The Stacks project

Lemma 73.49.4. Consider a commutative diagram

\[ \xymatrix{ X \ar[rr]_ f \ar[rd]_ p & & Y \ar[ld]^ q \\ & Z } \]

of algebraic spaces. Assume that

  1. $p$ is locally of finite presentation,

  2. $p$ is flat,

  3. $p$ is closed, and

  4. $q$ 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 73.23.6 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 63.4.8) 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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