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.
Proof. We have seen the equivalence of (1) and (2) in Lemma 29.49.12. It is immediate that (2) implies (3). To finish we assume (3) holds and we prove (1). Observe that $\mathcal{O}_{X, x} = \kappa (x)$ and $\mathcal{O}_{Y, y} = \kappa (y)$ by Algebra, Lemma 10.25.1. By Lemma 29.49.2 there is a rational map $f : U \to Y$ which sends $x \in U$ to $y$ and induces the given isomorphism $\mathcal{O}_{Y, y} \cong \mathcal{O}_{X, x}$. Thus $f$ is a birational morphism and hence induces an isomorphism on nonempty opens by Lemma 29.50.5. This finishes the proof. $\square$
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