The p-adic Galois Cohomology of Valuation Fields

2025/02/20 by Tongmu He, He, Tongmu
Mathematics · Pharmacology, Toxicology and Pharmaceutics · #11S25 #13F30 #14F30 (primary) #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Alkaloids: synthesis and pharmacology #Commutative Algebra (math.AC) #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2502.14858

openalex publication_date 2025/02/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

Related

• Galois cohomology of p-adic fields and (φ, τ)-modules
• Recovering function fields from their integral ℓ-adic cohomology with the Galois action
• Optimisation du théorème d'Ax-Sen-Tate et application à un calcul de cohomologie galoisienne p-adique
• Perfectoidness via Sen Theory and Applications to Shimura Varieties
• Galois cohomology of a p-adic field via (Φ,Γ)-modules in the imperfect residue field case