Hom sheaf of a torsion sheaf

Question

Let $\mathcal{F}$ be a torsion sheaf in a smooth projective scheme $X$.

Is it true that $\mathcal{Hom}(\mathcal{F}, \mathcal{O}_{X}) = 0$?

If so, where can I find the demonstration of this result?

I would appreciate any help.

Thank you very much.

Answer 1

Let $f : \mathcal F \to \mathcal O_X$ non-zero and $s \in F(X)$ so that the image of $s$ is not zero. Since $\mathcal F$ is torsion, there is $t \in \mathcal O_X$ so that $ts = 0$. It implies that $s$ is zero outside the locus $Z(t)$. In particular, $f(s)$ is zero outside $Z(t)$ and by density $f(s)$ is zero everywhere, a contradiction.