Explicit isomorphisms for a Herr-type complex over a metabelian extension

TL;DR

This paper constructs explicit isomorphisms between a complex associated to Galois representations over certain Banach algebras and their Galois cohomology, extending previous results to a broader algebraic setting.

Contribution

It introduces explicit isomorphisms for a Herr-type complex over a Banach algebra in a metabelian extension, generalizing earlier finite extension cases.

Findings

Constructed a complex over false-Tate extensions for Galois representations.

Established explicit isomorphisms between the complex's cohomology and Galois cohomology.

Extended previous finite extension results to a more general Banach algebra setting.

Abstract

Let $S$ be a Banach algebra over $\mathbb{Q}_p$ whose residue fields are finite extensions of $\mathbb{Q}_p$. Given an arithmetic family $V$ of Galois representations, i.e., a finite free $S$-module $V$ with a continuous action of the absolute Galois group of a $p$-adic number field, we construct a complex associated to $V$ over false-Tate extensions and construct explicit isomorphisms between its cohomology and the Galois cohomology. This recovers earlier results by Tavares Ribeiro when $S$ is a finite extension of $\mathbb{Q}_p$.