## Tag `01RO`

Chapter 28: Morphisms of Schemes > Section 28.47: Birational morphisms

Definition 28.47.1. Let $X$, $Y$ be schemes. Assume $X$ and $Y$ have finitely many irreducible components. We say a morphism $f : X \to Y$ is

birationalif

- $f$ 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.

The code snippet corresponding to this tag is a part of the file `morphisms.tex` and is located in lines 11646–11663 (see updates for more information).

```
\begin{definition}
\label{definition-birational}
\begin{reference}
\cite[(2.2.9)]{EGA1}
\end{reference}
Let $X$, $Y$ be schemes. Assume $X$ and $Y$ have finitely many
irreducible components. We say a morphism $f : X \to Y$ is
{\it birational} if
\begin{enumerate}
\item $f$ 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
\item 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.
\end{enumerate}
\end{definition}
```

## References

[EGA1, (2.2.9)]

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