Galois cohomology of $p$-adic fields and $(\varphi, \tau)$-modules

TL;DR

This paper develops explicit Herr complexes based on $(, au)$-modules to compute Galois cohomology of $p$-adic representations, offering new tools for understanding $p$-adic Galois actions and applications to $p$-divisible groups.

Contribution

It introduces explicit Herr complexes using $(, au)$-modules, providing an alternative to $(, abla)$-modules for Galois cohomology computations.

Findings

Constructed explicit Herr complexes with $(, au)$-modules.

Applied the complexes to study $p$-divisible groups.

Enhanced computational methods for Galois cohomology.

Abstract

We construct various explicit Herr complexes that compute the Galois cohomology of a $p$-adic representation of the absolute Galois group of a complete discrete valuation field of characteristic $0$ with a perfect residue field of characteristic $p$, using the associated $(\varphi,\tau)$-modules (defined by Xavier Caruso), instead of $(\varphi,\Gamma)$-modules. We also give an application to $p$-divisible groups.