The Stacks project

110.46 A ring map which identifies local rings which is not ind-étale

Note that the ring map $R \to B(R)$ constructed in Section 110.45 is a colimit of finite products of copies of $R$. Hence $R \to B(R)$ is ind-Zariski, see Pro-étale Cohomology, Definition 61.4.1. Next, consider the ring map $A \to B^{twist}(A)$ constructed in Section 110.45. Since this ring map is Zariski locally on $\mathop{\mathrm{Spec}}(A)$ isomorphic to an ind-Zariski ring map $R \to B(R)$ we conclude that it identifies local rings (see Pro-étale Cohomology, Lemma 61.4.6). The discussion in Section 110.45 shows there is a section $B^{twist}(A) \to A$ whose kernel is not generated by idempotents. Now, if $A \to B^{twist}(A)$ were ind-étale, i.e., $B^{twist}(A) = \mathop{\mathrm{colim}}\nolimits A_ i$ with $A \to A_ i$ étale, then the kernel of $A_ i \to A$ would be generated by an idempotent (Algebra, Lemmas 10.143.8 and 10.143.9). This would contradict the result mentioned above.

Lemma 110.46.1. There is a ring map $A \to B$ which identifies local rings but which is not ind-étale. A fortiori it is not ind-Zariski.

Proof. See discussion above. $\square$


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Your email address will not be published.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 09AN. Beware of the difference between the letter 'O' and the digit '0'.