The Stacks project

Lemma 66.4.2. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $x \in |X|$. The following are equivalent:

  1. there exists a family of schemes $U_ i$ and étale morphisms $\varphi _ i : U_ i \to X$ such that $\coprod \varphi _ i : \coprod U_ i \to X$ is surjective, and such that for each $i$ the fibre of $|U_ i| \to |X|$ over $x$ is finite, and

  2. for every affine scheme $U$ and étale morphism $\varphi : U \to X$ the fibre of $|U| \to |X|$ over $x$ is finite.

Proof. The implication (2) $\Rightarrow $ (1) is trivial. Let $\varphi _ i : U_ i \to X$ be a family of étale morphisms as in (1). Let $\varphi : U \to X$ be an étale morphism from an affine scheme towards $X$. Consider the fibre product diagrams

\[ \xymatrix{ U \times _ X U_ i \ar[r]_-{p_ i} \ar[d]_{q_ i} & U_ i \ar[d]^{\varphi _ i} \\ U \ar[r]^\varphi & X } \quad \quad \xymatrix{ \coprod U \times _ X U_ i \ar[r]_-{\coprod p_ i} \ar[d]_{\coprod q_ i} & \coprod U_ i \ar[d]^{\coprod \varphi _ i} \\ U \ar[r]^\varphi & X } \]

Since $q_ i$ is étale it is open (see Remark 66.4.1). Moreover, the morphism $\coprod q_ i$ is surjective. Hence there exist finitely many indices $i_1, \ldots , i_ n$ and a quasi-compact opens $W_{i_ j} \subset U \times _ X U_{i_ j}$ which surject onto $U$. The morphism $p_ i$ is étale, hence locally quasi-finite (see remark on étale morphisms above). Thus we may apply Morphisms, Lemma 29.55.10 to see the fibres of $p_{i_ j}|_{W_{i_ j}} : W_{i_ j} \to U_ i$ are finite. Hence by Properties of Spaces, Lemma 64.4.3 and the assumption on $\varphi _ i$ we conclude that the fibre of $\varphi $ over $x$ is finite. In other words (2) holds. $\square$


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