Lemma 76.49.4. Consider a commutative diagram
\xymatrix{ X \ar[rr]_ f \ar[rd]_ p & & Y \ar[ld]^ q \\ & Z }
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
A = \{ x \in |X| : X\text{ flat at }x \text{ over }Y\} .
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
|W| = |Z| \setminus |p|\left(|X| \setminus A\right)
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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