Lemma 87.14.4 (0APV)—The Stacks project

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Table of contents
Part 5: Topics in Geometry
Chapter 87: Formal Algebraic Spaces
Section 87.14: Completion along a closed subset
Lemma 87.14.4 (cite)

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Lemma 87.14.4. Let $S$ be a scheme. Let $f : X' \to X$ be a morphism of algebraic spaces over $S$. Let $T \subset |X|$ be a closed subset and let $T' = |f|^{-1}(T) \subset |X'|$. Then

\[ \xymatrix{ X'_{/T'} \ar[r] \ar[d] & X' \ar[d]^ f \\ X_{/T} \ar[r] & X } \]

is a cartesian diagram of sheaves. In particular, the morphism $X'_{/T'} \to X_{/T}$ is representable by algebraic spaces.

Proof. Namely, suppose that $Y \to X$ is a morphism from a scheme into $X$ such that $|Y|$ maps into $T$. Then $Y \times _ X X' \to X$ is a morphism of algebraic spaces such that $|Y \times _ X X'|$ maps into $T'$. Hence the functor $Y \times _{X_{/T}} X'_{/T'}$ is represented by $Y \times _ X X'$ and we see that the lemma holds. $\square$

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