Rings of Integers of Type K(π, 1)

Alexander Schmidt

Received: October 17, 2007

Communicated by Peter Schneider

Abstract. We investigate the Galois group G_{S}(p) of the maximal pextension unramified outside a finite set S of primes of a number field in the (tame) case, when no prime dividing p is in S. We show that the cohomology of G_{S}(p) is ‘often’ isomorphic to the étale cohomology of the scheme Spec(O_{k} ∖ S), in particular, G_{S}(p) is of cohomological dimension 2 then.

2000 Mathematics Subject Classification: 11R34, 12G10

Keywords and Phrases: Galois cohomology, restricted ramification, cohomological dimension

1 Introduction

We call a connected locally noetherian scheme Y a ‘K(π, 1)’ for a prime number p if the higher homotopy groups of the p-completion Y_{et}^{(p)} of its étale homotopy type Y_{et} vanish. In this paper we consider the case of an arithmetic curve, where the K(π, 1)-property is linked with open questions in the theory of Galois groups with restricted ramification of number fields: Let k be a number field, S a finite set of nonarchimedean primes of k and p a prime number. For simplicity, we assume that p is odd or that k is totally imaginary. By a p-extension we understand a Galois extension whose Galois group is a (pro-) p-group. Let k_{S}(p) denote the maximal p-extension of k unramified outside S and put G_{S}(p) = Gal(k_{S}(p)|k). A systematic study of this group had been started by Šafarevič, and was continued by Koch, Kuz’min, Wingberg and many others; see [NSW], VIII, §7 for basic properties of G_{S}(p). In geometric terms (and omitting the base point), we have

G_{S}(p) ≅ π_{1}((Spec(O_{k}) ∖ S)_{et}^{(p)}).

As is well known to the experts, if S contains the set S_{p} of primes dividing p, then Spec(O_{k}) ∖ S is a K(π, 1) for p (see Proposition 2.3 below). In particular, if S ⊃ S_{p}, then G_{S}(p) is of cohomological dimension less or equal to 2.

The group G_{S}(p) is most mysterious in the tame case, i.e. when S ∩S_{p} = ∅. In this case, examples when Spec(O_{k}) ∖ S is not a K(π, 1) are easily constructed. On the contrary, until recently not a single K(π, 1)-example was known. The following properties of the group G_{S}(p) were known so far:

• G_{S}(p) is a ‘fab-group’, i.e. U^{ab} is finite for each open subgroup U ⊂ G.

• G_{S}(p) can be infinite (Golod-Šafarevič).

• G_{S}(p) is a finitely presented pro-p-group (Koch). A conjecture of Fontaine and Mazur [FM] asserts that G_{S}(p) has no infinite p-adic analytic quotients.

In 2005, Labute considered the case k = ℚ and found finite sets S of prime numbers (called strictly circular sets) with p ∉ S such that G_{S}(p) has cohomological dimension 2. In [S1] the author showed that, in the examples given by Labute, Spec(ℤ) ∖ S is a K(π, 1) for p. The objective of this paper is a systematic study of the K(π, 1)-property. Our focus is on the tame case, where we conjecture that rings of integers of type K(π, 1) are cofinal in the following sense:

Conjecture 1. Let k be a number field and let p be a prime number. Assume that p ≠ 2 or that k is totally imaginary. Let S be a finite set of primes of k with S ∩S_{p} = ∅. Let, in addition, a set T of primes of Dirichlet density δ(T) = 1 be given. Then there exists a finite subset T_{1} ⊂ T such that Spec(O_{k}) ∖ (S ∪ T_{1}) is a K(π, 1) for p.

Of course we may assume that T ∩ S_{p} = ∅ in the conjecture. Our main result is the following

Theorem 1. Conjecture 1 is true if the number field k does not contain a primitive p-th root of unity and the class number of k is prime to p.

Explicit examples of rings of integers of type K(π, 1) can be found in [La], [S1] (for k = ℚ) and in [Vo] (for k imaginary quadratic).

The K(π, 1)-property has strong consequences. We write X = Spec(O_{k}) and assume in all results that p ≠ 2 or that k is totally imaginary. Primes 𝔭 ∈ S ∖ S_{p} with μ_{p} ⊄ k_{𝔭} are redundant in the sense that removing these primes from S does not change (X ∖ S)_{et}^{(p)}, see section 4. In the tame case, we may therefore restrict our considerations to sets of primes whose norms are congruent to 1 modulo p. These are the results.

