Gersten-type conjecture for henselian local rings of normal crossing varieties (2402.18042v2)

Abstract: Let $n\geq 0$ be an integer. For a normal crossing variety $Y$ over the spectrum of a field $k$ of positive characteristic $p>0$, K.Sato defined an \'{e}tale logarithmic Hodge-Witt sheaf $\lambda^{n}_{Y, r}$ on the \'{e}tale site $Y_{\mathrm{\acute{e}t}}$ which agrees with $W_{r}\Omega^{n}_{Y, \log}$ in the case where $Y$ is smooth over $\operatorname{Spec}(k)$. In this paper, we prove the Gersten-type conjecture for $\lambda^{n}_{r}$ over the henselization of the local ring $\mathcal{O}{Y, y}$ of $Y$ at a point $y\in Y$. As an application, we prove the relative version of the Gersten-type conjecture for the $p$-adic \'{e}tale Tate twist $\mathfrak{T}{1}(n)$ over the henselization of the local ring $\mathcal{O}_{\mathfrak{X}, x}$ of a semistable family $\mathfrak{X}$ over the spectrum of a discrete valuation ring $B$ of mixed characteristic $(0, p)$ at a point $x\in \mathfrak{X}$ in the case where $B$ contains $p$-th roots of unity. Moreover, we prove a generalization of Artin's theorem about the Brauer groups.