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Cohomology of topological groups with applications to the Weil group

Part of: (Co)homology theory Topological and differentiable algebraic systems Categories and geometry Arithmetic algebraic geometry

Published online by Cambridge University Press:  01 May 2008

M. Flach
M. Flach*
Affiliation:
Department of Mathematics, Caltech, Pasadena, CA 91125, USA (email: flach@its.caltech.edu)
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Abstract

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We establish various properties of the definition of cohomology of topological groups given by Grothendieck, Artin and Verdier in SGA4, including a Hochschild–Serre spectral sequence and a continuity theorem for compact groups. We use these properties to compute the cohomology of the Weil group of a totally imaginary field, and of the Weil-étale topology of a number ring recently introduced by Lichtenbaum (both with integer coefficients).

Keywords

cohomologytopological groupsWeil group

MSC classification

Secondary: 14F20: Étale and other Grothendieck topologies and (co)homologies 11G40: $L$-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture 18F10: Grothendieck topologies 22A99: None of the above, but in this section
Type
Research Article
Information
Compositio Mathematica , Volume 144 , Issue 3 , May 2008 , pp. 633 - 656
Copyright
Copyright © Foundation Compositio Mathematica 2008

References

The author is supported by grant DMS-0401403 from the National Science Foundation.