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

$X$ 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 $s$ of $S$ and $\mathcal{O}_{X, x}$ and $\mathcal{O}_{Y, y}$ are isomorphic as $\mathcal{O}_{S, s}$-algebras.

## Comments (0)