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I am interested in the following question: for $f$ being a morphism of schemes, which conditions ensure that $Rf^*_{et}$ is conservative? This is true if $f$ has a section or if $f$ is an \'etale covering; so this is also true for any smooth surjective $f$. Could I say that this statement is well-known?:) Is there a canonical reference for this fact (or for any its nice generalization?)? Actually, I would also like to apply this statement for a pro-smooth surjective $f$; are there any nice references for the properties of such morphisms?

Upd. Thank you very much for the comments; it was silly for me to forget about surjectivity.

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  • $\begingroup$ The "$R$" in "$Rf^*$" is a bit confusing for me...Anyway, if $f$ is an open immersion (which is etale), this doesn't seem to be true to me. $\endgroup$
    – shenghao
    Commented Mar 16, 2011 at 21:32
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    $\begingroup$ If you work in the setting of étale sheaves, then the functor $f^{\star}$ is conservative for any surjective morphism of schemes $f$; see Proposition 9.1 of Exposé VIII in SGA 4, which provide the statement at the level of categories of sheaves (this is a trivial consequence of the fact that geometric points are the points of the étale site in the toposic sense; see Theorem 7.9 in loc. cit.). The statement at the level of derived categories follows immediately from there, because the functors of type $f^\star$ are exact. I don't see why being étale or smooth implies the property you want. $\endgroup$
    – D.-C. Cisinski
    Commented Mar 16, 2011 at 21:39

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