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arXiv · 2502.14858

The $p$-adic Galois Cohomology of Valuation Fields

Tongmu He
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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.

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Tongmu He. 2025-02-20. The $p$-adic Galois Cohomology of Valuation Fields. https://arxiv.org/abs/2502.14858

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