Lemma 66.6.7. Let S be a scheme. Let X be an algebraic space over S. Let U be a scheme. Let \varphi : U \to X be an étale morphism such that the projections R = U \times _ X U \to U are quasi-compact; for example if \varphi is quasi-compact. Then the fibres of
are finite.
Proof. Denote R = U \times _ X U, and s, t : R \to U the projections. Let u \in U be a point, and let x \in |X| be its image. The fibre of |U| \to |X| over x is equal to s(t^{-1}(\{ u\} )) by Lemma 66.4.3, and the fibre of |R| \to |X| over x is t^{-1}(s(t^{-1}(\{ u\} ))). Since t : R \to U is étale and quasi-compact, it has finite fibres (as its fibres are disjoint unions of spectra of fields by Morphisms, Lemma 29.36.7 and quasi-compact). Hence we win. \square
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