The Stacks project

Lemma 76.23.5. Let $S$ be a scheme. Let $f : X \to Y$ and $Y \to Z$ be morphisms of algebraic spaces over $S$. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module. Assume

  1. $X$ is locally of finite presentation over $Z$,

  2. $\mathcal{F}$ an $\mathcal{O}_ X$-module of finite presentation,

  3. $\mathcal{F}$ is flat over $Z$, and

  4. $Y$ is locally of finite type over $Z$.

Then the set

\[ A = \{ x \in |X| : \mathcal{F} \text{ flat at }x \text{ over }Y\} . \]

is open in $|X|$ and its formation commutes with arbitrary base change: If $Z' \to Z$ is a morphism of algebraic spaces, and $A'$ is the set of points of $X' = X \times _ Z Z'$ where $\mathcal{F}' = \mathcal{F} \times _ Z Z'$ is flat over $Y' = Y \times _ Z Z'$, then $A'$ is the inverse image of $A$ under the continuous map $|X'| \to |X|$.

Proof. One way to prove this is to translate the proof as given in More on Morphisms, Lemma 37.16.4 into the category of algebraic spaces. Instead we will prove this by reducing to the case of schemes. Namely, choose a diagram as in Lemma 76.23.1 part (3) such that $a$, $b$, and $c$ are surjective. It follows from the definitions that this reduces to the corresponding theorem for the morphisms of schemes $U \to V \to W$, the quasi-coherent sheaf $a^*\mathcal{F}$, and the point $u \in U$. The only minor point to make is that given a morphism of algebraic spaces $Z' \to Z$ we choose a scheme $W'$ and a surjective étale morphism $W' \to W \times _ Z Z'$. Then we set $U' = W' \times _ W U$ and $V' = W' \times _ W V$. We write $a', b', c'$ for the morphisms from $U', V', W'$ to $X', Y', Z'$. In this case $A$, resp. $A'$ are images of the open subsets of $U$, resp. $U'$ associated to $a^*\mathcal{F}$, resp. $(a')^*\mathcal{F}'$. This indeed does reduce the lemma to More on Morphisms, Lemma 37.16.4. $\square$

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