Lemma 29.54.7. Let $X$ be a reduced scheme with finitely many irreducible components. Then the normalization morphism $X^\nu \to X$ is birational.

Proof. The normalization induces a bijection of irreducible components by Lemma 29.54.5. Let $\eta \in X$ be a generic point of an irreducible component of $X$ and let $\eta ^\nu \in X^\nu$ be the generic point of the corresponding irreducible component of $X^\nu$. Then $\eta ^\nu \mapsto \eta$ and to finish the proof we have to show that $\mathcal{O}_{X, \eta } \to \mathcal{O}_{X^\nu , \eta ^\nu }$ is an isomorphism, see Definition 29.50.1. Because $X$ and $X^\nu$ are reduced, we see that both local rings are equal to their residue fields (Algebra, Lemma 10.25.1). On the other hand, by the construction of the normalization as the normalization of $X$ in $Y = \coprod \mathop{\mathrm{Spec}}(\kappa (\eta ))$ we see that we have $\kappa (\eta ) \subset \kappa (\eta ^\nu ) \subset \kappa (\eta )$ and the proof is complete. $\square$

## 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Your email address will not be published.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 0BXC. Beware of the difference between the letter 'O' and the digit '0'.