
Grothendieck, A., Artin, M., Verdier, J.L.: Théorie des Topos et cohomologie étale des schémas (SGA4). In: Lectures Notes in Mathematics, vols. 269, 270, 305. Springer (1972)
Johnstone, P.T.: Sketches of an elephant: a topos theory compendium. In: Vol. 1, 2. Oxford Logic Guides, 43. The Clarendon Press, Oxford University Press, New York (2002)
Lichtenbaum S.: The Weil-étale topology on schemes over finite fields. Compos. Math. 141(3), 689–702 (2005)
Lichtenbaum S.: The Weil-étale topology for number rings. Ann. Math. 170(2), 657–683 (2009)
Moerdijk I.: Prodiscrete groups and Galois toposes. Nederl. Akad. Wetensch. Indag. Math. 51(2), 219–234 (1989)
Morin, B.: Sur le topos Weil-étale d’un corps de nombres. Thesis (2008)
Morin, B.: On the Weil-étale cohomology of number fields. Trans. Am. Math. Soc. (to appear)
Morin, B.: The Weil-étale fundamental group of a number field I. Kyushu J. Math. 65 (2011, to appear)
Mostert P.S.: Sections in principal fibre spaces. Duke Math. J. 23, 57–71 (1956)
Neukirch J., Schmidt A., Wingberg K.: Cohomology of Number Fields. 2nd edn. Grundlehren der Mathematischen Wissenschaften, 323. Springer, Berlin (2008)
Tate, J.: Number theoretic background. Automorphic forms, representations and L-functions. In: Proceedings of Symposium on Pure Mathematics. Oregon State University, Corvallis, 1977, Part 2, 3–26, Proceedings of Symposium on Pure Mathematics, vol. XXXIII. American Mathematical Society, Providence (1979)
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Morin, B. The Weil-étale fundamental group of a number field II. Sel. Math. New Ser. 17, 67–137 (2011). https://doi.org/10.1007/s00029-010-0041-z
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DOI: https://doi.org/10.1007/s00029-010-0041-z