Is it true for any $O_X$-module $F$ that
$$
Hom_{O_X}(F,O_X)\cong F(X)= \Gamma(X,F)
$$
And is it in general true that the sheafification of $F(X)$ is $F$ ?
I think that it is true for quasi-coherent sheaves but it can be generalized at least the first part?
2026-03-25 12:55:17.1774443317
About Hom and global section functor for $O_X$ modules
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No. However we have $Hom(O_X,F) \cong \Gamma(X,F)$. For a counter-example to your equality, we have $Hom(F,O_X) = 0$ when $F$ is a skyscraper sheaf and $X$ is of dimension at least $1$.
It's not clear what do you mean by "the sheafification of $F(X)$". If $X$ is affine then the data of a quasi-coherent sheaf is equivalent to the data of its global sections. In general it's not the case.