Lemma 67.20.12. 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 étale morphism with U affine. To prove the lemma we have to show that \mathop{\mathrm{Spec}}(A) \times _ X U is affine, see Lemma 67.20.3. 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 étale. 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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