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

Note that the ring map $R \to B(R)$ constructed in Section 109.44 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 109.44. 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 109.44 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 109.45.1. 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. $\square$

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