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Let $L/K$ be a finite Galois extension of complete discrete valued fields ofcharacteristic $p$. 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 ofWitt vectors of length $n$ with coefficients in $\sO_L$. We show that theproabelian group ${H^1(G,W_n(\sO_L))}_{n\in \N}$ is zero. This is anequicharacteristic analogue of Hesselholt's conjecture.

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