Lemma 29.9.3. Let X and Y be schemes over a base scheme S. Given points x \in X and y \in Y, there is a point of X \times _ S Y mapping to x and y under the projections if and only if x and y lie above the same point of S.
Proof. The condition is obviously necessary, and the converse follows from the proof of Schemes, Lemma 26.17.5. \square
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