Lemma 10.155.12. Let $R \to S$ be a ring map. Let $\mathfrak q \subset S$ be a prime lying over $\mathfrak p \subset R$. Choose separable algebraic closures $\kappa (\mathfrak p) \subset \kappa _1^{sep}$ and $\kappa (\mathfrak q) \subset \kappa _2^{sep}$. Let $R^{sh}$ and $S^{sh}$ be the corresponding strict henselizations of $R_\mathfrak p$ and $S_\mathfrak q$. Given any commutative diagram

The local ring map $R^{sh} \to S^{sh}$ of Lemma 10.155.10 identifies $S^{sh}$ with the strict henselization of $R^{sh} \otimes _ R S$ at a prime lying over $\mathfrak q$ and the maximal ideal $\mathfrak m^{sh} \subset R^{sh}$.

## Comments (2)

Comment #4681 by Peng DU on

Comment #4806 by Johan on

There are also: