On cohomology of Witt vectors of algebraic integers and a conjecture of Hesselholt

TL;DR

This paper proves Hesselholt's conjecture that the first cohomology group of the Witt ring of integers in Galois extensions of local fields is zero, confirming a significant prediction in algebraic number theory.

Contribution

The paper establishes the conjecture for all Galois extensions, extending previous partial results and advancing understanding of Witt vectors in local field extensions.

Findings

Hesselholt's conjecture is proven for all Galois extensions.

The result confirms the vanishing of the first cohomology group in this context.

The proof applies to complete discrete valued fields with residue characteristic p.

Abstract

Let $K$ be a complete discrete valued field of characteristic zero with residue field $k_K$ of characteristic $p > 0$. Let $L/K$ be a finite Galois extension with the Galois group $G$ and suppose that the induced extension of residue fields $k_L/k_K$ is separable. In his paper, Hesselholt conjectured that $H^1(G,W(\sO_L))$ is zero, where $\sO_L$ is the ring of integers of $L$ and $W(\sO_L)$ is the Witt ring of $\sO_L$ w.r.t. the prime $p$. He partially proved this conjecture for a large class of extensions. In this paper, we prove Hesselholt's conjecture for all Galois extensions.