Lemma 15.42.2. Let $\varphi : R \to S$ be a ring map. Assume

1. $\varphi$ is regular,

2. $S$ is Noetherian, and

3. $R$ is Noetherian and normal.

Then $S$ is normal.

Proof. For Noetherian rings being normal is the same as having properties $(S_2)$ and $(R_1)$, see Algebra, Lemma 10.157.4. Hence we may apply Algebra, Lemmas 10.163.4 and 10.163.5. $\square$

Comment #4259 by DS on

typo "reduced" for "normal" in the proof.

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.

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 0BFK. Beware of the difference between the letter 'O' and the digit '0'.