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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


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