Lemma 29.4.5. Let $X$ be a scheme. Let $Z, Y \subset X$ be closed subschemes. 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
is a cartesian diagram of schemes, 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 29.2.2. Let $U = \mathop{\mathrm{Spec}}(A)$ be an affine open of $X$ and let $Z \cap U$ and $Y \cap U$ correspond to the ideals $I \subset A$ and $J \subset A$. Then $Z \cap Y \cap U$ corresponds to $I + J \subset A$. Since $A/I \otimes _ A A/J = A/(I + J)$ we see that the diagram is cartesian by our description of fibre products of schemes in Schemes, Section 26.17. $\square$
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