The Stacks project

Lemma 66.9.1. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. In each of the following cases $X$ is a scheme:

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

  4. add more here.

Proof. 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 60.9.2 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 27.10.5). 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 25.6.8). The étale morphisms $s, t : R \to U$ induce finite residue field extensions. Hence $s$ and $t$ are finite by Algebra, Lemma 10.53.4 (small detail omitted). Thus Groupoids, Proposition 38.23.9 shows that $X = U/R$ is an affine scheme.

Proof of (2) – almost identical to the proof of (4). 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 60.9.2 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 61.23.2 and 61.39.9. Similarly, $R \to \mathop{\mathrm{Spec}}(k)$ is locally of finite type. Hence by Varieties, Lemma 32.20.2 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 25.6.8. Applying Varieties, Lemma 32.20.2 once more we see that $R$ is finite over $k$. Hence $s, t$ are finite, see Morphisms, Lemma 28.42.14. Thus Groupoids, Proposition 38.23.9 shows that the open subspace $U/R$ of $X$ is an affine scheme. Since the schematic locus of $X$ is an open subspace (see Properties of Spaces, Lemma 60.13.1), and since $U \to X$ was an arbitrary étale morphism from an affine scheme we conclude that $X$ is a scheme.

Proof of (1). By Cohomology of Spaces, Lemma 63.10.1 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 63.16.7. $\square$


Comments (2)

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.


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