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TOPOLOGY ON COHOMOLOGY OF LOCAL FIELDS

Part of: Algebraic number theory: local and $p$-adic fields Algebraic geometry: Foundations

Published online by Cambridge University Press:  25 August 2015

KĘSTUTIS ČESNAVIČIUS
KĘSTUTIS ČESNAVIČIUS*
Affiliation:
Department of Mathematics, University of California, Berkeley, CA 94720-3840, USA; [email protected]

Abstract

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Arithmetic duality theorems over a local field $k$ are delicate to prove if $\text{char}\,k>0$. In this case, the proofs often exploit topologies carried by the cohomology groups $H^{n}(k,G)$ for commutative finite type $k$-group schemes $G$. These ‘Čech topologies’, defined using Čech cohomology, are impractical due to the lack of proofs of their basic properties, such as continuity of connecting maps in long exact sequences. We propose another way to topologize $H^{n}(k,G)$: in the key case when $n=1$, identify $H^{1}(k,G)$ with the set of isomorphism classes of objects of the groupoid of $k$-points of the classifying stack $\mathbf{B}G$ and invoke Moret-Bailly’s general method of topologizing $k$-points of locally of finite type $k$-algebraic stacks. Geometric arguments prove that these ‘classifying stack topologies’ enjoy the properties expected from the Čech topologies. With this as the key input, we prove that the Čech and the classifying stack topologies actually agree. The expected properties of the Čech topologies follow, and these properties streamline a number of arithmetic duality proofs given elsewhere.

MSC classification

Primary: 11S99: None of the above, but in this section
Secondary: 11S25: Galois cohomology 14A20: Generalizations (algebraic spaces, stacks)
Type
Research Article
Information
Forum of Mathematics, Sigma , Volume 3 , 2015 , e16
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Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author 2015

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