Remark on equicharacteristic analogue of Hesselholt's conjecture on cohomology of Witt vectors

TL;DR

This paper proves that for a finite Galois extension of complete discrete valued fields of characteristic p, the first cohomology group of Witt vectors vanishes, providing an equicharacteristic analogue of Hesselholt's conjecture.

Contribution

It establishes the vanishing of the proabelian group of first cohomology for Witt vectors in an equicharacteristic setting, extending Hesselholt's conjecture.

Findings

Proabelian group ${H^1(G,W_n( ext{O}_L))}_{n o ext{N}}$ is zero.

Provides an analogue of Hesselholt's conjecture in characteristic p.

Advances understanding of cohomology of Witt vectors in equicharacteristic fields.

Abstract

Let $L/K$ be a finite Galois extension of complete discrete valued fields of characteristic $p$. Assume that the induced residue field extension $k_L/k_K$ is separable. For an integer $n\geq 0$, let $W_n(\sO_L)$ denote the ring of Witt vectors of length $n$ with coefficients in $\sO_L$. We show that the proabelian group ${H^1(G,W_n(\sO_L))}_{n\in \N}$ is zero. This is an equicharacteristic analogue of Hesselholt's conjecture.