The Stacks project

\begin{equation*} \DeclareMathOperator\Coim{Coim} \DeclareMathOperator\Coker{Coker} \DeclareMathOperator\Ext{Ext} \DeclareMathOperator\Hom{Hom} \DeclareMathOperator\Im{Im} \DeclareMathOperator\Ker{Ker} \DeclareMathOperator\Mor{Mor} \DeclareMathOperator\Ob{Ob} \DeclareMathOperator\Sh{Sh} \DeclareMathOperator\SheafExt{\mathcal{E}\mathit{xt}} \DeclareMathOperator\SheafHom{\mathcal{H}\mathit{om}} \DeclareMathOperator\Spec{Spec} \newcommand\colim{\mathop{\mathrm{colim}}\nolimits} \newcommand\lim{\mathop{\mathrm{lim}}\nolimits} \newcommand\Qcoh{\mathit{Qcoh}} \newcommand\Sch{\mathit{Sch}} \newcommand\QCohstack{\mathcal{QC}\!\mathit{oh}} \newcommand\Cohstack{\mathcal{C}\!\mathit{oh}} \newcommand\Spacesstack{\mathcal{S}\!\mathit{paces}} \newcommand\Quotfunctor{\mathrm{Quot}} \newcommand\Hilbfunctor{\mathrm{Hilb}} \newcommand\Curvesstack{\mathcal{C}\!\mathit{urves}} \newcommand\Polarizedstack{\mathcal{P}\!\mathit{olarized}} \newcommand\Complexesstack{\mathcal{C}\!\mathit{omplexes}} \newcommand\Pic{\mathop{\mathrm{Pic}}\nolimits} \newcommand\Picardstack{\mathcal{P}\!\mathit{ic}} \newcommand\Picardfunctor{\mathrm{Pic}} \newcommand\Deformationcategory{\mathcal{D}\!\mathit{ef}} \end{equation*}

Remark 10.150.4. We can also construct $R^{sh}$ from $R^ h$. Namely, for any finite separable subextension $\kappa \subset \kappa ' \subset \kappa ^{sep}$ there exists a unique (up to unique isomorphism) finite ├ętale local ring extension $R^ h \subset R^ h(\kappa ')$ whose residue field extension reproduces the given extension, see Lemma 10.148.7. Hence we can set

\[ R^{sh} = \bigcup \nolimits _{\kappa \subset \kappa ' \subset \kappa ^{sep}} R^ h(\kappa ') \]

The arrows in this system, compatible with the arrows on the level of residue fields, exist by Lemma 10.148.7. This will produce a henselian local ring by Lemma 10.149.7 since each of the rings $R^ h(\kappa ')$ is henselian by Lemma 10.148.4. By construction the residue field extension induced by $R^ h \to R^{sh}$ is the field extension $\kappa \subset \kappa ^{sep}$. Hence $R^{sh}$ so constructed is strictly henselian. By Lemma 10.149.2 the $R$-algebra $R^{sh}$ is a colimit of ├ętale $R$-algebras. Hence the uniqueness of Lemma 10.149.6 shows that $R^{sh}$ is the strict henselization.


Comments (0)

There are also:

  • 4 comment(s) on Section 10.150: Henselization and strict henselization

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Your email address will not be published.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