Grothendieck–Murre tame fundamental group of a formal neighbourhood: kernel and extension class for intersecting boundary components (LNM 208, Theorem 7.3.1 without disjointness)

Grothendieck and Murre (Lecture Notes in Mathematics 208, 1971, §7) computed the tame fundamental group $\pi_1^D(\mathcal{S},\xi^*)$ of the formal neighbourhood of an irreducible divisor $D_0$: it is an extension of the tame fundamental group of the open part of the special fibre by a procyclic group $K$, and the extension class is the image of the Kummer (self-intersection) class of $D_0$. The determination of the kernel — the equality $\operatorname{Im}(\beta)=\ker(j)$, which identifies $K$ as the profinite tame inertia group $\widehat{\mathbb{Z}}'(1)$ modulo the image of the second homotopy group of the special fibre — is proved only under the additional hypothesis that the non-central components of the divisor on the special fibre are pairwise disjoint. Without that hypothesis, which is automatic in the classical relative-curve applications but fails in dimension at least three, only the general inclusion $\operatorname{Im}(\beta)\subseteq\ker(j)$ and the surjection of $\widehat{\mathbb{Z}}'(1)$ onto $K$ are known. No resolution of the general (disjointness-free) case is known in the literature to date. Callegaro–Gaiffi–Lochak (2015), working in a genuinely intersecting setting (wonderful compactifications of complements of complex reflection arrangements — smooth compactifications whose boundary divisor has normal crossings, with boundary components meeting whenever the corresponding flats are nested), explicitly use only the part valid without disjointness and pose the kernel determination as an open problem they call hard. The problem of determining $\ker(j)$ (equivalently $K$) and completing the extension-class description for intersecting non-central components therefore remains open.

Background

For a connected locally noetherian scheme, the finite étale covers (finite covering maps in the algebraic sense) are classified by a profinite group — a topological group that is an inverse limit of finite groups — called the étale fundamental group. If $U\subset X$ is a dense open whose complement is a divisor, one can classify instead the finite covers of $U$ that are tamely ramified along $X\setminus U$: at each codimension-one point of the boundary, the inertia groups of the cover (the subgroups of the Galois group acting trivially on the residue fields at the points above it) must have order prime to the residue characteristic. This yields the tame fundamental group, whose local model is the Kummer cover $\tau^n=t$ over the punctured disk. Grothendieck and Murre's 1971 monograph (Lecture Notes in Mathematics 208) studies a refinement motivated by Mumford's computation of the fundamental group of a neighbourhood of a surface singularity via its link: instead of the complement of the whole divisor, take the formal neighbourhood of a single irreducible component $D_0$ — the completion of the ambient scheme along $D_0$, carrying all infinitesimal information transverse to it — and classify its finite covers tamely ramified along the remaining components.

Precisely, the standing assumptions (loc. cit. 5.1.1) are: $\mathcal{S}$ is a connected locally noetherian regular formal scheme (locally of the form $\operatorname{Spf}A$ for an adic noetherian ring $A$); $D_0,\dots,D_r$ are regular divisors with normal crossings on $\mathcal{S}$ (each locally defined by one equation from a regular system of parameters of the local ring — a minimal set of generators of the maximal ideal of a regular local ring — the components meeting transversally); and the ideal $J(D_0)$ of $D_0$ is an ideal of definition of the formal scheme (its vanishing defines the closed fibre; in the motivating situation $\mathcal{S}$ is the completion of an ambient regular scheme along the irreducible divisor $D_0$, and then the closed fibre $S_0$ is $D_0$ itself). Writing $D_{i,0}$ for the divisor cut out on $S_0$ by $D_i$, and $U_0=S_0\setminus\bigcup_{i\neq0}D_{i,0}$ with $D'_0=\sum_{i\neq0}D_{i,0}$, the tame fundamental group $\pi_1^D(\mathcal{S},\xi^*)$ of the formal neighbourhood — based at a geometric point $\xi^*$ (a morphism from the spectrum of a separably closed field into $\mathcal{S}$) compatible with a geometric point $\xi_0$ of $U_0$ — sits in an exact sequence $1\to K\to\pi_1^D(\mathcal{S},\xi^*)\to\pi_1^{D'_0}(S_0,\xi_0)\to 1,$ where $\pi_1^{D'_0}(S_0,\xi_0)$ is the tame fundamental group of $S_0$ ramified along $D'_0$. The kernel $K$ is procyclic (topologically generated by a single element): there is a canonical surjection $j$ onto it from the profinite tame inertia group $\mathcal{M}^t(\xi_0)\cong\widehat{\mathbb{Z}}'(1)$ — one copy of the product of the $\ell$-adic integers over all primes $\ell$ different from the residue characteristic, twisted by the cyclotomic character (the natural action of the absolute Galois group of the residue field on roots of unity) — recording the normal direction to $S_0$ inside $\mathcal{S}$.

