Lemma 76.49.4. Consider a commutative diagram
of algebraic spaces. Assume that
$p$ is locally of finite presentation,
$p$ is flat,
$p$ is closed, and
$q$ is locally of finite type.
Then there exists an open subspace $W \subset Z$ such that a morphism $Z' \to Z$ factors through $W$ if and only if the base change $f_{Z'} : X_{Z'} \to Y_{Z'}$ is flat.
Proof. By Lemma 76.23.6 the set
is open in $|X|$ and its formation commutes with arbitrary base change. Let $W \subset Z$ be the open subspace (see Properties of Spaces, Lemma 66.4.8) with underlying set of points
i.e., $z \in |Z|$ is a point of $W$ if and only if the whole fibre of $|X| \to |Z|$ over $z$ is contained in $A$. This is open because $p$ is closed. Since the formation of $A$ commutes with arbitrary base change it follows that $W$ works. $\square$
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