Lemma 66.31.4. Let $S$ be a scheme. Let $X \to Y \to Z$ be morphisms of algebraic spaces over $S$. Let $\mathcal{F}$ be a quasi-coherent sheaf on $X$. Let $x \in |X|$ with image $y \in |Y|$.

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

Let $x : \mathop{\mathrm{Spec}}(K) \to X$ be a representative of $x$. If

$\mathcal{F}$ is flat at $x$ over $Y$,

$x^*\mathcal{F} \not= 0$, and

$\mathcal{F}$ is flat at $x$ over $Z$,

then $Y$ is flat at $y$ over $Z$.

Let $\overline{x}$ be a geometric point of $X$ lying over $x$ with image $\overline{y}$ in $Y$. If $\mathcal{F}_{\overline{x}}$ is a faithfully flat $\mathcal{O}_{Y, \overline{y}}$-module and $\mathcal{F}$ is flat at $x$ over $Z$, then $Y$ is flat at $y$ over $Z$.

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