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

Abstract

Let n≥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 étale logarithmic Hodge-Witt sheaf λnY,r on the étale site Ye´t which agrees with WrΩnY,log in the case where Y is smooth over Spec(k). In this paper, we prove the Gersten-type conjecture for λnr over the henselization of the local ring OY,y of Y at a point y∈Y. As an application, we prove the relative version of the Gersten-type conjecture for the p-adic étale Tate twist T1(n) over the henselization of the local ring OX,x of a semistable family X over the spectrum of a discrete valuation ring B of mixed characteristic (0,p) at a point x∈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.