Let $f:X\rightarrow Y$ be a morphism of schemes.

1. When $PicY\rightarrow PicX$ is an embedding , and $f_{*}\mathscr{O}_{X}$ is invertible, it is the structure sheaf of $Y$.

2. In the proof of Zariski's Main Theorem, we have: If $f$ is birational, finite, integral, and $Y$ is normal, then $f_{*}\mathscr{O}_{X}$ is the structure sheaf of $Y$.

My questions are

1) what What exactly prevent $f_{*}\mathscr{O}_{X}$ to be a structure sheaf?

2) Is there any necessary and sufficient condition(s) guarantee that $f_{*}\mathscr{O}_{X}$ is a structure sheaf?

Let $f:X\rightarrow Y$ be a morphism of schemes.

1. When $PicY\rightarrow PicX$ is an embedding, $f_{*}\mathscr{O}_{X}$ is the structure sheaf of $Y$.

2. In the proof of Zariski's Main Theorem, we have: If $f$ is birational, finite, integral, and $Y$ is normal, then $f_{*}\mathscr{O}_{X}$ is the structure sheaf of $Y$.

My questions are

1) what exactly prevent $f_{*}\mathscr{O}_{X}$ to be a structure sheaf?

2) Is there any necessary and sufficient condition(s) guarantee that $f_{*}\mathscr{O}_{X}$ is a structure sheaf?