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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