Proposition 79.12.11. Let S be a scheme. Let f : X \to Y be a morphism of algebraic spaces over S which is separated and locally of finite type. Then (X/Y)_{fin} is an algebraic space. Moreover, the morphism (X/Y)_{fin} \to Y is étale.
Proof. By Lemma 79.12.3 we may replace X by the open subscheme which is locally quasi-finite over Y. Hence we may assume that f is separated and locally quasi-finite. We will check the three conditions of Spaces, Definition 65.6.1. Condition (1) follows from Lemma 79.12.1. Condition (2) follows from Lemma 79.12.7. Finally, condition (3) follows from Lemma 79.12.10. Thus (X/Y)_{fin} is an algebraic space. Moreover, that lemma shows that there exists a commutative diagram
\xymatrix{ U \ar[rr] \ar[rd] & & (X/Y)_{fin} \ar[ld] \\ & Y }
with horizontal arrow surjective and étale and south-east arrow étale. By Properties of Spaces, Lemma 66.16.3 this implies that the south-west arrow is étale as well. \square
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