Lemma 76.37.3. Let S be a scheme. Let f : X \to Y be a morphism of algebraic spaces over S which is flat and locally of finite type. Let V \subset Y be an open subspace such that |V| \subset |Y| is dense and such that X_ V \to V has relative dimension \leq d. If also either
f is locally of finite presentation, or
V \to Y is quasi-compact,
then f : X \to Y has relative dimension \leq d.
Proof. We may replace Y by its reduction, hence we may assume Y is reduced. Then V is scheme theoretically dense in Y, see Morphisms of Spaces, Lemma 67.17.7. By definition the property of having relative dimension \leq d can be checked on an étale covering, see Morphisms of Spaces, Sections 67.33. Thus it suffices to prove f has relative dimension \leq d after replacing X by a scheme surjective and étale over X. Similarly, we can replace Y by a scheme surjective and étale and over Y. The inverse image of V in this scheme is scheme theoretically dense, see Morphisms of Spaces, Section 67.17. Since a scheme theoretically dense open of a scheme is in particular dense, we reduce to the case of schemes which is More on Flatness, Lemma 38.11.3. \square
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