Given a $\mathbb{C}$-scheme $S$, two $S$-schemes $X$ and $Y$ that are flat over $S$ and a coherent sheaf of $O\_Y$-modules $F$.


Assume we have a (faithfully) flat $S$-morphism $\pi: X \rightarrow Y$ and we know that $\pi^{\*}F$ is a coherent sheaf of $O\_X$-modules that is flat over $S$.


Can we conclude that $F$ is flat over $S$?


Can we drop the faithfully? Or is this wrong in this generality, do we need to assume more, to get such a result?


My situation is the follwing $X=A\times S$, $Y=B\times S$, for two smooth projective schemes $A$ and $B$ and $F$ is of the form $\pi\_{\*}G$ for a coherent sheaf $G$ on $X$ flat over $S$, with the property $\pi^{\*}\pi\_{\*}G\cong G$ 



Can relative flatness of a sheaf be tested using (faithfully) flat morphisms?
You can assume less: $\pi$ faithfully flat, $F$ quasi-coherent, no assumptions on structure morphisms. Write down the functors and see immediately their how faithfulness and exactness depend on each other's.


If you need a reference, EGA IV$\_2$ 2.2.11 (iii). 


