The Stacks project

Lemma 86.29.5. Let $S$ be a scheme. Let $X$ be a countably indexed affine formal algebraic space over $S$. Let $f : Y \to X$ be a closed immersion of formal algebraic spaces over $S$. Then $Y$ is a countably indexed affine formal algebraic space and $f$ corresponds to $A \to A/K$ where $A$ is an object of $\textit{WAdm}^{count}$ (Section 86.21) and $K \subset A$ is a closed ideal.

Proof. By Lemma 86.10.4 we see that $X = \text{Spf}(A)$ where $A$ is an object of $\textit{WAdm}^{count}$. Since a closed immersion is representable and affine, we conclude by Lemma 86.19.9 that $Y$ is an affine formal algebraic space and countably index. Thus applying Lemma 86.10.4 again we see that $Y = \text{Spf}(B)$ with $B$ an object of $\textit{WAdm}^{count}$. By Lemma 86.27.5 we conclude that $f$ is given by a morphism $A \to B$ of $\textit{WAdm}^{count}$ which is taut and has dense image. To finish the proof we apply Lemma 86.5.10. $\square$


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