I am sorry, but I am quite new to Ext groups of sheaves. However, I have a closed embedding of projective $\mathbb{C}$-schemes $\iota : X \hookrightarrow Y$ and was wondering if $$\iota_*:\mathrm{Ext}^*_X(\mathcal{F},\mathcal{G}) \to \mathrm{Ext}^*_Y (\iota_*\mathcal{F},\iota_*\mathcal{G})$$ was an iso, respectively if there are certain properties of that map that I may exploit to control said map.
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6$\begingroup$ A simple counterexample is given by the inclusion $pt \hookrightarrow \mathbb P^1$ (we might just as well replace $\mathbb P^1$ with $\mathbb A^1$). Obviously the ext groups are trivial on a point, but the skyscraper sheaf does have higher self-exts, as computed by the Koszul complex. I expect that your map is almost never an isomorphism for this reason. $\endgroup$– Sam GunninghamCommented Jan 17, 2019 at 14:56
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$\begingroup$ thanks a lot! then I will give up that hope. $\endgroup$– FelixCommented Jan 17, 2019 at 14:58
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First, there is an adjunction isomorphism $$ Ext^\bullet(i_*F,i_*G) \cong Ext^\bullet(Li^*(i_*F),G), $$ where $Li^*$ is the derived pullback functor. Furthermore, if $X$ in $Y$ is a locally complete intersection, then $$ L_pi^*(i_*F) \cong F \otimes \Lambda^pN^\vee_{X/Y}. $$ These two observations combine into a spectral sequence $$ E_2^{p,q} = Ext^q(F \otimes \Lambda^pN^\vee_{X/Y}, G), $$ whose first row is formed by $Ext^\bullet(F,G)$ and that converges to $Ext^n(i_*F,i_*G)$.
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$\begingroup$ thanks a lot! that helps quite a bit! In particular since I am working with derived categories, and need that ext for controlling them. $\endgroup$– FelixCommented Jan 18, 2019 at 8:21