The Stacks project

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

  1. $f$ is separated, locally of finite type, and flat,

  2. $f^{-1}(V) \to V$ is an isomorphism, and

  3. $V \to Y$ 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 ├ętale 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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