On étale hypercohomology of henselian regular local rings with values in $p$-adic étale Tate twists

Abstract

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 étale hypercohomology groups $H^{n+1}_{\textrm{ét}} (R, \mathfrak{T}_r (n)) \simeq H^{1}_{\textrm{ét}} (k, W_r \Omega^n_{\log})$ for any integers $n \geqslant 0$ and $r \gt 0$ where $\mathfrak{T}_r (n)$ is the p-adic Tate twist and $W_r \Omega^n_{\log}$ 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.