Lemma 29.50.7. Let $S$ be a scheme. Let $X$ and $Y$ be integral schemes locally of finite type over $S$. Let $x \in X$ and $y \in Y$ be the generic points. The following are equivalent

$X$ and $Y$ are $S$-birational,

there exist nonempty opens of $X$ and $Y$ which are $S$-isomorphic, and

$x$ and $y$ map to the same point $s \in S$ and $\kappa (x) \cong \kappa (y)$ as $\kappa (s)$-extensions.

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