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It is well known that if $(A, \mathfrak m)$ is a Henselian local ring with residue field $\kappa$, then base-change from $A$ to $\kappa$ determines an equivalence of categories

$$F: \{\text{Finite étale $A$-algebras}\} \longrightarrow \{\text{Finite étale $\kappa$-algebras}\}.$$

In fact, $A$ and $\kappa$ have canonically isomorphic étale fundamental groups, which implies the above equivalence of categories.

A few questions:

  1. Is the converse false? Namely, is there an example of a local ring $(A, \mathfrak m)$ such that the canonical map $\pi_1(\kappa) \to \pi_1(A)$ is an isomorphism, yet $A$ is not Henselian?

  2. Is there an example of a local ring $(A, \mathfrak m)$ such that the functor $F$ is an equivalence of categories, but $A$ is not Henselian?

  3. Is there an example of a ring satisfying (2) but not (1)? A profinite group is not determined up to isomorphism by its category of continuous permutation representations, so in principle, this is not completely ruled out.

Thank you for any insight!

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  • $\begingroup$ It is possibly of use that strictly henselian rings are those that occur as stalks of the étale site. Also of some use may be the equivalent conditions described at stacks.math.columbia.edu/tag/04GE. I will give the bounty to whoever answers 1, 2, or 3. $\endgroup$ Apr 18, 2019 at 0:45

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