Theorem 2. Let S be a finite non-empty set of primes of k whose norms are congruent to 1 modulo p. If X ∖ S is a K(π, 1) for p and G_{S}(p) ≠ 1, then the following hold.

(i) cd G_{S}(p) = 2, scd G_{S}(p) = 3.

(ii) G_{S}(p) is a duality group.

Rings of Integers of Type K(π, 1) 443

The dualizing module D of G_{S}(p) is given by D = tor_{p}C_{S}(k_{S}(p)), i.e. it is the subgroup of p-torsion elements in the S-idèle class group of k_{S}(p).

Remarks: 1. If X ∖ S is a K(π, 1) for p and G_{S}(p) = 1, then k is imaginary quadratic, #S = 1 and p = 2 or 3. See Proposition 7.4 for a more precise statement.

2. We have a natural exact sequence

0 → μ_{p^{∞}}(k_{S}(p)) → ⨁_{𝔭∈S} Ind _{Gal(k_{S}(p)|k)}^{G_{𝔭}(k_{S}(p)|k)} μ_{p^{∞}}(k_{S}(p)_{𝔭}) →

tor_{p}C_{S}(k_{S}(p)) → O_{k_{S}(p),S}^{×} ⊗ ℚ_{p}/ℤ_{p} → 0 ,

where O_{k_{S}(p),S}^{×} is the group of S-units of k_{S}(p) and μ_{p^{∞}}(K) denotes the group of all p-power roots of unity in a field K. Note that μ_{p^{∞}}(k_{S}(p)) is finite, while, by Theorem 3 below, for 𝔭 ∈ S the field k_{S}(p)_{𝔭} contains all p-power roots of unity.

3. In the wild case S ⊃ S_{p}, where X ∖ S is always a K(π, 1) for p, G_{S}(p) is of cohomological dimension 1 or 2. The strict cohomological dimension is conjecturally equal to 2 (=Leopoldt’s conjecture for each finite subextension of k in k_{S}(p)). In the wild case, G_{S}(p) is often, but not always a duality group, cf. [NSW] Prop. 10.7.13.

Allowing ramification at a prime 𝔭 does not mean that the ramification is realized globally. Therefore it is a natural and interesting question how far we get locally at the primes in S when going up to k_{S}(p). See [NSW] X, §3 for results in the wild case. In the tame case, we have the following

Theorem 3. Let S be a finite non-empty set of primes of k whose norms are congruent to 1 modulo p. If X ∖ S is a K(π, 1) for p and G_{S}(p) ≠ 1, then

k_{S}(p)_{𝔭} = k_{𝔭}(p)

for all primes 𝔭 ∈ S, i.e. k_{S}(p) realizes the maximal p-extension of the local field k_{𝔭}.

Remark: Under the assumptions of the theorem, let 𝔮 ∉ S. Then either 𝔮 splits completely in k_{S}(p), or k_{S}(p) realizes the maximal unramified p-extension k_{𝔮}^{nr}(p). We do not know whether the completely split case actually occurs.

The next result addresses the question of enlarging the set S without destroying the K(π, 1)-property.

Theorem 4. Let S′ be a finite non-empty set of primes of k whose norms are congruent to 1 modulo p and let S ⊂ S′ be a nonempty subset. Assume that X ∖ S is a K(π, 1) for p and that G_{S}(p) ≠ 1. If each 𝔮 ∈ S′ ∖ S does not split completely in k_{S}(p), then X ∖ S′ is a K(π, 1) for p. Furthermore, in this case, the arithmetic form of Riemann’s existence theorem holds: the natural homomorphism

∗_{𝔭∈S′\S(k_{S}(p))} T_{𝔭}(k_{S′}(p)|k_{S}(p)) ⟶ Gal(k_{S′}(p)|k_{S}(p))

is an isomorphism, i.e. Gal(k_{S′}(p)|k_{S}(p)) is the free pro-p product of a bundle of inertia groups.

Finally, we deduce a statement on universal norms of unit groups.

