Lemma 62.5.8. Let $f : X \to S$ be a morphism of schemes which is locally of finite type. Let $r \geq 0$ be an integer. Let $g : S' \to S$ be a surjective morphism of schemes. Set $S'' = S' \times _ S S'$ and let $f' : X' \to S'$ and $f'' : X'' \to S''$ be the base changes of $f$. Let $x \in X$ with $\text{trdeg}_{\kappa (f(x))}(\kappa (x)) = r$.

There exists an $x' \in X'$ mapping to $x$ with $\text{trdeg}_{\kappa (f'(x'))}(\kappa (x')) = r$.

If $x'_1, x'_2 \in X'$ are both as in (1), then there exists an $x'' \in X''$ with $\text{trdeg}_{\kappa (f''(x''))}(\kappa (x'')) = r$ and $\text{pr}_ i(x'') = x'_ i$.

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