Lemma 71.10.7. Let $k$ be a field with algebraic closure $\overline{k}$. Let $X$ be an algebraic space over $k$ such that

1. $X$ is decent and locally of finite type over $k$,

2. $X_{\overline{k}}$ is a scheme, and

3. any finite set of $\overline{k}$-rational points of $X_{\overline{k}}$ are contained in an affine.

Then $X$ is a scheme.

Proof. If $k \subset K$ is an extension, then the base change $X_ K$ is decent (Decent Spaces, Lemma 67.6.5) and locally of finite type over $K$ (Morphisms of Spaces, Lemma 66.23.3). By Lemma 71.10.1 it suffices to prove that $X$ becomes a scheme after base change to the perfection of $k$, hence we may assume $k$ is a perfect field (this step isn't strictly necessary, but makes the other arguments easier to think about). By covering $X$ by quasi-compact opens we see that it suffices to prove the lemma in case $X$ is quasi-compact (small detail omitted). In this case $|X|$ is a sober topological space (Decent Spaces, Proposition 67.12.4). Hence it suffices to show that every closed point in $|X|$ is contained in the schematic locus of $X$ (use Properties of Spaces, Lemma 65.13.1 and Topology, Lemma 5.12.8).

Let $x \in |X|$ be a closed point. By Decent Spaces, Lemma 67.14.6 we can find a closed immersion $\mathop{\mathrm{Spec}}(l) \to X$ representing $x$. Then $\mathop{\mathrm{Spec}}(l) \to \mathop{\mathrm{Spec}}(k)$ is of finite type (Morphisms of Spaces, Lemma 66.23.2) and we conclude that $l$ is a finite extension of $k$ by the Hilbert Nullstellensatz (Algebra, Theorem 10.34.1). It is separable because $k$ is perfect. Thus the scheme

$\mathop{\mathrm{Spec}}(l) \times _ X X_{\overline{k}} = \mathop{\mathrm{Spec}}(l) \times _{\mathop{\mathrm{Spec}}(k)} \mathop{\mathrm{Spec}}(\overline{k}) = \mathop{\mathrm{Spec}}(l \otimes _ k \overline{k})$

is the disjoint union of a finite number of $\overline{k}$-rational points. By assumption (3) we can find an affine open $W \subset X_{\overline{k}}$ containing these points.

By Lemma 71.10.2 we see that $X_{k'}$ is a scheme for some finite extension $k'/k$. After enlarging $k'$ we may assume that there exists an affine open $U' \subset X_{k'}$ whose base change to $\overline{k}$ recovers $W$ (use that $X_{\overline{k}}$ is the limit of the schemes $X_{k''}$ for $k' \subset k'' \subset \overline{k}$ finite and use Limits, Lemmas 32.4.11 and 32.4.13). We may assume that $k'/k$ is a Galois extension (take the normal closure Fields, Lemma 9.16.3 and use that $k$ is perfect). Set $G = \text{Gal}(k'/k)$. By construction the $G$-invariant closed subscheme $\mathop{\mathrm{Spec}}(l) \times _ X X_{k'}$ is contained in $U'$. Thus $x$ is in the schematic locus by Lemmas 71.10.3 and 71.10.4. $\square$

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