The Stacks project

Lemma 66.11.9. Let $S$ be a scheme. Let $X$ be a decent algebraic space over $S$. Let $\overline{x}$ be a geometric point of $X$ lying over $x \in |X|$. The ├ętale local ring $\mathcal{O}_{X, \overline{x}}$ of $X$ at $\overline{x}$ (Properties of Spaces, Definition 64.22.2) is the strict henselization of the henselian local ring $\mathcal{O}_{X, x}^ h$ of $X$ at $x$.

Proof. Follows from Lemma 66.11.8, Properties of Spaces, Lemma 64.22.1 and the fact that $(R^ h)^{sh} = R^{sh}$ for a local ring $(R, \mathfrak m, \kappa )$ and a given separable algebraic closure $\kappa ^{sep}$ of $\kappa $. This equality follows from Algebra, Lemma 10.153.6. $\square$


Comments (0)


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Your email address will not be published.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