Lemma 88.21.11. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of formal algebraic spaces. Assume $X$ and $Y$ are locally Noetherian, $f$ locally of finite type, and $f$ a monomorphism. Then $f$ is rig surjective if and only if every adic morphism $\text{Spf}(R) \to Y$ where $R$ is a complete discrete valuation ring factors through $X$.
Proof. One direction is trivial. For the other, suppose that $\text{Spf}(R) \to Y$ is an adic morphism such that there exists an extension of complete discrete valuation rings $R \subset R'$ with $\text{Spf}(R') \to \text{Spf}(R) \to X$ factoring through $Y$. Then $\mathop{\mathrm{Spec}}(R'/\mathfrak m_ R^ n R') \to \mathop{\mathrm{Spec}}(R/\mathfrak m_ R^ n)$ is surjective and flat, hence the morphisms $\mathop{\mathrm{Spec}}(R/\mathfrak m_ R^ n) \to X$ factor through $X$ as $X$ satisfies the sheaf condition for fpqc coverings, see Formal Spaces, Lemma 87.32.1. In other words, $\text{Spf}(R) \to Y$ factors through $X$. $\square$
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