In this section we collect a number of results of the form: If $f : X \to Y$ is a flat morphism of algebraic spaces and $f$ satisfies some property over a dense open of $Y$, then $f$ satisfies the same property over all of $Y$.
Lemma 76.37.1. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module. Let $V \subset Y$ be an open subspace. Assume
$f$ is locally of finite presentation,
$\mathcal{F}$ is of finite type and flat over $Y$,
$V \to Y$ is quasi-compact and scheme theoretically dense,
$\mathcal{F}|_{f^{-1}V}$ is of finite presentation.
Then $\mathcal{F}$ is of finite presentation.
Proof. It suffices to prove the pullback of $\mathcal{F}$ to a scheme surjective and étale over $X$ is of finite presentation. Hence we may assume $X$ is a scheme. 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). Thus we reduce to the case of schemes which is More on Flatness, Lemma 38.11.1. $\square$
Lemma 76.37.2. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Let $V \subset Y$ be an open subspace. Assume
$f$ is locally of finite type and flat,
$V \to Y$ is quasi-compact and scheme theoretically dense,
$f|_{f^{-1}V} : f^{-1}V \to V$ is locally of finite presentation.
Then $f$ is of locally of finite presentation.
Proof. The proof is identical to the proof of Lemma 76.37.1 except one uses More on Flatness, Lemma 38.11.2. $\square$
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. $\square$
Lemma 76.37.4. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$ which is flat and proper. Let $V \to Y$ be an open subspace with $|V| \subset |Y|$ dense such that $X_ V \to V$ is finite. If also either $f$ is locally of finite presentation or $V \to Y$ is quasi-compact, then $f$ is finite.
Proof. By Lemma 76.37.3 the fibres of $f$ have dimension zero. By Morphisms of Spaces, Lemma 67.34.6 this implies that $f$ is locally quasi-finite. By Morphisms of Spaces, Lemma 67.51.1 this implies that $f$ is representable. We can check whether $f$ is finite étale locally on $Y$, hence we may assume $Y$ is a scheme. Since $f$ is representable, we reduce to the case of schemes which is More on Flatness, Lemma 38.11.4. $\square$
Lemma 76.37.5. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Let $V \subset Y$ be an open subspace. If
$f$ is separated, locally of finite type, and flat,
$f^{-1}(V) \to V$ is an isomorphism, and
$V \to Y$ is quasi-compact and scheme theoretically dense,
then $f$ is an open immersion.
Proof. Applying Lemma 76.37.2 we see that $f$ is locally of finite presentation. Applying Lemma 76.37.3 we see that $f$ has relative dimension $\leq 0$. By Morphisms of Spaces, Lemma 67.34.6 this implies that $f$ is locally quasi-finite. By Morphisms of Spaces, Lemma 67.51.1 this implies that $f$ is representable. By Descent on Spaces, Lemma 74.11.14 we can check whether $f$ is an open immersion étale locally on $Y$. Hence we may assume that $Y$ is a scheme. Since $f$ is representable, we reduce to the case of schemes which is More on Flatness, Lemma 38.11.5. $\square$
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