Variations on the Bloch–Ogus theorem

Abstract

Let $R$ be a semi-local regular ring of geometric type over a field $k$. Let $\mathcal{U}=\mathrm{Spec}R$ be the semi-local scheme. Consider a smooth proper morphism $p:Y\to \mathcal{U}$. Let $Y_{k(u)}$ be the fiber over the generic point of a subvariety $u$ of $\mathcal{U}$. We prove that the Gersten-type complex for étale cohomology

is exact, where $C$ is a locally constant sheaf with finite stalks of $\mathbb{Z}/n\mathbb{Z}$-modules on $Y_{et}$ and $n$ is an integer prime to $\mathrm{char}(k)$.