It is known that any morphism is flat at an open set of points. I'd like to know if there is a relative version of this fact. 


Let $f: X \rightarrow Y$ and $g:Y \rightarrow S$ be morphisms of varieties, and let $h=g \circ f$ be their composition. Suppose that $h$ and $g$ are flat. Is it true that the set 
$\{x\in X| f|\_{h ^{-1}(h(x))} : h ^{-1}(h(x)) \to g ^{-1}(h(x)) \text{ is flat at } x \}$
is open?



Relative generic flatness.
With these assumptions, the set in question is the set of points in $X$ where $f$ is flat; hence it is open. This is "flatness by fibers", EGA IV (11.3.10) (applied with $\mathcal{F}=\mathcal{O}\_X$).


