Gersten-type conjecture for henselian local rings of normal crossing varieties

Abstract

Let $n\geq 0$ and $r>0$ be integers. Let $\mathcal{O}_{X, x}^{h}$ be the henselization of the local ring $\mathcal{O}_{X, x}$ of a scheme $X$ at a point $x\in X$. 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 \'{e}tale sheaves which satisfy some properties over $\mathcal{O}_{Y, y}^{h}$. For example, $\lambda_{Y, r}^{n}$ and $\mu_{l}^{\otimes n}$ satisfy these properties where $\mu_{l}$ is the \'{e}tale sheaf of $l$-th roots of unity for an integer $l$ which is prime to the characteristic of $Y$. Let $B$ be a discrete valuation ring of mixed characteristic $(0, p)$ and $\mathfrak{X}$ a semistable family over $\operatorname{Spec}(B)$. Suppose that $B$ contains $p$-th roots of unity. As an application of the Gersten-type conjecture for $\lambda^{n}_{r}$, we prove the relative version of the Gersten-type conjecture for the $p$-adic \'{e}tale Tate twist $\mathfrak{T}_{1}(n)$ over $\mathcal{O}_{\mathfrak{X}, x}^{h}$. Moreover, we prove a generalization of Artin's theorem about the Brauer groups.