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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