Variations on the Bloch-Ogus Theorem

Let $R$ be a semi-local regular ring of geometric type over a field $k$. Let $\U=\spec R$ be the semi-local scheme. Consider a smooth proper morphism $p:Y \ra \U$. Let $Y_{k(u)}$ be the fiber over the generic point of a subvariety $u$ of $\U$. We prove that the Gersten-type complex for étale cohomology $$ 0 \ra \het^q(Y,C) \ra \het^q(Y_{k(\U)},C) \ra \coprod_{u\in \U^{(1)}} \het^{q-1}(Y_{k(u)},C(-1)) \ra\ldots $$ is exact, where $C$ is a locally constant sheaf with finite stalks of $\zz/n\zz$-modules on $Y_{et}$ and $n$ is an integer prime to $\chrt(k)$.

2000 Mathematics Subject Classification: 14F20, 16E05

Keywords and Phrases: étale cohomology, arithmetic resolution

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