  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.
