Lemma 38.20.9. Let $f : X \to S$ be a morphism of schemes which is locally of finite type. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module of finite type. Let $n \geq 0$. The following are equivalent

for $s \in S$ the closed subset $Z \subset X_ s$ of points where $\mathcal{F}$ is not flat over $S$ (see Lemma 38.10.4) satisfies $\dim (Z) < n$, and

for $x \in X$ such that $\mathcal{F}$ is not flat at $x$ over $S$ we have $\text{trdeg}_{\kappa (f(x))}(\kappa (x)) < n$.

If this is true, then it remains true after any base change.

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