Open Access

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

Research in the Mathematical Sciences20152:20

Received: 15 January 2015

Accepted: 13 May 2015

Published: 16 October 2015

The Erratum to this article has been published in Research in the Mathematical Sciences 2015 2:25


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.


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