
Lemma 10.148.11. Let $R \to S$ be a ring map with $S$ henselian local. Given

1. an étale ring map $R \to A$,

2. a prime $\mathfrak q$ of $A$ lying over $\mathfrak p = R \cap \mathfrak m_ S$,

3. a $\kappa (\mathfrak p)$-algebra map $\tau : \kappa (\mathfrak q) \to S/\mathfrak m_ S$,

then there exists a unique homomorphism of $R$-algebras $f : A \to S$ such that $\mathfrak q = f^{-1}(\mathfrak m_ S)$ and $f \bmod \mathfrak q = \tau$.

Proof. Consider $A \otimes _ R S$. This is an étale algebra over $S$, see Lemma 10.141.3. Moreover, the kernel

$\mathfrak q' = \mathop{\mathrm{Ker}}(A \otimes _ R S \to \kappa (\mathfrak q) \otimes _{\kappa (\mathfrak p)} \kappa (\mathfrak m_ S) \to \kappa (\mathfrak m_ S))$

of the map using the map given in (3) is a prime ideal lying over $\mathfrak m_ S$ with residue field equal to the residue field of $S$. Hence by Lemma 10.148.3 there exists a unique splitting $\tau : A \otimes _ R S \to S$ with $\tau ^{-1}(\mathfrak m_ S) = \mathfrak q'$. Set $f$ equal to the composition $A \to A \otimes _ R S \to S$. $\square$

There are also:

• 2 comment(s) on Section 10.148: Henselian local rings

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 08HQ. Beware of the difference between the letter 'O' and the digit '0'.