Theorem 5. Let S be a finite non-empty set of primes of k whose norms are congruent to 1 modulo p. Assume that X ∖ S is a K(π, 1) for p and that G_{S}(p) ≠ 1. Then

\underleftarrow{lim}_{K⊂k_{S}(p)} O_{K}^{×} ⊗ ℤ_{p} = 0 = \underleftarrow{lim}_{K⊂k_{S}(p)} O_{K,S}^{×} ⊗ ℤ_{p},

where K runs through all finite subextensions of k in k_{S}(p), O_{K}^{×} and O_{K,S}^{×} are the groups of units and S-units of K, respectively, and the transition maps are the norm maps.

The structure of this paper is as follows. First we give the necessary definitions and make some calculations of étale cohomology groups for which we couldn’t find an appropriate reference. In section 4, we deal with the first obstruction against the K(π, 1)-property, the h^{2}-defect. Then we recall Labute’s results on mild pro-p-groups, which we use in the proof of Theorem 1 given in section 6. In the last three sections we prove Theorems 2–5.

The author thanks Denis Vogel for valuable comments on a preliminary version of this paper.

2 First observations

We tacitly assume all schemes to be connected and omit base points from the notation. Let Y be a locally noetherian scheme and let p be a prime number. We denote by Y_{et}^{(p)} the p-completion of the étale homotopy type of Y, see [AM], [Fr]. By Ỹ(p) we denote the universal pro-p-covering of Y. The projection Ỹ(p) → Y is Galois with group π_{1}^{et}(Y)(p) = π_{1}(Y_{et}^{(p)}), cf. [AM], (3.7). Any discrete p-torsion π_{1}^{et}(Y)(p)-module M defines a locally constant sheaf on Y_{et}, which we denote by the same letter. The Hochschild-Serre spectral sequence defines natural homomorphisms

ϕ_{M,i} : H^{i}(π_{1}^{et}(Y)(p), M) ⟶ H_{et}^{i}(Y, M), i ≥ 0.

Since H_{et}^{1}(Ỹ(p), M) = 0, the map ϕ_{M,i} is an isomorphism for i = 0 and 1, and is injective for i = 2. For a pro-p-group G we denote by K(G, 1) the associated Eilenberg-MacLane space ([AM], (2.6)).

Proposition 2.1. The following conditions are equivalent:

(i) The classifying map Y_{et}^{(p)} ⟶ K(π_{1}^{et}(Y)(p), 1) is a weak equivalence.

(ii) π_{i}(Y_{et}^{(p)}) = 0 for all i ≥ 2.

(iii) H_{et}^{i}(Ỹ(p), ℤ/pℤ) = 0 for all i ≥ 1.

(iv) ϕ_{ℤ/pℤ,i} : H^{i}(π_{1}^{et}(Y)(p), ℤ/pℤ) ⟶ H_{et}^{i}(Y, ℤ/pℤ) is an isomorphism for all i ≥ 0.

(v) ϕ_{M,i} : H^{i}(π_{1}^{et}(Y)(p), M) ⟶ H_{et}^{i}(Y, M) is an isomorphism for all i ≥ 0 and any discrete p-torsion π_{1}^{et}(Y)(p)-module M.

Proof. The equivalences (i)⇔(ii)⇔(v) are the content of [AM], (4.3), (4.4). The equivalence (iii)⇔(iv) follows in a straightforward manner from the Hochschild-Serre spectral sequence. The implication (v)⇒(iv) is trivial. Assume that (iv) holds. As π_{1}^{et}(Y)(p) is a pro-p-group, any finite p-primary π_{1}^{et}(Y)(p)-module M has a composition series with graded pieces isomorphic to ℤ/pℤ with trivial π_{1}^{et}(Y)(p)-action ([NSW], Corollary 1.7.4). Hence, if M is finite, the five-lemma implies that ϕ_{M,i} is an isomorphism for all i. An arbitrary discrete p-primary π_{1}^{et}(Y)(p)-module is the filtered inductive limit of finite p-primary π_{1}^{et}(Y)(p)-modules. Since group cohomology ([NSW], Proposition 1.5.1) and étale cohomology ([AGV], VII, 3.3) commute with filtered inductive limits, ϕ_{M,i} is an isomorphism for all i and all discrete p-torsion π_{1}^{et}(Y)(p)-modules M. This implies (v) and completes the proof.

Definition. We say that Y is a ‘K(π, 1) for p’ if the equivalent conditions of Proposition 2.1 are satisfied.

Now let k be a number field, S a finite set of nonarchimedean primes of k and p a prime number. We put X = Spec(O_{k}). The following observation is straightforward.

