The Stacks project

Lemma 29.54.8. A finite (or even integral) birational morphism $f : X \to Y$ of integral schemes with $Y$ normal is an isomorphism.

Proof. Let $V \subset Y$ be an affine open with inverse image $U \subset X$ which is an affine open too. Since $f$ is a birational morphism of integral schemes, the homomorphism $\mathcal{O}_ Y(V) \to \mathcal{O}_ X(U)$ is an injective map of domains which induces an isomorphism of fraction fields. As $Y$ is normal, the ring $\mathcal{O}_ Y(V)$ is integrally closed in the fraction field. Since $f$ is finite (or integral) every element of $\mathcal{O}_ X(U)$ is integral over $\mathcal{O}_ Y(V)$. We conclude that $\mathcal{O}_ Y(V) = \mathcal{O}_ X(U)$. This proves that $f$ is an isomorphism as desired. $\square$


Comments (0)


Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0AB1. Beware of the difference between the letter 'O' and the digit '0'.