The $p$-adic Galois Cohomology of Valuation Fields
Tongmu He
TL;DR
This work extends the classical p-adic Galois cohomology results of Tate and Hyodo from discretely valued fields to height-1 valuation fields, including both arithmetic and geometric cases. It replaces higher ramification and local class field theory techniques with Gabber–Ramero cotangent-complex methods, leverages perfectoid towers, and establishes canonical comparisons between modules of differentials and Galois cohomology, with differential Tate twists playing a central role. The main contributions are explicit descriptions of H^q(G_F, eoverline{F})(n) in geometric and arithmetic settings, reductions to cyclotomic towers via ramified base-change theorems, and a prismatic–based Appendix that provides an alternative, conceptual route to the results. These results lay groundwork for p-adic Hodge theory on general valuation fields and offer concrete tools for analyzing p-adic cohomology beyond the discretely valued case. The framework encompasses finite-transcendence cases over Q_p and yields a robust analogue of the Hodge–Tate spectral structure in both arithmetic and geometric contexts.
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.