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Vanishing and comparison theorems in rigid analytic geometry

Part of: (Co)homology theory Arithmetic problems. Diophantine geometry

Published online by Cambridge University Press:  26 December 2019

David Hansen
David Hansen*
Affiliation:
Max Planck Institute for Mathematics, Vivatsgasse 7,Bonn53111, Germany email dhansen@mpim-bonn.mpg.de
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Abstract

We prove a rigid analytic analogue of the Artin–Grothendieck vanishing theorem. Precisely, we prove (under mild hypotheses) that the geometric étale cohomology of any Zariski-constructible sheaf on any affinoid rigid space $X$ vanishes in all degrees above the dimension of $X$. Along the way, we show that branched covers of normal rigid spaces can often be extended across closed analytic subsets, in analogy with a classical result for complex analytic spaces. We also prove some new comparison theorems relating the étale cohomology of schemes and rigid analytic varieties, and give some applications of them. In particular, we prove a structure theorem for Zariski-constructible sheaves on characteristic-zero affinoid spaces.

Keywords

rigid analytic spacesétale cohomologyArtin–Grothendieck vanishingcomparison theorems

MSC classification

Primary: 14G22: Rigid analytic geometry 14F05: Sheaves, derived categories of sheaves and related constructions 14F20: Étale and other Grothendieck topologies and (co)homologies
Type
Research Article
Information
Compositio Mathematica , Volume 156 , Issue 2 , February 2020 , pp. 299 - 324
Copyright
© The Author 2019

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