The Stacks project

Lemma 76.23.8. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$ which is locally of finite presentation. Let $\mathcal{F}$ be a finitely presented $\mathcal{O}_ X$-module flat over $Y$. Then the set

\[ \{ x \in |X| : \mathcal{F}\text{ free in an étale neighbourhood of }x\} \]

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

Proof. Openness holds trivially. Let $Y' \to Y$ be a morphism of algebraic spaces, set $X' = Y' \times _ Y X$, and let $x' \in |X'|$ be a point lying over $x \in |X|$. By Lemma 76.23.7 we see that $x$ is in our set if and only if $(\mathcal{F}_{\overline{y}})_{\overline{x}}$ is a flat $\mathcal{O}_{X_{\overline{y}}, \overline{x}}$-module. Simiarly, $x'$ is in the analogue of our set for the pullback $\mathcal{F}'$ of $\mathcal{F}$ to $X'$ if and only if $(\mathcal{F}'_{\overline{y}'})_{\overline{x}'}$ is a flat $\mathcal{O}_{X'_{\overline{y}'}, \overline{x}'}$-module (with obvious notation). These two assertions are equivalent by Lemma 76.23.1 applied to the morphism $\text{id} : X \to X$ over $Y$. Thus the statement on base change holds. $\square$

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