The Stacks project

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

  1. 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

  2. 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.

Proof. Let $x \in X$ be a point over $s \in S$. Then the dimension of the closure of $\{ x\} $ in $X_ s$ is $\text{trdeg}_{\kappa (s)}(\kappa (x))$ by Varieties, Lemma 33.20.3. Conversely, if $Z \subset X_ s$ is a closed subset of dimension $d$, then there exists a point $x \in Z$ with $\text{trdeg}_{\kappa (s)}(\kappa (x)) = d$ (same reference). Therefore the equivalence of (1) and (2) holds (even fibre by fibre). The statement on base change follows from Morphisms, Lemmas 29.25.7 and 29.28.3. $\square$

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