The Stacks project

Lemma 15.49.8. Let $R$ be a Noetherian local ring which is a G-ring. Then the henselization $R^ h$ and the strict henselization $R^{sh}$ are G-rings.

Proof. We will use the criterion of Lemma 15.49.7. Let $\mathfrak q \subset R^ h$ be a prime and set $\mathfrak p = R \cap \mathfrak q$. Set $\mathfrak q_1 = \mathfrak q$ and let $\mathfrak q_2, \ldots , \mathfrak q_ t$ be the other primes of $R^ h$ lying over $\mathfrak p$, so that $R^ h \otimes _ R \kappa (\mathfrak p) = \prod \nolimits _{i = 1, \ldots , t} \kappa (\mathfrak q_ i)$, see Lemma 15.44.13. Using that $(R^ h)^\wedge = R^\wedge $ (Lemma 15.44.3) we see

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

Hence $(R^ h)^\wedge \otimes _{R^ h} \kappa (\mathfrak q_ i)$ is geometrically regular over $\kappa (\mathfrak p)$ by assumption. Since $\kappa (\mathfrak q_ i)$ is separable algebraic over $\kappa (\mathfrak p)$ it follows from Algebra, Lemma 10.160.6 that $(R^ h)^\wedge \otimes _{R^ h} \kappa (\mathfrak q_ i)$ is geometrically regular over $\kappa (\mathfrak q_ i)$.

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

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

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 $(R^{sh})^\wedge \otimes _{R^{sh}} \kappa (\mathfrak r_ i)$ is regular over $\kappa (\mathfrak p)$ by Lemma 15.40.4 as $R^\wedge \otimes _ R \kappa (\mathfrak p)$ is regular over $\kappa (\mathfrak p)$ by assumption. Since $\kappa (\mathfrak r_ i)$ is separable algebraic over $\kappa (\mathfrak p)$ it follows from Algebra, Lemma 10.160.6 that $(R^{sh})^\wedge \otimes _{R^{sh}} \kappa (\mathfrak r_ i)$ is geometrically regular over $\kappa (\mathfrak r_ i)$. $\square$


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