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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