The Stacks project

Lemma 65.6.7. 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 ├ętale 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

\[ |U| \to |X| \quad \text{and}\quad |R| \to |X| \]

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 65.4.3, and the fibre of $|R| \to |X|$ over $x$ is $t^{-1}(s(t^{-1}(\{ u\} )))$. Since $t : R \to U$ is ├ętale 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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