The Stacks project

Lemma 29.50.6. Let $S$ be a scheme. Let $X$ and $Y$ be irreducible schemes locally of finite presentation over $S$. Let $x \in X$ and $y \in Y$ be the generic points. The following are equivalent

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

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

  3. $x$ and $y$ map to the same point $s$ of $S$ and $\mathcal{O}_{X, x}$ and $\mathcal{O}_{Y, y}$ are isomorphic as $\mathcal{O}_{S, s}$-algebras.

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). 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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