Spectral sequence for Ext 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.

Well, a useless thing to say is that in general you have the Grothendieck spectral sequence (cf answer by Matt Emerton Contravariant Grothendieck Spectral Sequence), (I'm thinking of A,B there as your $Rf_*F, Rf_*G$).
But as your morphism is affine then this does not help one bit.
I guess it's impossible a priori to give any comparison: essentially because $f_*$ and $\underline{Hom}$ aren't compatible (which, via $E^\vee \otimes F = \underline{Hom}(E,F)$ is another way of saying that $f_*$ and $\otimes$ aren't compatible).
But I would very like to be proven wrong, as I'm also similarly stuck!
