The Stacks project

[Theorem 3.2, ArtinII]

Theorem 87.29.1. Let $S$ be a scheme. Let $X$ be a locally Noetherian algebraic space over $S$. Let $T \subset |X|$ be a closed subset. Let $\mathfrak X = X_{/T}$ be the formal completion of $X$ along $T$. Let

\[ \mathfrak f : \mathfrak X' \to \mathfrak X \]

be a formal modification (Definition 87.24.1). Then there exists a unique proper morphism $f : X' \to X$ which is an isomorphism over the complement of $T$ in $X$ whose completion $f_{/T}$ recovers $\mathfrak f$.

Proof. This follows from Theorem 87.27.4 and Lemma 87.28.4. $\square$

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