# Can relative flatness of a sheaf be tested using (faithfully) flat morphisms?

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 $\pi^{*}\pi_{*}G\cong G$

• No, you cannot drop the "faithfully." For example, if $X$ is the empty scheme, it is flat, but it detects nothing. The empty scheme is a good heuristic for testing whether faithfulness is relevant. In fact, given the characterization of faithfully flat morphisms among flat morphisms as the surjective ones (ie, the ones with nonempty fibers), it is the whole story. – Ben Wieland Sep 28 '15 at 18:48

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).