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

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 \'{e}tale logarithmic Hodge-Witt sheaf λⁿY, r on the \'{e}tale site Y_et 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 \'{e}tale 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.