New methods for \((\varphi, \Gamma)\)-modules

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  1. Special Collection in Honor of Robert F. Coleman

Abstract

We provide new proofs of two key results of p-adic Hodge theory: the Fontaine-Wintenberger isomorphism between Galois groups in characteristic 0 and characteristic p, and the Cherbonnier–Colmez theorem on decompletion of \((\varphi , \Gamma )\)-modules. These proofs are derived from joint work with Liu on relative p-adic Hodge theory, and are closely related to the theory of perfectoid algebras and spaces, as in the work of Scholze.

Keywords

p-adic Hodge theory Perfectoid fields Field of norms equivalence Witt vectors \((\varphi , \Gamma )\)-modules Cherbonnier–Colmez theorem 

Mathematics Subject Classification

Primary 11S20 Secondary 11S15 13F35 

Notes

Acknowledgements

Financial support was provided by NSF CAREER grant DMS-0545904, DARPA grant HR0011-09-1-0048, MIT (NEC Fund), UC San Diego (Warschawski Professorship). The author thanks Ruochuan Liu, Ryan Rodriguez, Peter Schneider, and Sarah Zerbes for helpful feedback.

Compliance with ethical guidelines

Competing interests The author declares that he has no competing interests.

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Copyright information

© Kedlaya. 2015

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of California San DiegoLa JollaUSA

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