The Stacks project

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

  1. $f$ is locally of finite presentation, or

  2. $V \to Y$ 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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