# Relation between tame fundamental group w.r.t. to D, and the fundamental group of the complement of D

### 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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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.
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