The Stacks project

Proof. Let $S$ be a scheme and let $X$ be a locally separated algebraic space over $S$. We may assume $S = \mathop{\mathrm{Spec}}(\mathbf{Z})$, see Properties of Spaces, Definition 60.3.1. Unadorned fibre products will be over $\mathbf{Z}$. Let $x \in |X|$. Choose a scheme $U$, an étale morphism $U \to X$, and a point $u \in U$ mapping to $x$ in $|X|$. As usual we identify $u = \mathop{\mathrm{Spec}}(\kappa (u))$. As $X$ is locally separated the morphism

\[ u \times _ X u \to u \times u \]

is an immersion (Morphisms of Spaces, Lemma 61.4.5). Hence More on Groupoids, Lemma 39.11.5 tells us that it is a closed immersion (use Schemes, Lemma 25.10.4). As $u \times _ X u \to u \times _ X U$ is a monomorphism (base change of $u \to U$) and as $u \times _ X U \to u$ is étale we conclude that $u \times _ X u$ is a disjoint union of spectra of fields (see Remark 62.4.1 and Schemes, Lemma 25.23.11). Since it is also closed in the affine scheme $u \times u$ we conclude $u \times _ X u$ is a finite disjoint union of spectra of fields. Thus $x$ can be represented by a monomorphism $\mathop{\mathrm{Spec}}(k) \to X$ where $k$ is a field, see Lemma 62.4.3.

Next, let $U = \mathop{\mathrm{Spec}}(A)$ be an affine scheme and let $U \to X$ be an étale morphism. To finish the proof it suffices to show that $F = U \times _ X \mathop{\mathrm{Spec}}(k)$ is finite. Write $F = \coprod _{i \in I} \mathop{\mathrm{Spec}}(k_ i)$ as the disjoint union of finite separable extensions of $k$. We have to show that $I$ is finite. Set $R = U \times _ X U$. As $X$ is locally separated, the morphism $j : R \to U \times U$ is an immersion. Let $U' \subset U \times U$ be an open such that $j$ factors through a closed immersion $j' : R \to U'$. Let $e : U \to R$ be the diagonal map. Using that $e$ is a morphism between schemes étale over $U$ such that $\Delta = j \circ e$ is a closed immersion, we conclude that $R = e(U) \amalg W$ for some open and closed subscheme $W \subset R$. Since $j'$ is a closed immersion we conclude that $j'(W) \subset U'$ is closed and disjoint from $j'(e(U))$. Therefore $\overline{j(W)} \cap \Delta (U) = \emptyset $ in $U \times U$. Note that $W$ contains $\mathop{\mathrm{Spec}}(k_ i \otimes _ k k_{i'})$ for all $i \not= i'$, $i, i' \in I$. By Lemma 62.15.1 we conclude that $I$ is finite as desired. $\square$


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