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Consider $F$ an $O_X$-module is true in general that $Hom_{O_X}(O_X,F) \equiv F(X)$ ? Morover I need to prove that if $F$ is a flasque sheaf then $H^n(X,F)=0$ for $n>0$.

I think is beacuse the fact that the section functor $\Gamma(U,-)$ keeps an exact sequence of sheaves $$ 0 \to F \to I \to H \to 0 $$ to an exact sequence $$ 0 \to \Gamma(U,F) \to \Gamma(U,I) \to \Gamma(U,H) \to 0 $$ This implies that the derived functors $R\Gamma$ are all $0$ it is correct? Thanks for the help!!

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  • $\begingroup$ For your first question, yes $\operatorname{Hom}_{\mathcal{O}_X}(\mathcal{O}_X,\mathcal{F})=\mathcal{F}(X)$. This is a good exercice to construct the bijection with these two groups. The second question is more difficult but there is a proof in any books on the subject. Do you have a difficulty with the proof ? $\endgroup$ – Roland Feb 28 at 15:24
  • $\begingroup$ The main problem was the first point but i wan try by myself, for the second part my purpose wa to understand better the proof in the Hartshorne's book. $\endgroup$ – andres Feb 28 at 22:45

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