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. $\square$

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