The chapter determines the extension in two steps. First (§7.1, Theorem 7.1.3, proved under the general standing assumptions): the class $k(e)\in H^2(U_0,\underline{K})$ of the extension is the image $j_*(c'(D_0))$ of the Kummer class $c'(D_0)$ of $D_0$ — the obstruction to taking $n$-th roots of a local equation of $D_0$ on $U_0$, i.e. the self-intersection class of $D_0$ in the formal scheme. The same class determines a homomorphism $\beta$ from the tame second homotopy group $\pi_2^{D'_0}(U_0,\xi_0)$ of the pair $(U_0,D'_0)$ (defined, in analogy with topology, from the universal tame cover of $U_0$) to $\mathcal{M}^t(\xi_0)$, and the inclusion $\operatorname{Im}(\beta)\subseteq\ker(j)$ holds in general. Second (§7.2, under the strengthened hypothesis that the divisors $D_{i,0}$, $i\neq0$, are pairwise disjoint): an explicit construction of Kummer-type Galois covers with prescribed inertia establishes the reverse inclusion, yielding Theorem 7.3.1: $\pi_2^{D'_0}(U_0,\xi_0)\xrightarrow{\beta}\mathcal{M}^t(\xi_0)\xrightarrow{j}\pi_1^D(\mathcal{S},\xi^*)\to\pi_1^{D'_0}(S_0,\xi_0)\to 1,$ the exact analogue for tame fundamental groups of the classical homotopy sequence of a circle-bundle over a tubular neighbourhood. In the curve examples of §7.3 this makes $\pi_1^D(\mathcal{S})$ concretely computable: for $S_0\cong\mathbb{P}^1$ with self-intersection number $d$, one obtains a cyclic group of order the prime-to-characteristic part of $d$.

The disjointness hypothesis deserves scrutiny. The non-central components $D_{i,0}$ are divisors on the scheme $S_0$, so they are automatically pairwise disjoint when $\dim S_0=1$ — the relative-curve case that drives the classical applications to moduli of curves, braid groups and tangential base points (base points of the fundamental group attached to formal directions along a divisor). When $\dim S_0\ge2$ — the first case of interest for three-dimensional bases such as wonderful compactifications of complements of hyperplane arrangements (smooth compactifications whose boundary divisor has normal crossings) — the components may meet, and the Kummer cover construction of §7.2, which prescribes the order of a global function along each $D_{i,0}$ separately, does not apply as written: at a crossing point the local prescriptions are coupled. What becomes of Theorem 7.3.1 without disjointness is exactly the open problem. For orientation: the complement-side theory — the tame fundamental group of $U=X\setminus D$ itself, isomorphic to a product of copies of $\widehat{\mathbb{Z}}'(1)$ semidirectly extended by $\pi_1(X)$ — is valid without any disjointness hypothesis and is widely used, and the modern reformulation in logarithmic geometry (a log scheme is a scheme equipped with a sheaf of commutative monoids recording the boundary) via Kummer étale covers of the log scheme attached to the pair $(X,D)$, together with the Fujiwara–Kato purity theorem (Zariski–Nagata purity for log regular log schemes, covered in Illusie's survey), likewise concerns the complement side. The question here concerns the formal-neighbourhood side, which is finer: $\pi_1^D(\mathcal{S},\xi^*)$ sees the self-intersection class of the special fibre and is the local invariant controlling divisorial inertia (the inertia subgroups attached to divisors) in specialisation arguments.

