Lemma 15.51.8. 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 15.51.7. To prove it for the strict henselization, it suffices to show that the formal fibres of $R^{sh}$ have $P$, see Lemma 15.51.4. 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 15.45.13. 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 15.45.3. Hence $R^\wedge \to (R^{sh})^\wedge$ is regular by Proposition 15.49.2. 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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