Lemma 68.15.2 (David Rydh). A locally separated algebraic space is decent.
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 66.3.1. 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
is an immersion (Morphisms of Spaces, Lemma 67.4.5). Hence More on Groupoids, Lemma 40.11.5 tells us that it is a closed immersion (use Schemes, Lemma 26.10.4). 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 68.4.1 and Schemes, Lemma 26.23.11). 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 68.4.3.
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 68.15.1 we conclude that I is finite as desired. \square
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