Timeline for When inverse image is conservative; a reference or a generalization?
Current License: CC BY-SA 2.5
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| Mar 17, 2011 at 9:24 | history | edited | Mikhail Bondarko | CC BY-SA 2.5 |
added 132 characters in body
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| Mar 16, 2011 at 21:39 | comment | added | D.-C. Cisinski | 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. | |
| Mar 16, 2011 at 21:32 | comment | added | shenghao | 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. | |
| Mar 16, 2011 at 20:21 | history | asked | Mikhail Bondarko | CC BY-SA 2.5 |