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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