Lemma 87.14.5. Let S be a scheme. Let X be an algebraic space over S. Let T \subset |X| 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 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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