The Stacks project

Lemma 68.4.6. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. The following are equivalent:

  1. there exist schemes $U_ i$ and étale morphisms $U_ i \to X$ such that $\coprod U_ i \to X$ is surjective and each $U_ i \to X$ has universally bounded fibres, and

  2. for every affine scheme $U$ and étale morphism $\varphi : U \to X$ the fibres of $U \to X$ are universally bounded.

Proof. The implication (2) $\Rightarrow $ (1) is trivial. Assume (1). Let $(\varphi _ i : U_ i \to X)_{i \in I}$ be a collection of étale morphisms from schemes towards $X$, covering $X$, such that each $\varphi _ i$ has universally bounded fibres. Let $\psi : U \to X$ be an étale morphism from an affine scheme towards $X$. For each $i$ consider the fibre product diagram

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

Since $q_ i$ is étale it is open (see Remark 68.4.1). Moreover, we have $U = \bigcup \mathop{\mathrm{Im}}(q_ i)$, since the family $(\varphi _ i)_{i \in I}$ is surjective. Since $U$ is affine, hence quasi-compact we can finite finitely many $i_1, \ldots , i_ n \in I$ and quasi-compact opens $W_ j \subset U \times _ X U_{i_ j}$ such that $U = \bigcup p_{i_ j}(W_ j)$. The morphism $p_{i_ j}$ 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 $p_{i_ j}|_{W_ j} : W_ j \to U_{i_ j}$ are universally bounded. Hence by Lemma 68.3.2 we see that the fibres of $W_ j \to X$ are universally bounded. Thus also $\coprod _{j = 1, \ldots , n} W_ j \to X$ has universally bounded fibres. Since $\coprod _{j = 1, \ldots , n} W_ j \to X$ factors through the surjective étale map $\coprod q_{i_ j}|_{W_ j} : \coprod _{j = 1, \ldots , n} W_ j \to U$ we see that the fibres of $U \to X$ are universally bounded by Lemma 68.3.5. In other words (2) holds. $\square$

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