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

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

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

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 $Y$.

## Comments (0)