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