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I would like to know if the following generalisation of Milne Étale Cohomology, Lemma III.1.16, p. 88 holds: (all cohomology groups with respect to the étale topology)

Let $Y \hookrightarrow X$ quasi-compact be a closed immersion with $Y$ irreducible of codimension $1$, $y$ the generic point of $Y$, and $Z_i \subset Y$ be a directed family of proper closed subschemes (edit: for the transition maps to be affine, one probably has to take a cofinal subsystem). Is then $\varprojlim_i(X \setminus Z_i) = \mathrm{Spec}\mathcal{O}_{X,y}$? (edit: this is of course nonsense. One additionally has to use excision to restrict to smaller and smaller open neighbourhoods of $Y \setminus Z_i$.)

And do we have ($\mathcal{F}$ an étale sheaf on $X$) $H^p_{\{y\}}(\mathrm{Spec}\mathcal{O}_{X,y}, \mathcal{F}) = \varinjlim_iH^p_{Y \setminus Z_i}(X\setminus Z_i,\mathcal{F})$?

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    $\begingroup$ What part of $Y \hookrightarrow X$ is quasi-compact? Is it the map, the source, or the target? $\endgroup$
    – S. Carnahan ♦
    Commented Feb 2, 2012 at 3:18
  • $\begingroup$ I think you can assume both to be quasi-compact. $\endgroup$
    – user19475
    Commented Feb 2, 2012 at 8:22

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