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

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 $\mathfrak f$.

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