The Stacks project

Lemma 29.47.4. Let $X$ be a scheme. The following are equivalent:

  1. The scheme $X$ is seminormal.

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

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

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

Moreover, if $X$ is seminormal then every open subscheme is seminormal. The same statements are true with “seminormal” replaced by “absolutely weakly normal”.

Proof. Combine Properties, Lemma 28.4.3 and Lemma 29.47.2. $\square$

Comments (0)

There are also:

  • 1 comment(s) on Section 29.47: Absolute weak normalization and seminormalization

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Your email address will not be published.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