The Stacks project

Lemma 66.14.5. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $Z, Y \subset X$ be closed subspaces. Let $Z \cap Y$ be the scheme theoretic intersection of $Z$ and $Y$. Then $Z \cap Y \to Z$ and $Z \cap Y \to Y$ are closed immersions and

\[ \xymatrix{ Z \cap Y \ar[r] \ar[d] & Z \ar[d] \\ Y \ar[r] & X } \]

is a cartesian diagram of algebraic spaces over $S$, i.e., $Z \cap Y = Z \times _ X Y$.

Proof. The morphisms $Z \cap Y \to Z$ and $Z \cap Y \to Y$ are closed immersions by Lemma 66.13.1. Since formation of the scheme theoretic intersection commutes with ├ętale localization we conclude the diagram is cartesian by the case of schemes. See Morphisms, Lemma 29.4.5. $\square$

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