This arises from lemma 7.11 and remark 7.12 from the book ‘Higher-Dimensional Algebraic Geometry’ by Debarre. Essentially, lemma 7.11 states the following:
“Let $f: Y\rightarrow X$ be a proper birational morphism between smooth varieties. Then for any effective divisor $E$ on $Y$ contained in the exceptional locus of $f$, one has $f_*\mathcal{O}(Y)(E) \cong \mathcal{O}_X$.”
Now suppose $F$ is any exceptional divisor of $f$. We write $F = F_1 -F_2$ as a difference of effective divisors, where $F_1$ and $F_2$ have no common components. Certainly, $f_*\mathcal{O}_Y(F) = f_*\mathcal{O}_Y(F_1-F_2)$ injects into the sheaf $f_*\mathcal{O}_Y(F_1) \cong \mathcal{O}_X$.
The author then states that ‘it is therefore a sheaf of ideals that defines a subscheme of $X$ supported on $f(F_2)$. It follows $f_*\mathcal{O}_Y(F) \cong \mathcal{O}_X$ iff $F$ is effective.’
My understanding is that if $f_* \mathcal{O}_Y(F) \cong \mathcal{O}_X$. Then $F_2$ must be zero necessarily. How do we conclude that in this case $F_2 =0$? Any help given would be greatly appreciated!
If $f_*\mathcal{O}_Y(F) \cong \mathcal{O}_X$ then by pullback-pushforward adjunction there is a morphism $$ \mathcal{O}_Y \cong f^*\mathcal{O}_X \to \mathcal{O}_Y(F). $$ Moreover, on the locus where $f$ is an isomorphism, this morphism is the identity. This shows that $F$ is an effective divisor supported on the exceptional locus of $f$.