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Given a closed immersion $i: Z \hookrightarrow X$ a coherent sheaf $\mathcal{F}$ on $X$ and a coherent sheaf $\mathcal{G}$ on $Z$, do we have $\mathrm{Ext}^n(\mathcal{F}, i_*\mathcal{G}) = \mathrm{Ext}^n(i^*\mathcal{F}, \mathcal{G})$? For $n = 0$ it is the usual adjunction, so can we deduce it by the usual "universal $\delta$-functor" argument?

Consider the exact $\delta$-functors $\mathrm{Coh}(Z) \to (Ab), \mathcal{G} \mapsto \mathrm{Ext}^n(\mathcal{F}, i_*\mathcal{G})$ and $\mathcal{G} \mapsto \mathrm{Ext}^n(i^*\mathcal{F}, \mathcal{G})$ (exact since $i_*$ is exact). They coincide for $n = 0$, so we just have to check if they are both effacable to coincide for every $n$.

Could we also derive this using derived categories?

Edit: It seems to be wrong: Take $i$ the inclusion of a closed point, then the RHS is always trivial, but I don't think the LHS is. E.g. $\mathrm{Ext}^1(k(x),k(x)) = T_x$ the Zariski tangent space. So where does the above "argument" go wrong?

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    It would be best if you wrote down in the question the argument in detail that you are thinking of, and *then* we can answer something useful. Otherwise, the answer is *yes for sufficiently general meanings of the word "usual"* and this does not help you or anyone!2011-11-27
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    (It is usually best if titles are not entirely composed of TeX formulas, by the way)2011-11-27
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    I have edited my question accordingly.2011-11-27

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