The Stacks project

Theorem 77.4.5. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module. Assume

  1. $X \to Y$ is locally of finite presentation,

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

  3. the set of weakly associated points of $Y$ is locally finite in $Y$.

Then $U = \{ x \in |X| : \mathcal{F}\text{ flat at }x\text{ over }Y\} $ is open in $X$ and $\mathcal{F}|_ U$ is an $\mathcal{O}_ U$-module of finite presentation and flat over $Y$.

Proof. Condition (3) means that if $V \to Y$ is a surjective ├ętale morphism where $V$ is a scheme, then the weakly associated points of $V$ are locally finite on the scheme $V$. (Recall that the weakly associated points of $V$ are exactly the inverse image of the weakly associated points of $Y$ by Divisors on Spaces, Definition 71.2.2.) Having said this the question is ├ętale local on $X$ and $Y$, hence we may assume $X$ and $Y$ are schemes. Thus the result follows from More on Flatness, Theorem 38.13.6. $\square$

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