Lemma 87.20.2. Let $S$ be a scheme. Let $X \to Y$ be a morphism of affine formal algebraic spaces which is representable by algebraic spaces, surjective, and flat. Then $X$ is countably indexed and classical if and only if $Y$ is countably indexed and classical.

**Proof.**
We have already seen the implication in one direction in Lemma 87.19.10. For the other direction, note that by Lemma 87.20.1 we may assume both $X$ and $Y$ are countably indexed. Thus $X = \text{Spf}(A)$ and $Y = \text{Spf}(B)$ for some weakly admissible topological $S$-algebras $A$ and $B$, see Lemma 87.10.4. By Lemma 87.9.10 the morphism $X \to Y$ corresponds to a continuous $S$-algebra homomorphism $\varphi : B \to A$. We see from Lemma 87.19.8 that $\varphi $ is taut. Let $J \subset B$ be an open ideal and let $I \subset A$ be the closure of $JA$. By Lemmas 87.16.4 and 87.4.11 we see that $\mathop{\mathrm{Spec}}(B/J) \times _ Y X = \mathop{\mathrm{Spec}}(A/I)$. Hence $B/J \to A/I$ is faithfully flat (since $X \to Y$ is surjective and flat). This means that $\varphi : B \to A$ is as in Section 87.8 (with the roles of $A$ and $B$ swapped). We conclude that the lemma holds by Lemma 87.8.2.
$\square$

## 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Your email address will not be published.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)