The $p$-adic Galois Cohomology of Valuation Fields

TL;DR

This paper computes the Galois cohomology of p-adic valuation fields extending pre-perfectoid fields, generalizing classical results with new proofs that avoid traditional ramification and class field theory methods.

Contribution

It provides a novel computation of Galois cohomology for valuation fields, extending classical results without relying on higher ramification groups or local class field theory.

Findings

Generalizes Tate and Hyodo's results on valuation fields

Offers a new proof avoiding traditional ramification tools

Utilizes Gabber-Ramero's cotangent complex computations

Abstract

We compute the Galois cohomology of any $p$-adic valuation field extension of a pre-perfectoid field. Moreover, we obtain a generalization and also a new proof of the classical results of Tate and Hyodo on discrete valuation fields, without using higher ramification group, local class field theory or Epp's elimination of ramifications. A key ingredient is Gabber-Ramero's computation of cotangent complexes for valuation rings.