The Stacks project

Remark 59.33.4. Let $S$ be a scheme. Let $s \in S$. If $S$ is locally Noetherian then $\mathcal{O}_{S, s}^ h$ is also Noetherian and it has the same completion:

\[ \widehat{\mathcal{O}_{S, s}} \cong \widehat{\mathcal{O}_{S, s}^ h}. \]

In particular, $\mathcal{O}_{S, s} \subset \mathcal{O}_{S, s}^ h \subset \widehat{\mathcal{O}_{S, s}}$. The henselization of $\mathcal{O}_{S, s}$ is in general much smaller than its completion and inherits many of its properties. For example, if $\mathcal{O}_{S, s}$ is reduced, then so is $\mathcal{O}_{S, s}^ h$, but this is not true for the completion in general. Insert future references here.


Comments (0)

There are also:

  • 2 comment(s) on Section 59.33: Stalks of the structure sheaf

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Your email address will not be published.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 03QM. Beware of the difference between the letter 'O' and the digit '0'.