Lemma 68.4.2. 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 68.4.1). 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 29.57.9 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 66.4.3 and the assumption on $\varphi _ i$ we conclude that the fibre of $\varphi$ over $x$ is finite. In other words (2) holds. $\square$

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 03JS. Beware of the difference between the letter 'O' and the digit '0'.