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https://arxiv.org/abs/math/9809114 Theorem 1.1 gives a fiberwise criterion for a Fourier Mukai functor to be fully faithful.

I am looking for a similar result on stacks with the maps being not necessarily schematic.

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    $\begingroup$ See arxiv.org/pdf/math/0210439.pdf $\endgroup$ Commented Sep 20, 2018 at 9:34
  • $\begingroup$ "schematic"="representable by schemes" (as opposed to "by algebraic spaces")? $\endgroup$
    – Qfwfq
    Commented Sep 20, 2018 at 9:46
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    $\begingroup$ @AliCaglayan: Yes I saw that article, but it only deals with the converse. i.e. the author assumes the functor is fully faithful. $\endgroup$
    – Adam Gal
    Commented Sep 20, 2018 at 9:56
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    $\begingroup$ @Qfwfq: yes, because in the schematic case you can reduce to the previous result. I imagine with some more work you can reduce also from more general situations but I was wondering if someone had not already done this. $\endgroup$
    – Adam Gal
    Commented Sep 20, 2018 at 9:57

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