Lemma 88.24.3 (0GDM)—The Stacks project

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Table of contents
Part 5: Topics in Geometry
Chapter 88: Algebraization of Formal Spaces
Section 88.24: Formal modifications
Lemma 88.24.3 (cite)

Lemma 88.24.3. Let $S$ , $f : X' \to X$ , $T \subset |X|$ , $U \subset X$ , $T' \subset |X'|$ , and $U' \subset X'$ be as in Section 88.23. If $X$ is locally Noetherian, $f$ is proper, and $U' \to U$ is an isomorphism, then $f_{/T} : X'_{/T'} \to X_{/T}$ is a formal modification.

Proof. By Formal Spaces, Lemmas 87.20.8 the source and target of the arrow are locally Noetherian formal algebraic spaces. The other conditions follow from Lemmas 88.23.4, 88.23.9, 88.23.10, and 88.23.11. $\square$

The Stacks project

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