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

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 $\mathbb{Q}_p$.