Lemma 67.49.4. Let S be a scheme. Let f : X \to Y be a morphism of algebraic spaces over S. Assume f is flat and locally of finite type and assume Y satisfies the equivalent conditions of Lemma 67.49.1. Then X satisfies the equivalent conditions of Lemma 67.49.1 and for x \in |X| we have: x has codimension 0 in X \Rightarrow f(x) has codimension 0 in Y.
Proof. The last statement follows from Lemma 67.49.3. Choose a surjective étale morphism V \to Y where V is a scheme. Choose a surjective étale morphism U \to X \times _ Y V where U is a scheme. It suffices to show that every quasi-compact open of U has finitely many irreducible components. We will use the results of Properties of Spaces, Lemma 66.11.1 without further mention. By what we've already shown, the codimension 0 points of U lie above codimension 0 points in U and these are locally finite by assumption. Hence it suffices to show that for v \in V of codimension 0 the codimension 0 points of the scheme theoretic fibre U_ v = U \times _ V v are locally finite. This is true because U_ v is a scheme locally of finite type over \kappa (v), hence locally Noetherian and we can apply Lemma 67.49.2 for example. \square
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