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.
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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.
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To Robert, forever quelling the rebellious provinces
An erratum to this article is available at http://dx.doi.org/10.1186/s40687-015-0045-6.
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Kedlaya, K.S. New methods for \((\varphi, \Gamma)\)-modules. Mathematical Sciences 2, 20 (2015). https://doi.org/10.1186/s40687-015-0031-z
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DOI: https://doi.org/10.1186/s40687-015-0031-z
Keywords
- p-adic Hodge theory
- Perfectoid fields
- Field of norms equivalence
- Witt vectors
- \((\varphi , \Gamma )\)-modules
- Cherbonnier–Colmez theorem