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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