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Let $F:\mathscr A\to\mathscr B$ be a functor between abelian categories with enough injectives, and let us assume $F$ is exact.

Question 2. When does $F$ preserve injective objects?

The motivating example for me comes from this answer, where it is explained that if $i:U\to X$ is an open immersion, then $i^\ast$ preserves injectives (because $i^\ast$ has a left adjoint which is exact). So the first thing I was asking myself (before getting to the more general Question 2!) was:

Question 1. If $f:X\to Y$ is a flat map, does $f^\ast:\textrm{QCoh}_Y\to \textrm{QCoh}_X$ preserve injectives?

Thank you in advance!

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  • $\begingroup$ Are $X,Y$ arbitrary schemes? Because then (global) injectives in $\mathsf{Qcoh}(X)$ are not well-behaved, and should not be confused with injectives in $\mathsf{Mod}(X)$ which happen to be quasi-coherent. $\endgroup$ – Martin Brandenburg Sep 28 '15 at 22:10
  • $\begingroup$ I am also ok with $f$ a proper (flat) map of finite type $\mathbb C$-schemes. Not arbitrary at all... What do you mean by not well-behaved? $\endgroup$ – Brenin Sep 28 '15 at 22:15
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    $\begingroup$ cf. mathoverflow.net/questions/6762 and mathoverflow.net/questions/89398 and mathoverflow.net/questions/58323 . But everything is fine in the Noetherian case :). Have you checked the case that $X,Y$ are affine first? $\endgroup$ – Martin Brandenburg Sep 28 '15 at 22:17
  • $\begingroup$ Ah! Well, I was tacitly assuming Noetherian, at least to have QCoh with enough injectives (although I admit I don't know if one really needs Noetherian assumptions). I should think more, but I do not seem to have a clue on the affine case either. $\endgroup$ – Brenin Sep 28 '15 at 22:32

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