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YOURS
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When will the pushforward of a structure sheaf is still 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?

YOURS
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