Lemma 87.37.5 (0GVU)—The Stacks project

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Table of contents
Part 5: Topics in Geometry
Chapter 87: Formal Algebraic Spaces
Section 87.37: Recompletion
Lemma 87.37.5 (cite)

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Lemma 87.37.5. Let $S$ be a scheme. Let $X$ be a formal algebraic space over $S$ . Let $T \subset |X_{red}|$ be a closed subset. The reduction $(X_{/T})_{red}$ of the completion $X_{/T}$ of $X$ along $T$ is the reduced induced closed subspace $Z$ of $X_{red}$ corresponding to $T$ .

Proof. It follows from Lemma 87.12.1, Properties of Spaces, Definition 66.12.5 (which uses Properties of Spaces, Lemma 66.12.3 to construct $Z$ ), and the definition of $X_{/T}$ that $Z$ and $(X_{/T})_{red}$ are reduced algebraic spaces characterized the same mapping property: a morphism $g : Y \to X$ whose source is a reduced algebraic space factors through them if and only if $|Y|$ maps into $T \subset |X|$ . $\square$

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