Corollary 2.2. Let k′ be a finite subextension of k in k_{S}(p) and let X′ = Spec(O_{k′}), S′ = S(k′). Then the following are equivalent.

(i) X ∖ S is a K(π, 1) for p,

(ii) X′ ∖ S′ is a K(π, 1) for p.

Proof. Both schemes have the same universal pro-p-covering.

We denote by S_{p} and S_{∞} the set of primes of k dividing p and the set of archimedean primes of k, respectively. For a set S of primes (which may contain archimedean places), let k_{S}(p) be the maximal p-extension of k unramified outside S and G_{S}(p) = Gal(k_{S}(p)|k). For a finite set S of nonarchimedean primes of k we have the identification

π_{1}^{et}((X ∖ S)_{et}^{(p)}) = G_{S∪S_{∞}}(p).

If p is odd or k is totally imaginary, then G_{S}(p) = G_{S∪S_{∞}}(p). The following proposition is given for sake of completeness. It deals with the ‘wild’ case S ⊃ S_{p}, and is well known.

Proposition 2.3. If S contains S_{p}, then X ∖ S is a K(π, 1) for p.

Proof. We verify condition (v) of Proposition 2.1. Let k_{S∪S_{∞}} be the maximal extension of k unramified outside S ∪S_{∞} and put G_{S∪S_{∞}} = Gal(k_{S∪S_{∞}}|k). For any p-primary discrete G_{S∪S_{∞}}(p)-module M the homomorphism ϕ_{M,i} factors as

H^{i}(G_{S∪S_{∞}}(p), M) → H^{i}(G_{S∪S_{∞}}, M) → H_{et}^{i}(X ∖ S, M).

By [NSW], Cor. 10.4.8, the left map is an isomorphism. That also the right map is an isomorphism follows in a straightforward manner by using the Kummer sequence, the Principal Ideal Theorem and known properties of the Brauer group, see for example [Zi], Prop. 3.3.1. or [Mi], II Prop. 2.9.

Remark: If p = 2 and k has real places it is useful to work with the modified étale site defined by T. Zink [Zi], which takes the real archimedean places of k into account. Proposition 2.3 holds true also for the modified étale site, see [S2], Thm. 6.

3 Calculation of étale cohomology groups

As a basis of our investigations, we need the calculation of the étale cohomology groups of open subschemes of Spec(O_{k}) with values in the constant sheaf ℤ/pℤ. Let p be a fixed prime number. All cohomology groups are taken with respect to the constant sheaf ℤ/pℤ, which we omit from the notation. Furthermore, we use the notation

h^{i}(−) = dim_{𝔽_{p}} H_{et}^{i}(−) (= dim_{𝔽_{p}} H_{et}^{i}(−, ℤ/pℤ) )

for the occurring cohomology groups. We start with some well-known local computations.

Proposition 3.1. Let k be a nonarchimedean local field of characteristic zero and residue characteristic ℓ. Let X = Spec(O_{k}) and let x be the closed point of X. Then the étale local cohomology groups H_{x}^{i}(X) vanish for i ≤ 1 and i ≥ 4, and

h_{x}^{2}(X) = \{ \begin{matrix}δ, & if ℓ ≠ p, \\ δ + [k : ℚ_{p}], & if ℓ = p, \end{matrix}

where δ = 1 if μ_{p} ⊂ k and zero otherwise. Furthermore, h_{x}^{3}(X) = δ. In particular, we have the Euler-Poincaré characteristic formula

∑_{i=0}^{3}(−1)^{i}h_{x}^{i}(X) = \{ \begin{matrix}0, & if ℓ ≠ p, \\ [k : ℚ_{p}], & if ℓ = p. \end{matrix}

Proof. As X is henselian, we have isomorphisms H_{et}^{i}(X) ≅ H_{et}^{i}(x) for all i, and therefore

h^{i}(X) = \{ \begin{matrix}1 & for i = 0, 1, \\ 0 & for i ≥ 2. \end{matrix}

Furthermore, X ∖ {x} = Spec(k), hence H_{et}^{i}(X ∖ {x}) ≅ H^{i}(k). The local duality theorem (cf. [NSW], Theorem 7.2.15) shows h^{2}(X ∖ {x}) = δ, and by [NSW], Corollary 7.3.9, we have

h^{1}(X ∖ \{x\}) = \{ \begin{matrix}1 + δ & if ℓ ≠ p, \\ 1 + δ + [k : ℚ_{p}] & if ℓ = p. \end{matrix}

Furthermore, the natural homomorphism H_{et}^{1}(X) → H_{et}^{1}(X ∖ {x}) is injective. Therefore the result of the proposition follows from the exact excision sequence

⋯ → H_{x}^{i}(X) → H_{et}^{i}(X) → H_{et}^{i}(X ∖ {x}) → H_{x}^{i+1}(X) → · · · .