Problem Statement

Work in the setting of Grothendieck–Murre, Lecture Notes in Mathematics 208, §§5–7: $\mathcal{S}$ a connected locally noetherian regular formal scheme, $D_0,\dots,D_r$ regular divisors with normal crossings on $\mathcal{S}$ (locally defined by equations from regular systems of parameters, meeting transversally), $J(D_0)$ an ideal of definition, so that the closed fibre of $\mathcal{S}$ is $S_0$ and $D_{i,0}$ denotes the divisor cut out on $S_0$ by $D_i$; $U_0=S_0\setminus\bigcup_{i\neq0}D_{i,0}$ and $D'_0=\sum_{i\neq0}D_{i,0}$; $\xi^*$ a compatible geometric point of $\mathcal{S}$ over a geometric point $\xi_0$ of $U_0$. Let $j\colon\mathcal{M}^t(\xi_0)\to\pi_1^D(\mathcal{S},\xi^*)$ be the canonical surjection onto the kernel $K$ of $\pi_1^D(\mathcal{S},\xi^*)\to\pi_1^{D'_0}(S_0,\xi_0)$, where $\mathcal{M}^t(\xi_0)\cong\widehat{\mathbb{Z}}'(1)$ is the profinite tame inertia group of the normal direction (the product of the $\ell$-adic integers over the primes $\ell$ different from the residue characteristic, cyclotomically twisted); and let $\beta\colon\pi_2^{D'_0}(U_0,\xi_0)\to\mathcal{M}^t(\xi_0)$ be the homomorphism, determined by the Kummer (self-intersection) class $c'(D_0)\in H^2(U_0,\mathcal{M}^t)$ of $D_0$ (the class measuring the obstruction to extracting $n$-th roots of a local equation of $D_0$ on $U_0$), from the tame second homotopy group of the pair $(U_0,D'_0)$ (built from the universal tame cover of $U_0$). The book proves $\operatorname{Im}(\beta)=\ker(j)$, and hence the exact sequence of Theorem 7.3.1, only under the additional hypothesis that the non-central components $D_{i,0}$, $i\neq0$, are pairwise disjoint.

The problem is to remove the disjointness hypothesis. When the non-central components are allowed to meet — the case that arises as soon as $\dim S_0\ge2$ — determine $\ker(j)\subseteq\mathcal{M}^t(\xi_0)$ intrinsically. In particular: (a) decide whether $\operatorname{Im}(\beta)=\ker(j)$ still holds, i.e. whether $\ker(j)$ is still generated by the self-intersection class of $D_0$; and (b) if not, describe the additional classes in $\ker(j)$ (expected to be supported on the intersection strata of the non-central components) by an intrinsic, checkable construction, and give the resulting complete determination of the kernel $K$ and — via the general formula $k(e)=j_*(c'(D_0))$ of Theorem 7.1.3 — of the extension class in $H^2(U_0,\underline{K})$. Equivalently: compute the tame fundamental group $\pi_1^D(\mathcal{S},\xi^*)$ of the formal neighbourhood, together with its inertia map $j$, for an irreducible divisor whose complementary components intersect, thereby extending Theorem 7.3.1 to general normal-crossings divisors. An explicit example (for instance a three-dimensional regular formal scheme with two meeting non-central components) in which $\ker(j)$ strictly contains $\operatorname{Im}(\beta)$, together with an intrinsic description of the difference, is a special case; the intrinsic determination of $\ker(j)$ (equivalently of $K$) in general and the resulting extension class is the question.

References

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