Let $f:\tilde{Z}\to Z$ be a proper birational map between (irreducible) varieties over $\mathbb{C}$ with $\tilde{Z}$ smooth and $Z$ normal. Let $D$ be a Weil divisor on $Z$. I am reading a paper that states $$\mathscr{H}om(f^*\mathcal{O}_Z(-D),\mathcal{O}_{\tilde{Z}})=f^*\mathcal{O}_Z(D)/torsion$$
Is this statement true in the generality I have suggested? How does one see this?
EDIT: Here are some ideas: Using $\mathcal{O}_Z(D)=\mathscr{H}om(\mathcal{O}_Z(-D),\mathcal{O}_{Z})$ (not sure if this is generally true with the hypotheses above, but I believe it is true in the situation I am concerned with) and adjunction of pullback and pushforward, and $f_*\mathcal{O}_{\tilde{Z}}=\mathcal{O}_Z$, it would be sufficient to show $(f^* f_* S)/torsion=S$ when $S$ is rank $1$ reflexive. By adjunction of pullback and pushforward there is a natural map $f^* f_* S \to S$ corresponding to the identity $f_* S \to f_* S$. Now I would need to prove this is surjective and it's kernel equal to the torsion. The latter part may use reflexitivity since torsion is the kernel of the map to a sheaf's double dual (but I'm not clear on how to use this).