The Stacks project

Lemma 88.29.2. Let $S$ be a scheme. Let $\mathfrak X' \to \mathfrak X$ be a formal modification (Definition 88.24.1) of locally Noetherian formal algebraic spaces over $S$. Given

  1. any adic Noetherian topological ring $A$,

  2. any adic morphism $\text{Spf}(A) \longrightarrow \mathfrak X$

there exists a proper morphism $X \to \mathop{\mathrm{Spec}}(A)$ of algebraic spaces and an isomorphism

\[ \text{Spf}(A) \times _{\mathfrak X} \mathfrak X' \longrightarrow X_{/Z} \]

over $\text{Spf}(A)$ of the base change of $\mathfrak X$ with the formal completion of $X$ along the “closed fibre” $Z = X \times _{\mathop{\mathrm{Spec}}(A)} \text{Spf}(A)_{red}$ of $X$ over $A$.

Proof. The morphism $\text{Spf}(A) \times _{\mathfrak X} \mathfrak X' \to \text{Spf}(A)$ is a formal modification by Lemma 88.24.4. Hence this follows from Theorem 88.29.1. $\square$

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