Lemma 10.155.13 (0C2Z)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 10: Commutative Algebra
Section 10.155: Henselization and strict henselization
Lemma 10.155.13 (cite)

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Lemma 10.155.13. Let $R \to S$ be a ring map. Let $\mathfrak q \subset S$ be a prime lying over $\mathfrak p \subset R$ such that $\kappa (\mathfrak p) \to \kappa (\mathfrak q)$ is an isomorphism. Choose a separable algebraic closure $\kappa ^{sep}$ of $\kappa (\mathfrak p) = \kappa (\mathfrak q)$. Then

\[ (S_\mathfrak q)^{sh} = (S_\mathfrak q)^ h \otimes _{(R_\mathfrak p)^ h} (R_\mathfrak p)^{sh} \]

Proof. This follows from the alternative construction of the strict henselization of a local ring in Remark 10.155.4 and the fact that the residue fields are equal. Some details omitted. $\square$

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