Lemma 10.154.17. Let $R \to S$ be a ring map. Let $\mathfrak q \subset S$ be a prime lying over $\mathfrak p \subset R$ such that $\kappa (\mathfrak p) \to \kappa (\mathfrak q)$ is an isomorphism. Choose a separable algebraic closure $\kappa ^{sep}$ of $\kappa (\mathfrak p) = \kappa (\mathfrak q)$. Then

\[ (S_\mathfrak q)^{sh} = (S_\mathfrak q)^ h \otimes _{(R_\mathfrak p)^ h} (R_\mathfrak p)^{sh} \]

**Proof.**
This follows from the alternative construction of the strict henselization of a local ring in Remark 10.154.4 and the fact that the residue fields are equal. Some details omitted.
$\square$

## 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Your email address will not be published.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.

## Comments (0)

There are also: