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If I have a morphism of schemes $f:X\rightarrow Y$ and sheaves $\mathcal F,G$ on $X$, then is there a spectral sequence which relates the Ext-groups

$\mathrm{Ext}(f_* \mathcal F, f_*\mathcal G)$ on $Y$ and $\mathrm{Ext}(\mathcal F, \mathcal G)$ on $X$?

I should add, that if this is not possible in general, that my morphism $f$ is actually an affine morphism and that I want to compare the two $\mathrm{Ext}$-groups in any way. The first thing I thought of was a spectral sequence, but perhaps there are other ways you know.

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    it's a shame no one gave an answer. have you made any progress with it?2011-11-23
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    I have to confess, no. But I quite agree with you and also think that these groups are pretty incomparable...2011-12-12
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    have had any success in some particular cases?2011-12-13

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