Definition 29.50.1. Let $X$, $Y$ be schemes. Assume $X$ and $Y$ have finitely many irreducible components. We say a morphism $f : X \to Y$ is *birational* if

$f$ induces a bijection between the set of generic points of irreducible components of $X$ and the set of generic points of the irreducible components of $Y$, and

for every generic point $\eta \in X$ of an irreducible component of $X$ the local ring map $\mathcal{O}_{Y, f(\eta )} \to \mathcal{O}_{X, \eta }$ is an isomorphism.

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