The Stacks project

Theorem 15.95.25 (Olivier). Let $A \to B$ be a local homomorphism of local rings. If $A$ is strictly henselian and $A \to B$ is weakly étale, then $A = B$.

Proof. We will show that for all $\mathfrak p \subset A$ there is a unique prime $\mathfrak q \subset B$ lying over $\mathfrak p$ and $\kappa (\mathfrak p) = \kappa (\mathfrak q)$. This implies that $B \otimes _ A B \to B$ is bijective on spectra as well as surjective and flat. Hence it is an isomorphism for example by the description of pure ideals in Algebra, Lemma 10.107.4. Hence $A \to B$ is a faithfully flat epimorphism of rings. We get $A = B$ by Algebra, Lemma 10.106.7.

Note that the fibre ring $B \otimes _ A \kappa (\mathfrak p)$ is a colimit of étale extensions of $\kappa (\mathfrak p)$ by Lemmas 15.95.7 and 15.95.16. Hence, if there exists more than one prime lying over $\mathfrak p$ or if $\kappa (\mathfrak p) \not= \kappa (\mathfrak q)$ for some $\mathfrak q$, then $B \otimes _ A L$ has a nontrivial idempotent for some (separable) algebraic field extension $L \supset \kappa (\mathfrak p)$.

Let $\kappa (\mathfrak p) \subset L$ be an algebraic field extension. Let $A' \subset L$ be the integral closure of $A/\mathfrak p$ in $L$. By Lemma 15.95.23 we see that $A'$ is a strictly henselian local ring whose residue field is a purely inseparable extension of the residue field of $A$. Thus $B \otimes _ A A'$ is a local ring by Lemma 15.95.24. On the other hand, $B \otimes _ A A'$ is integrally closed in $B \otimes _ A L$ by Lemma 15.95.22. Since $B \otimes _ A A'$ is local, it follows that the ring $B \otimes _ A L$ does not have nontrivial idempotents which is what we wanted to prove. $\square$


Comments (0)


Post a comment

Your email address will not be published. Required fields are marked.

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.

All contributions are licensed under the GNU Free Documentation License.




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 092Z. Beware of the difference between the letter 'O' and the digit '0'.