Higher fields of norms and $(\phi ,\Gamma )$-modules

Abstract

We describe a generalisation of the Fontaine-Wintenberger theory of the “field of norms” functor to local fields with imperfect residue field, generalising work of Abrashkin for higher dimensional local fields. We also compute the cohomology of associated $p$-adic Galois representations using $(\varphi ,\Gamma )$-modules.