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.
$\square$

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