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

TL;DR

This paper develops a new framework linking Galois representations of product groups over fields with imperfect residue fields to multivariable -tale (, \u03b3)-modules, extending existing theories to more complex residue field cases.

Contribution

It introduces an equivalence of categories connecting Galois representations of product groups with multivariable (, )-modules over Laurent series rings for imperfect residue fields.

Findings

Established an equivalence of categories for imperfect residue fields.

Extended multivariable (, )-module theory to new residue field contexts.

Provided a framework for analyzing Galois representations in mixed characteristic settings.

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 $\Bbb{F}_p$-representations of the product of $\Delta$ copies of the absolute Galois group of $K$ and multivariable \' etale $(\varphi, \Gamma)$-modules over a multivariable Laurent series ring over $k_K$.