Suppose $\pi : X \to Y$ satisfies that pullback on quasi-coherent sheaves is exact, how do I prove that $\pi$ is flat via the local definition; i.e stalkwise $O_{X,p}$ is a flat $O_{Y,q}$ module whenever $p \to q$?

I managed to prove the other direction (which is usually the harder one).

For this direction, what I want to say is that if $Spec(A)$ is openly embedded in $X$ and goes to $Spec(B)$ which is openly embedded in $Y$ then the pullback functor on those is also exact, but this requires being able to extend quasicoherent sheafs on $Spec(B)$ to all of $Y$.

This is sometimes possible; pushforward does the job when $Y$ is separated: then the map is affine.

Is there a general solution? I don't mind assuming stuff are separated, only that I'm missing softer easier arguments that generate quasicoherent sheafs.

As a user who deleted their answer commented- flatness is just the functor of pullback over ALL SHEAVES (not just quasicoherent) is exact right? It kills me I can't find this online

  • $\begingroup$ One possible idea: see if you can reduce exactness for general sheaves of $\mathcal{O}_X$-modules to the quasicoherent case. After all, you're looking at when some map of sheaves is/isn't injective, and you can test this one section at a time. (No guarantee that this works or that there won't be more to pay attention to, but it's rolling around my head and the only way for me to get rid of it is to type it here.) $\endgroup$
    – KReiser
    Jun 29 '21 at 7:11

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