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

Abstract

Let $n\ge 0$ 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 étale logarithmic Hodge-Witt sheaf ${\lambda }_{Y,r}^n$ on the étale site $Y_{\acute{\mathrm{e}}\mathrm{t}}$ which agrees with $W_r{\Omega }_{Y,\log }^n$ in the case where $Y$ is smooth over $\mathrm{Spec}(k)$. In this paper, we prove the Gersten-type conjecture for étale 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 étale 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 $\mathrm{Spec}(B)$. Suppose that $B$ contains $p$-th roots of unity. As an application of the Gersten-type conjecture for ${\lambda }_r^n$, we prove the relative version of the Gersten-type conjecture for the $p$-adic étale 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.

75 pages, Abstract changed