Lemma 76.23.4. Let $S$ be a scheme. Let $f : X \to Y$ and $Y \to Z$ be a morphism of algebraic spaces over $S$. Assume

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

$X$ is flat over $Z$,

for every $z \in |Z|$ the fibre of $X$ over $z$ is flat over the fibre of $Y$ over $z$, and

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

Then $f$ is flat. If $f$ is also surjective, then $Y$ is flat over $Z$.

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