Now let k be a number field, S a finite set of nonarchimedean primes of k and X = Spec(O_{k}). We assume for simplicity that p is odd or that k is totally imaginary, so that we can ignore the archimedean places of k for cohomological considerations. We introduce the following notation

r_{1} the number of real places of k

r_{2} the number of complex places of k

r = r_{1} + r_{2}, the number of archimedean places of k S_{p} the set of places of k dividing p δ equal to 1 if μ_{p} ⊂ k and zero otherwise δ_{𝔭} equal to 1 if μ_{p} ⊂ k_{𝔭} and zero otherwise Cl(k) the ideal class group of k Cl_{S}(k) the S-ideal class group of k h_{k} = #Cl(k), the class number of k nA = ker(A →^{·n} A), where A is an abelian group and n ∈ ℕ A/n = coker(A →^{·n} A), where A is an abelian group and n ∈ ℕ.

Proposition 3.2. Assume that p ≠ 2 or that k is totally imaginary. Then H_{et}^{i}(X ∖ S) = 0 for i ≥ 4, and

χ(X ∖ S) := ∑_{i=0}^{3}(−1)^{i}h^{i}(X ∖ S) = r − ∑_{𝔭∈S∩S_{p}}[k_{𝔭} : ℚ_{p}] .

In particular,

χ(X ∖ S) = \{ \begin{matrix}r, & if S ∩ S_{p} = ∅ , \\ −r_{2}, & if S ⊃ S_{p} . \end{matrix}

Proof. The assertion for S = S_{p} is well known, see [Mi], II Theorem 2.13 (a). Consider the exact excision sequence

⋯ → ⨁_{𝔭∈S} H_{𝔭}^{i}(X_{𝔭}) → H_{et}^{i}(X) → H_{et}^{i}(X ∖ S) → ⨁_{𝔭∈S} H_{𝔭}^{i+1}(X_{𝔭}) → ⋯,

where X_{𝔭} = Spec(O_{k,𝔭}) is the spectrum of the completion of O_{k} at 𝔭. Using this excision sequence for S = S_{p}, Proposition 3.1 implies the result for S = ∅, noting that ∑_{𝔭∈S_{p}}[k_{𝔭} : ℚ_{p}] − r_{2} = [k : ℚ] − r_{2} = r. The result for arbitrary S follows from the case S = ∅, the above excision sequence and from Proposition 3.1.

In order to give formulae for the individual cohomology groups, we consider the Kummer group (cf. [NSW], VIII, §6)

V_{S}(k) := {a ∈ k^{×} | a ∈ k_{𝔭}^{×p} for 𝔭 ∈ S and a ∈ U_{𝔭}k_{𝔭}^{×p} for 𝔭 ∉ S}/k^{×p},

where U_{𝔭} denotes the unit group of the local field k_{𝔭} (convention: U_{𝔭} = k_{𝔭}^{×} if 𝔭 is archimedean).^{1} V_{S}(k) is a finite group. More precisely, we have the following

Proposition 3.3. There exists a natural exact sequence

0 ⟶ O_{k}^{×}/p ⟶ V_{∅}(k) ⟶ _{p}Cl(k) ⟶ 0 .

In particular,

dim_{𝔽_{p}} V_{∅}(k) = dim_{𝔽_{p}} _{p}Cl(k) + dim_{𝔽_{p}} O_{k}^{×}/p = dim_{𝔽_{p}} _{p}Cl(k) + r − 1 + δ.

If S is arbitrary and 𝔭 ∉ S is an additional prime of k, then we have a natural exact sequence

0 ⟶ V_{S∪{𝔭}}(k) ⟶^{ϕ} V_{S}(k) ⟶ U_{𝔭}k_{𝔭}^{×p}/k_{𝔭}^{×p}.

