Lemma 87.21.7. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of locally Noetherian formal algebraic spaces which is representable by algebraic spaces, étale, and surjective. Then $f$ is rig-surjective.

Proof. Let $p : \text{Spf}(R) \to Y$ be an adic morphism where $R$ is a complete discrete valuation ring. Let $Z = \text{Spf}(R) \times _ Y X$. Then $Z \to \text{Spf}(R)$ is representable by algebraic spaces, étale, and surjective. Hence $Z$ is nonempty. Pick a nonempty affine formal algebraic space $V$ and an étale morphism $V \to Z$ (possible by our definitions). Then $V \to \text{Spf}(R)$ corresponds to $R \to A^\wedge$ where $R \to A$ is an étale ring map, see Formal Spaces, Lemma 86.19.13. Since $A^\wedge \not= 0$ (as $V \not= \emptyset$) we can find a maximal ideal $\mathfrak m$ of $A$ lying over $\mathfrak m_ R$. Then $A_\mathfrak m$ is a discrete valuation ring (More on Algebra, Lemma 15.44.4). Then $R' = A_\mathfrak m^\wedge$ is a complete discrete valuation ring (More on Algebra, Lemma 15.43.5). Applying Formal Spaces, Lemma 86.9.10. we find the desired morphism $\text{Spf}(R') \to V \to Z \to X$. $\square$

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