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.
$\square$

## Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like `$\pi$`

). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

## Comments (0)