Lemma 10.155.8 (08HU)—The Stacks project

Lemma 10.155.8. Let $R \to S$ be a ring map. Let $\mathfrak q \subset S$ be a prime lying over $\mathfrak p \subset R$ . Let $R \to R^ h$ and $S \to S^ h$ be the henselizations of $R_\mathfrak p$ and $S_\mathfrak q$ . The local ring map $R^ h \to S^ h$ of Lemma 10.155.6 identifies $S^ h$ with the henselization of $R^ h \otimes _ R S$ at the unique prime lying over $\mathfrak m^ h$ and $\mathfrak q$ .

Proof. By Lemma 10.155.7 we see that $R^ h$ , resp. $S^ h$ are filtered colimits of étale $R$ , resp. $S$ -algebras. Hence we see that $R^ h \otimes _ R S$ is a filtered colimit of étale $S$ -algebras $A_ i$ (Lemma 10.143.3). By Lemma 10.154.5 we see that $S^ h$ is a filtered colimit of étale $R^ h \otimes _ R S$ -algebras. Since moreover $S^ h$ is a henselian local ring with residue field equal to $\kappa (\mathfrak q)$ , the statement follows from the uniqueness result of Lemma 10.154.7. $\square$

Comments (2)

Comment #5874 by Rankeya Datta on January 08, 2021 at 01:25

The notation $\frak{m}^h$ feels a little strange to use in the statement of the Lemma because the notation $\frak{m}$ does not appear anywhere else in the statement. That being said, I understand that $(\frak{p}R_{\frak{p}})^h$ looks clumsy...

Comment #6085 by Johan on April 29, 2021 at 18:16

Going to leave this alone for now

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