## Relative generic flatness.

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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?

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## 1 Answer

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 $\mathcal{F}=\mathcal{O}_X$).

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I know that a map $f:X \to Y$ is flat in an open subset $U \subset X$. I just do not undestand why it implies what I need.  Do you claim that $\\{x\in X| f|_{h ^{-1}(h(x))} : h ^{-1}(h(x)) \to g ^{-1}(h(x)) \text{ is flat at } x \\} =$ $= \\{ x \in X| f \text{ is flat at } x \\}$ ? Why?  – Rami Feb 9 2012 at 21:34
I think I got it. EGA IV (11.3.10)  says that  $\\{x\in X| f|_{h ^{-1}(h(x))} : h ^{-1}(h(x)) \to g ^{-1}(h(x)) \text{ is flat at } x \\} = \\{ x \in X| f \text{ is flat at } x \\}$.   and EGA IV (11.3.1) says that the later is open.  Please correct me if I'm wrong.  Thank you very much.  – Rami Feb 10 2012 at 2:45
That's right. The "open" bit is also stated at the end of (11.3.10). – Laurent Moret-Bailly Feb 10 2012 at 8:39
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