An Alternative Proof of Hesselholt's Conjecture on Galois Cohomology of Witt Vectors of Algebraic Integers

TL;DR

This paper presents a simpler alternative proof of Hesselholt's conjecture, which states that certain Galois cohomology groups of Witt vectors vanish, confirming a key property in algebraic number theory.

Contribution

The paper offers a more straightforward proof of Hesselholt's conjecture on the vanishing of Galois cohomology groups of Witt vectors, complementing prior proofs by Hogadi and Pisolkar.

Findings

Confirmed the vanishing of the pro-abelian Galois cohomology group

Provided a simpler proof method compared to previous work

Strengthened understanding of Witt vectors in Galois cohomology

Abstract

Let $K$ be a complete discrete valuation field of characteristic zero with residue field $k_K$ of characteristic $p>0$. Let $L/K$ be a finite Galois extension with Galois group $G=\Gal(L/K)$ and suppose that the induced extension of residue fields $k_L/k_K$ is separable. Let $\mathbb{W}_n(\cdot)$ denote the ring of $p$-typical Witt vectors of length $n$. Hesselholt conjectured that the pro-abelian group $\{H^1(G,\mathbb{W}_n(\mathcal{O}_L))\}_{n\geq 1}$ is isomorphic to zero. Hogadi and Pisolkar have recently provided a proof of this conjecture. In this paper, we provide an alternative proof of Hesselholt's conjecture which is simpler in several respects.