The Stacks project

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

\[ \xymatrix{ \kappa _1^{sep} \ar[r]_{\phi } & \kappa _2^{sep} \\ \kappa (\mathfrak p) \ar[r]^{\varphi } \ar[u] & \kappa (\mathfrak q) \ar[u] } \]

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 $\mathfrak m^{sh} \subset R^{sh}$.

Proof. The proof is identical to the proof of Lemma 10.155.8 except that it uses Lemma 10.155.11 instead of Lemma 10.155.7. $\square$

Comments (2)

Comment #4681 by Peng DU on

In the statement, it uses , which is not introduced.

BTW, this will answer the follow question in MathOverflow: Let be a finite morphism of schemes, let , here is a separable closure of the residue field . By some results about strict henselian local rings one can see This result (tag/08HV) will give if .

This further gives a decomposition ( stands for separable degree)

There are also:

  • 8 comment(s) on Section 10.155: Henselization and strict henselization

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