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Theorem (Global Splitting): Let $X$ be an integral separated normal scheme flat and of finite type over $\mathbb Z$. Let $\phi: Y\rightarrow X$ be a connected etale covering which splits completely over all closed points of $X$. Then Generalized Chebotare density implies that $\phi$ is an isomorphism.

The above theorem can be found in Moritz Kerz's paper "Higher class field theory and the connected component"(Proposition 3.1).

Question:

1) If $X$ is a spectrum of an excellent regular henselian local ring, and $\phi: Y\rightarrow X$ is a connected non-abelian Galois covering which splits completely over all closed points of $X$, is $\phi$ an isomorphism?

2) For general $X$, does the above theorem true? How far away can one generalize the above theorem? Counterexamples are welcomed...

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  • $\begingroup$ For your first question, you need to add some kind of connectedness hypothesis for $Y$; otherwise there are trivial counterexamples. For your second question, what do you mean by "general $X$"? Is $X$ henselian? $\endgroup$
    – Jason Starr
    Commented Feb 8, 2016 at 11:57
  • $\begingroup$ yeah, of course Y should be connected... In the second question I did not assume $X$ to be henselian...is there a counterexample for henselian trait? $\endgroup$
    – ely
    Commented Feb 8, 2016 at 12:00
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    $\begingroup$ For a counter-example to Q1, why not take $X = \mathrm{Spec}(\mathbb{Z}_p)$ and $Y \to X$ a totally ramified non-abelian Galois covering? Due to being totally ramified, the fibre over the closed point is just the the trivial extension of residue fields, hence clearly split. $\endgroup$
    – Daniel Loughran
    Commented Feb 8, 2016 at 22:20
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    $\begingroup$ @DanielLoughran. I had the same thought, but I suspect the OP meant to add a hypothesis that the cover is finite and etale. $\endgroup$
    – Jason Starr
    Commented Feb 8, 2016 at 23:55
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    $\begingroup$ stacks.math.columbia.edu/tag/04GG part 7 or 8 is the statement you want. Etale morphisms to henselian local rings have sections, and connected etale with section implies isomorphism. $\endgroup$
    – Will Sawin
    Commented Feb 10, 2016 at 4:51

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