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*}

Local rings of dimension zero are henselian.

Lemma 10.148.10. Let $(R, \mathfrak m)$ be a local ring of dimension $0$. Then $R$ is henselian.

Proof. Let $R \to S$ be a finite ring map. By Lemma 10.148.3 it suffices to show that $S$ is a product of local rings. By Lemma 10.35.21 $S$ has finitely many primes $\mathfrak m_1, \ldots , \mathfrak m_ r$ which all lie over $\mathfrak m$. There are no inclusions among these primes, see Lemma 10.35.20, hence they are all maximal. Every element of $\mathfrak m_1 \cap \ldots \cap \mathfrak m_ r$ is nilpotent by Lemma 10.16.2. It follows $S$ is the product of the localizations of $S$ at the primes $\mathfrak m_ i$ by Lemma 10.52.5. $\square$


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