I. Panin, K. Zainoulline: Variations on the Bloch-Ogus Theorem

panin@pdmi.ras.ru, kirill@ihes.fr

Submission: 2002, Mar, 13

In the present paper we discuss questions concerning the arithmetic resolution for \'etale cohomology. In particular, consider a smooth quasi-projective variety $X$ over a field $k$ together with the local scheme $\U=\spec \Ox_{X,x}$ at a point $x\in X$. Let $p:Y \ra \U$ be a smooth projective morphism and let $Y_{k(u)}$ denote its fiber over the generic point of a subvariety $u$ of $\U$. We prove there is a Gersten-type exact sequence $$ 0 \ra \het^q(Y,F) \ra \het^q(Y_{k(\U)},F) \ra \coprod_{u\in \U^{(1)}} \het^{q-1}(Y_{k(u)},F(-1)) \ra $$ of \'etale cohomology with coefficients in a locally constant \'etale sheaf $F$ of $\zz/n\zz$-modules on $Y$ which has finite stalks and $(n, char(k))=1$

2000 Mathematics Subject Classification:

Keywords and Phrases:

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