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Let $f\colon X\to Y$ be a morphism of complex analytic spaces (though I'm very happy to restrict to complex manifolds).

Theorem (Grauert). The pushforward $f_*\colon\mathcal{O}_X\text{-mod}\to\mathcal{O}_Y\text{-mod}$ preserves coherence if $f$ is proper.

Question(s).

  1. What conditions can we place on $f$ to ensure that the pullback $f^*\colon\mathcal{O}_Y\text{-mod}\to\mathcal{O}_X\text{-mod}$ preserves coherence?
  2. Can we say anything about when such sufficient conditions are also necessary?
  3. (Bonus: Can we also answer question 2 for the case of pushforward/proper?)

(I was sure that this question would already be on MO, and I did search but couldn't find anything)

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This is always true. By Oka's coherence theorem, $\mathcal O_X$ and $\mathcal O_Y$ are coherent. Therefore, a quasi-coherent sheaf $\mathscr F$ is coherent if and only if it is of finite presentation [Tag 01BZ]. For any morphism of ringed spaces, pullback preserves quasi-coherence [Tag 01BG] and finite presentation [Tag 01BQ].

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    $\begingroup$ of course! Tag [01BZ] was exactly the piece of information that I was looking for, thank you :-) $\endgroup$
    – Tim
    Commented 13 hours ago

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