The Stacks project

Lemma 66.20.12. Let $S$ be a scheme. Let $X$ be a quasi-separated algebraic space over $S$. Let $A$ be an Artinian ring. Any morphism $\mathop{\mathrm{Spec}}(A) \to X$ is affine.

Proof. Let $U \to X$ be an ├ętale morphism with $U$ affine. To prove the lemma we have to show that $\mathop{\mathrm{Spec}}(A) \times _ X U$ is affine, see Lemma 66.20.3. Since $X$ is quasi-separated the scheme $\mathop{\mathrm{Spec}}(A) \times _ X U$ is quasi-compact. Moreover, the projection morphism $\mathop{\mathrm{Spec}}(A) \times _ X U \to \mathop{\mathrm{Spec}}(A)$ is ├ętale. Hence this morphism has finite discrete fibers and moreover the topology on $\mathop{\mathrm{Spec}}(A)$ is discrete. Thus $\mathop{\mathrm{Spec}}(A) \times _ X U$ is a scheme whose underlying topological space is a finite discrete set. We are done by Schemes, Lemma 26.11.8. $\square$


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