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 $\operatorname{H}^{n+1}_{\mathrm{\acute{e}t}}(R, \mathfrak{T}_{r}(n)) \simeq \operatorname{H}^{1}_{\mathrm{\acute{e}t}}(k, W_{r}Ω_{\log}^{n})$ for any integers $n\geq 0$ and $r>0$ where $\mathfrak{T}_{r}(n)$ is the $p$-adic Tate twist and $W_{r}Ω_{\log}^{n}$ 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.

Theorems & Definitions (22)

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