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.