Theorem 41.10.1. Let $Y$ be a locally Noetherian scheme. Let $f : X \to Y$ be a morphism which is locally of finite type. Let $\mathcal{F}$ be a coherent $\mathcal{O}_ X$-module. The set of points in $X$ where $\mathcal{F}$ is flat over $Y$ is an open set. In particular the set of points where $f$ is flat is open in $X$.

**Proof.**
See More on Morphisms, Theorem 37.15.1.
$\square$

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