Lemma 65.10.3. Let S be a scheme. Let U be a scheme over S. Let j = (s, t) : R \to U \times _ S U be an étale equivalence relation on U over S. If the quotient U/R is an algebraic space, then U \to U/R is étale and surjective. Hence (U, R, U \to U/R) is a presentation of the algebraic space U/R.
Proof. Denote c : U \to U/R the morphism in question. Let T be a scheme and let a : T \to U/R be a morphism. We have to show that the morphism (of schemes) \pi : T \times _{a, U/R, c} U \to T is étale and surjective. The morphism a corresponds to an fppf covering \{ \varphi _ i : T_ i \to T\} and morphisms a_ i : T_ i \to U such that a_ i \times a_{i'} : T_ i \times _ T T_{i'} \to U \times _ S U factors through R, and such that c \circ a_ i = a \circ \varphi _ i. Hence
T_ i \times _{\varphi _ i, T} T \times _{a, U/R, c} U = T_ i \times _{c \circ a_ i, U/R, c} U = T_ i \times _{a_ i, U} U \times _{c, U/R, c} U = T_ i \times _{a_ i, U, t} R.
Since t is étale and surjective we conclude that the base change of \pi to T_ i is surjective and étale. Since the property of being surjective and étale is local on the base in the fpqc topology (see Remark 65.4.3) we win. \square
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