Lemma 79.12.3. Let S be a scheme. Let f : X \to Y be a morphism of algebraic spaces over S which is locally of finite type. Let X' \subset X be the maximal open subspace over which f is locally quasi-finite, see Morphisms of Spaces, Lemma 67.34.7. Then (X/Y)_{fin} = (X'/Y)_{fin}.
Proof. Lemma 79.12.2 gives us an injective map (X'/Y)_{fin} \to (X/Y)_{fin}. Morphisms of Spaces, Lemma 67.34.7 assures us that formation of X' commutes with base change. Hence everything comes down to proving that if Z \subset X is an open subspace such that f|_ Z : Z \to Y is finite, then Z \subset X'. This is true because a finite morphism is locally quasi-finite, see Morphisms of Spaces, Lemma 67.45.8. \square
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