The Stacks project

87.29 Artin's theorem on dilatations

In this section we use a different font for formal algebraic spaces to stress the similarity of the statements with the corresponding statements in [ArtinII]. Here is the first main theorem of this chapter.


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. $\square$

Here is the characterization of formal modifcations as promised in Section 87.24.

Lemma 87.29.2. Let $S$ be a scheme. Let $\mathfrak X' \to \mathfrak X$ be a formal modification (Definition 87.24.1) 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 87.24.4. Hence this follows from Theorem 87.29.1. $\square$

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