Multivariable $(\phi ,\Gamma )$-modules and Representations of Products of Galois Groups: The Case of Imperfect Residue Field

Abstract

Let $K$ be a complete discretely valued field with mixed characteristic $(0,p)$ and imperfect residue field $k_K$. Let $\Delta$ be a finite set. We construct an equivalence of categories between finite dimensional ${\mathbb{F}}_p$-representations of the product of $\Delta$ copies of the absolute Galois group of $K$ and multivariable \' etale $(\phi ,\Gamma )$-modules over a multivariable Laurent series ring over $k_K$.

final version, to appear in Bull. Soc. Math. France