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Proving that $H^1(X,\mathcal{Hom}(\mathcal{G},\mathcal{E})) \cong \text{Ext}^1(\mathcal{G},\mathcal{E})$ holds for locally free sheaves

This isomorphism holds for any degree of cohomology and it only requires that the first sheaf $\mathcal{G}$ is locally free. This relies on the fact that $\mathcal{H}om(\mathcal{G},-)$ is exact. There ...
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