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*}

Lemma 15.50.8. Let $R$ be a Noetherian local ring which is a $P$-ring where $P$ satisfies (B), (C), (D), and (E). Then the henselization $R^ h$ and the strict henselization $R^{sh}$ are $P$-rings.

Proof. We have seen this for the henselization in Lemma 15.50.7. To prove it for the strict henselization, it suffices to show that the formal fibres of $R^{sh}$ have $P$, see Lemma 15.50.4. Let $\mathfrak r \subset R^{sh}$ be a prime and set $\mathfrak p = R \cap \mathfrak r$. Set $\mathfrak r_1 = \mathfrak r$ and let $\mathfrak r_2, \ldots , \mathfrak r_ s$ be the other primes of $R^{sh}$ lying over $\mathfrak p$, so that $R^{sh} \otimes _ R \kappa (\mathfrak p) = \prod \nolimits _{i = 1, \ldots , s} \kappa (\mathfrak r_ i)$, see Lemma 15.44.13. Then we see that

\[ \prod \nolimits _{i = 1, \ldots , t} (R^{sh})^\wedge \otimes _{R^{sh}} \kappa (\mathfrak r_ i) = (R^{sh})^\wedge \otimes _{R^{sh}} (R^{sh} \otimes _ R \kappa (\mathfrak p)) = (R^{sh})^\wedge \otimes _ R \kappa (\mathfrak p) \]

Note that $R^\wedge \to (R^{sh})^\wedge $ is formally smooth in the $\mathfrak m_{(R^{sh})^\wedge }$-adic topology, see Lemma 15.44.3. Hence $R^\wedge \to (R^{sh})^\wedge $ is regular by Proposition 15.48.2. We conclude that property $P$ holds for $\kappa (\mathfrak p) \to (R^{sh})^\wedge \otimes _ R \kappa (\mathfrak p)$ by (C) and our assumption on $R$. Using property (B), using the decomposition above, and looking at local rings we conclude that property $P$ holds for $\kappa (\mathfrak p) \to (R^{sh})^\wedge \otimes _{R^{sh}} \kappa (\mathfrak r)$. Since $\kappa (\mathfrak r)/\kappa (\mathfrak p)$ is separable algebraic, it follows from (E) that $P$ holds for $\kappa (\mathfrak r) \to (R^{sh})^\wedge \otimes _{R^{sh}} \kappa (\mathfrak r)$. $\square$


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