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 $H_{\acute{\mathrm{e}}\mathrm{t}}^{n+1}(R,{\mathfrak{T}}_r(n))\simeq H_{\acute{\mathrm{e}}\mathrm{t}}^1(k,W_r{\Omega }_{\log }^n)$ for any integers $n\ge 0$ 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.

30 pages, Final version, Revised after the referee report. Added Remark 3.1 and references