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

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.

Keywords

Gersten-type conjecture, local-global principle, $p$-adic Tate twist

2010 Mathematics Subject Classification

14F20, 14F30, 14F42

The full text of this article is unavailable through your IP address: 172.17.0.1

Received 10 April 2023

Received revised 7 August 2023

Accepted 8 August 2023

Published 18 September 2024