The Stacks project

Theorem 76.23.3. 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, and

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

Let $x \in |X|$ and let $y \in |Y|$ and $z \in |Z|$ be the images of $x$. If $\mathcal{F}_{\overline{x}} \not= 0$, then the following are equivalent:

  1. $\mathcal{F}$ is flat over $Z$ at $x$ and the restriction of $\mathcal{F}$ to its fibre over $z$ is flat at $x$ over the fibre of $Y$ over $z$, and

  2. $Y$ is flat over $Z$ at $y$ and $\mathcal{F}$ is flat over $Y$ at $x$.

Moreover, the set of points $x$ where (1) and (2) hold is open in $\text{Supp}(\mathcal{F})$.

Proof. Choose a diagram as in Lemma 76.23.1 part (3). 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 $u \in U$. Thus the theorem follows from the corresponding result for schemes which is More on Morphisms, Theorem 37.16.2. $\square$

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