# About $O_X$-modules?

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!!

• 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 ? – Roland Feb 28 at 15:24
• 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. – andres Feb 28 at 22:45