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Motivation

I was re-reading parts of Grothendieck-Murre, and these questions came up naturally.

The situation in chapter 9 is that $S'=Spec(A)$ where $A$ is a complete local noetherian ring of dimension $2$ with an algebraically closed residue field, and $D$ is some divisor in $A$. Then they take a "desingularization" of $S'$, $T'$, such that $T'$ maps isomorphically to $S'$ away from $D$, and the divisor that goes to $D$ is normal crossings. Call the divisor that maps to $D$ also $D$. They first show that $\pi_1^{(p)\,D}(T')$ is isomorphic to $\pi_1^{(p)}$ of the completion of $T'$ with respect to $D$ where this $\pi_1$ is the tame $\pi_1$ with respect to $D$. I assume this goes through even if $D$ is not normal crossings (if I'm wrong about this, I would really like to know).

Then they prove that $\pi_1^{(p)}(S'\setminus D)(=\pi_1^{(p)}(T'\setminus D))$ is isomorphic to $\pi_1^{(p),D}(T')$.

The question that arises is: why did they take this detour through this desingularization? Which step doesn't go through in the case that $D$ is not normal crossings? As I said, I think it's unlikely that it is the step that the tame fundamental groups are the same when completing, but I would like to know if I'm wrong.

Question

Let $S'=Spec(A)$ where $A$ is a complete local noetherian ring of dimension $2$ with an algebraically closed residue field, and $D$ is some divisor in $A$. Could there exist a divisor, $D$ (necessarily not normal crossings), such that $\pi_1^{(p)}(S'\setminus D)$ is not isomorphic to $\pi_1^{(p),D}(S')$?

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1 Answer 1

up vote 5 down vote accepted

Your question as stated does not strictly speaking make sense. In Grothendieck-Murre sect.2.4 the tame fundamental group $\pi_1^D(S,\xi)$ is only defined when $S$ is normal and $D$ is a DNC. Although in 2.2.2 they define tamely ramified covers in greater generality, they observe in Remark 2.2.3(4) that this is "certainly not the correct" definition when $D$ is not a DNC. However if you were to use their definition 2.2.2 (which requires in addition that $S'$ be normal) to define the tame $\pi_1$, then the groups in your question would be isomorphic for any $D$. Indeed, you have an equivalence of the relevant categories of coverings, given by restriction to $S'\setminus D$ in one direction and by normalisation in the other.

The aim of Chapter 9 Grothendieck-Murre is to prove that $\pi_1^{(p)}$ of a connected open subscheme of $S'$ (not necessarily normal) is topologically finitely presented. For this you really do need to pass to the desingularisation, so as to bring Kummer theory of tame covers to bear.

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Thanks! Is the "correct" generalization of tame known in the non-normal case, or is it still a matter of debate? What do we expect the correct definition to abide by? –  James D. Taylor Oct 21 '10 at 15:32
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The natural framework for tame covers to non-DNC situations is logarithmic geometry. There is a log-etale fundamental group, which classifies Kummer log-etale covers of log schemes, and which in a DNC situation is just tame $\pi_1$. Unfortunately there is not much in the literature about log-$\pi_1$ that I know of - it was one of the topics of Isabelle Vidal's thesis, but that part of it remains unpublished. There is a bunch of papers on log-etale cohomology, mostly by Nakayama. –  Tony Scholl Oct 21 '10 at 16:29
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There's a section on the log-etale fundamental group in Jacob Stix' thesis, which can be found on his homepage. –  Lars Oct 21 '10 at 23:04

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