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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