Lemma 71.9.1. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Assume $X$ satisfies at least one of the following conditions

1. $X$ is quasi-separated and $\dim (X) = 0$,

2. $X$ is locally of finite type over a field $k$ and $\dim (X) = 0$,

3. $X$ is Noetherian and $\dim (X) = 0$, or

Then $X$ is a separated scheme and any quasi-compact open of $X$ is affine.

Proof. If we prove that any quasi-compact open of $X$ is affine, then $X$ is a separated scheme. Thus we may assume $X$ is quasi-compact and we aim to show that $X$ is affine. Cases (2) and (3) follow immediately from case (1) but we will give a separate proofs of (2) and (3) as these proofs use significantly less theory.

Proof of (3). Let $U$ be an affine scheme and let $U \to X$ be an étale morphism. Set $R = U \times _ X U$. The two projection morphisms $s, t : R \to U$ are étale morphisms of schemes. By Properties of Spaces, Definition 65.9.2 we see that $\dim (U) = 0$ and $\dim (R) = 0$. Since $R$ is a locally Noetherian scheme of dimension $0$, we see that $R$ is a disjoint union of spectra of Artinian local rings (Properties, Lemma 28.10.5). Since we assumed that $X$ is Noetherian (so quasi-separated) we conclude that $R$ is quasi-compact. Hence $R$ is an affine scheme (use Schemes, Lemma 26.6.8). The étale morphisms $s, t : R \to U$ induce finite residue field extensions. Hence $s$ and $t$ are finite by Algebra, Lemma 10.54.4 (small detail omitted). Thus Groupoids, Proposition 39.23.9 shows that $X = U/R$ is an affine scheme.

Proof of (2) – almost identical to the proof of (3). Let $U$ be an affine scheme and let $U \to X$ be a surjective étale morphism. Set $R = U \times _ X U$. The two projection morphisms $s, t : R \to U$ are étale morphisms of schemes. By Properties of Spaces, Definition 65.9.2 we see that $\dim (U) = 0$ and similarly $\dim (R) = 0$. On the other hand, the morphism $U \to \mathop{\mathrm{Spec}}(k)$ is locally of finite type as the composition of the étale morphism $U \to X$ and $X \to \mathop{\mathrm{Spec}}(k)$, see Morphisms of Spaces, Lemmas 66.23.2 and 66.39.9. Similarly, $R \to \mathop{\mathrm{Spec}}(k)$ is locally of finite type. Hence by Varieties, Lemma 33.20.2 we see that $U$ and $R$ are disjoint unions of spectra of local Artinian $k$-algebras finite over $k$. The same thing is therefore true of $U \times _{\mathop{\mathrm{Spec}}(k)} U$. As

$R = U \times _ X U \longrightarrow U \times _{\mathop{\mathrm{Spec}}(k)} U$

is a monomorphism, we see that $R$ is a finite(!) union of spectra of finite $k$-algebras. It follows that $R$ is affine, see Schemes, Lemma 26.6.8. Applying Varieties, Lemma 33.20.2 once more we see that $R$ is finite over $k$. Hence $s, t$ are finite, see Morphisms, Lemma 29.44.14. Thus Groupoids, Proposition 39.23.9 shows that $X = U/R$ is an affine scheme.

Cohomological proof of (1). By Cohomology of Spaces, Lemma 68.10.1 we have vanishing of higher cohomology groups for all quasi-coherent sheaves $\mathcal{F}$ on $X$. Hence $X$ is affine (in particular a scheme) by Cohomology of Spaces, Proposition 68.16.7.

Geometric proof of (1). Choose a stratification

$\emptyset = U_{n + 1} \subset U_ n \subset U_{n - 1} \subset \ldots \subset U_1 = X$

and étale morphisms $f_ p : V_ p \to U_ p$ as in Decent Spaces, Lemma 67.8.6 (we will use all their properties below). Then $\dim (V_ p) = 0$ by our definition of dimension of algebraic spaces. Thus Properties, Lemma 28.10.6 applies to each $V_ p$. Then $f_ p^{-1}(U_{p + 1}) \subset V_ p$ is quasi-compact open and hence is affine as well as closed. It follows that $|T_ p| \subset |U_ p|$ (see locus citatus) is open as well as closed. Hence $X$ is a disjoint union of open and closed subspaces whose reduced structures are schemes. It follows that $X$ is a scheme (Limits of Spaces, Lemma 69.15.3). Then the proof is finished by the case of schemes that we already referenced above. $\square$

Comment #261 by on

In the last sentence, arbtrary should be arbitrary. And morphisms should be morphism, again in the last sentence.

By the way, throughout the Stacks project the capitalisation of artinian (or Artinian) is inconsistent.

Comment #7005 by Laurent Moret-Bailly on

Of course, in (1) quasi-compactness is not needed. Also, in all these cases $X$ is separated, and affine if it is quasi-compact.

Comment #7343 by Laurent Moret-Bailly on

I think you get a non-cohomological proof by using the stratification result Tag 07ST, the case of schemes (Tag 0CKV), and an obvious induction.

Comment #7394 by on

@#7343: OK, I added a proof here. But in the future, can you just type in the arguments in latex in the comment box (or email them to me)?

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