The Stacks project

Lemma 27.7.2. Let $X$ be a scheme. The following are equivalent:

  1. The scheme $X$ is normal.

  2. For every affine open $U \subset X$ the ring $\mathcal{O}_ X(U)$ is normal.

  3. There exists an affine open covering $X = \bigcup U_ i$ such that each $\mathcal{O}_ X(U_ i)$ is normal.

  4. There exists an open covering $X = \bigcup X_ j$ such that each open subscheme $X_ j$ is normal.

Moreover, if $X$ is normal then every open subscheme is normal.

Proof. This is clear from the definitions. $\square$


Comments (0)

There are also:

  • 2 comment(s) on Section 27.7: Normal schemes

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Your email address will not be published.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 033J. Beware of the difference between the letter 'O' and the digit '0'.