The Stacks project

Lemma 37.16.8. Let $f : X \to S$ be a morphism of schemes which is locally of finite presentation. Let $\mathcal{F}$ be a finitely presented $\mathcal{O}_ X$-module flat over $S$. Then the set

\[ \{ x \in X : \mathcal{F}\text{ free in a neighbourhood of }x\} \]

is open in $X$ and its formation commutes with arbitrary base change $S' \to S$.

Proof. Openness holds trivially. Let $x \in X$ mapping to $s \in S$. By Lemma 37.16.7 we see that $x$ is in our set if and only if $\mathcal{F}|_{X_ s}$ is flat at $x$ over $X_ s$. Clearly this is also equivalent to $\mathcal{F}$ being flat at $x$ over $X$ (because this statement is implied by freeness of $\mathcal{F}_ x$ and implies flatness of $\mathcal{F}|_{X_ s}$ at $x$ over $X_ s$). Thus the base change statement follows from Lemma 37.16.5 applied to $\text{id} : X \to X$ over $S$. $\square$


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