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For a variety $X$ I define its Zariski-etale site as follows: the category is the category of etale $X$-schemes, and the coverings are Zariski ones. Note that this topology is more coarse than the Nisnevich and the etale one.

Now, for a closed embedding $i:Z\to X$ I would like to study $i^\ast_{Zar-et}$. What can one say about this functor; did anybody study this setting already? In particular, I have the following question.

  1. Let $S$ be a Nisnevich (or Zariski-etale) sheaf over $X$. Is it true that $H^*(Y, i^\ast_{Zar-et}S)$ could be computed as the limit with respect to Zariski hypercoverings $Y.$ of the cohomology with coefficients in $S$ of the simplicial scheme whose components are the henselizations of the components of $Y.$ in $X$? This should be an easy consequence of the Verdier Hypercovering Theorem; yet I wonder whether I apply it correctly.

  2. If $S$ is a flabby Nisnevich sheaf, is it true that $H^j(Y, i^\ast_{Zar-et}S)=0$ for $j>0$?

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