Let $R$ be the henselization of a local ring of a semistable family over the spectrum of a discrete valuation ring of mixed characteristic $(0, p)$ and $k$ the residue field of $R$. In this paper, we prove an isomorphism of \'{ e}tale hypercohomology groups $\operatorname{ H}^{ n+1}_{ \mathrm{ \acute{ e}t}}(R, \mathfrak{ T}_{ r}(n)) \simeq \operatorname{ H}^{ 1}_{ \mathrm{ \acute{ e}t}}(k, W_{ r}\Omega_{ \log}^{ n})$ for any integers $n\geq 0$ and $r>0$ where $\mathfrak{ T}_{ r}(n)$ is the $p$-adic Tate twist and $W_{ r}\Omega_{ \log}^{ n}$ is the logarithmic Hodge-Witt sheaf. As an application, we prove the local-global principle for Galois cohomology groups over function fields of curves over an excellent henselian discrete valuation ring of mixed characteristic.
Categories: math.NT