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Why is $H^p(X, \mathcal{E}xt^q(F, I))=0$ for $p>0$ and $I$ an injective sheaf and $F$ an arbitrary sheaf? I'm trying to check the hypothesis in the grothendieck spectral sequence applied to the functor $\mathcal{E}xt(F, -)$ and global sections functor, and so I'm trying to see why the sheaf $\mathcal{E}xt(F, I)$ is acylic for the the global sections functor, which I think translates to my question in the first sentence.

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A note here: To apply the Grothendieck spectral sequence, you only need the hypotheses to hold for $\mathcal{H}om$ and $\Gamma$, not $\mathcal{E}xt$ and $\Gamma$. (I.e., both should be left-exact, and $\mathcal{H}om$ should take injectives to $\Gamma$-acyclics, which it does since it takes injectives to flabby sheaves.) – Charles Staats Aug 2 '10 at 18:55
up vote 6 down vote accepted

$\mathcal{E}xt^q(F, I)$ is 0 for $q > 0$. On the other hand it follows easily by considering extensions by 0 that $\mathcal{H}om(F, I)$ is flabby, hence acyclic.

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