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Compatible systems and ramification

Part of: Arithmetic algebraic geometry Algebraic number theory: local and $p$-adic fields (Co)homology theory

Published online by Cambridge University Press:  21 October 2019

Qing Lu  and
Weizhe Zheng
Qing Lu
Affiliation:
School of Mathematical Sciences, Beijing Normal University, Beijing 100875, China School of Mathematical Sciences, University of the Chinese Academy of Sciences, Beijing 100049, China email qlu@bnu.edu.cn
Weizhe Zheng
Affiliation:
Morningside Center of Mathematics and Hua Loo-Keng Key Laboratory of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China University of the Chinese Academy of Sciences, Beijing 100049, China email wzheng@math.ac.cn
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Abstract

We show that compatible systems of $\ell$-adic sheaves on a scheme of finite type over the ring of integers of a local field are compatible along the boundary up to stratification. This extends a theorem of Deligne on curves over a finite field. As an application, we deduce the equicharacteristic case of classical conjectures on $\ell$-independence for proper smooth varieties over complete discrete valuation fields. Moreover, we show that compatible systems have compatible ramification. We also prove an analogue for integrality along the boundary.

Keywords

ℓ-independenceintegralityvaluative criterionlocal fieldwild ramification

MSC classification

Primary: 14F20: Étale and other Grothendieck topologies and (co)homologies
Secondary: 11G25: Varieties over finite and local fields 11S15: Ramification and extension theory
Type
Research Article
Information
Compositio Mathematica , Volume 155 , Issue 12 , December 2019 , pp. 2334 - 2353
Copyright
© The Authors 2019 

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Footnotes

The first author was partially supported by National Natural Science Foundation of China Grants 11371043 and 11501541. The second author was partially supported by National Natural Science Foundation of China Grants 11621061, 11688101 and 11822110 and the National Center for Mathematics and Interdisciplinary Sciences, Chinese Academy of Sciences.

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