For 𝔭 ∉ S_{p}, we have dim_{𝔽_{p}} coker(ϕ) ≤ δ_{𝔭}.

Proof. Sending an a ∈ V_{∅}(k) to the class in Cl(k) of the fractional ideal 𝔞 with (a) = 𝔞^{p} yields a surjective homomorphism V_{∅}(k) → _{p}Cl(k) with kernel O_{k}^{×}/p. This, together with Dirichlet’s Unit Theorem, shows the first statement. The second exact sequence follows immediately from the definitions. There are natural isomorphisms

U_{𝔭}k_{𝔭}^{×p}/k_{𝔭}^{×p} ≅ U_{𝔭}/U_{𝔭} ∩ k_{𝔭}^{×p} = U_{𝔭}/U_{𝔭}^{p}.

For 𝔭 ∉ S_{p} we have dim_{𝔽_{p}} U_{𝔭}/U_{𝔭}^{p} = δ_{𝔭}, showing the last statement.

1In terms of flat cohomology, we have V_{S}(k) = ker (H_{fl}^{1} (X ∖ S, μ_{p}) → ⨁_{𝔭∈S} H^{1}(k_{𝔭}, μ_{p})).

Documenta Mathematica 12 (2007) 441–471

The étale cohomology groups of X ∖ S have the following dimensions.

Theorem 3.4. Let S be a finite set of nonarchimedean primes of k. Assume p ≠ 2 or that k is totally imaginary. Then H_{et}^{i}(X ∖ S) = 0 for i ≥ 4 and

\begin{aligned}h^{0}(X ∖ S) & = & 1 , \\ h^{1}(X ∖ S) & = & 1 + ∑_{𝔭∈S} δ_{𝔭} − δ + dim_{𝔽_{p}} V_{S} + ∑_{𝔭∈S∩S_{p}}[k_{𝔭} : ℚ_{p}] − r, \\ h^{2}(X ∖ S) & = & ∑_{𝔭∈S} δ_{𝔭} − δ + dim_{𝔽_{p}} V_{S} + θ , \\ h^{3}(X ∖ S) & = & θ . \end{aligned}

Here θ is equal to 1 if δ = 1 and S = ∅, and zero in all other cases.

Proof. The statement on h^{0} is trivial and the vanishing of H^{i} for i ≥ 4 was already part of Proposition 3.2. Artin-Verdier duality (see [Ma], 2.4 or [Mi], Theorem 3.1) shows

H_{et}^{3}(X)^{∨} ≅ Hom_{X}(ℤ/pℤ, 𝔾_{m}) = μ_{p}(k).

Consider the exact excision sequence

⨁_{𝔭∈S} H_{𝔭}^{3}(X_{𝔭}) →^{α} H_{et}^{3}(X) →^{β} H_{et}^{3}(X ∖ S)→ ⨁_{𝔭∈S} H_{𝔭}^{4}(X_{𝔭}).

By Proposition 3.1, the right hand group is zero, hence β is surjective. By the local duality theorem (see [Ma], 2.4, [Mi], II Corollary 1.10), the dual map to α is the natural inclusion

μ_{p}(k) → ⨁_{𝔭∈S} μ_{p}(k_{𝔭}),

which is injective, unless δ = 1 and S = ∅. Therefore h^{3}(X ∖ S) = 1 if δ = 1 and S = ∅, and zero otherwise. Using the isomorphism H^{1}(G_{S}(p)) →^{∼} H_{et}^{1}(X ∖ S), the statement on h^{1} follows from the corresponding formula for the first cohomology of G_{S}(p), see [NSW], Theorem 8.7.11. Finally, the result for h^{2} follows by using the Euler-Poincaré characteristic formula in Proposition 3.2.

Corollary 3.5. Assume that δ = 0 or S ≠ ∅. Then X ∖ S is a K(π, 1) for p if and only if the following conditions (i) and (ii) are satisfied.

(i) ϕ_{2} : H^{2}(G_{S}(p)) ↪ H_{et}^{2}(X ∖ S) is an isomorphism,

(ii) cd G_{S}(p) ≤ 2.

Proof. The given conditions are obviously necessary. Furthermore, ϕ_{0} and ϕ_{1} are isomorphisms and H_{et}^{i}(X ∖ S) = 0 for i ≥ 3 by Theorem 3.4. Therefore (i) and (ii) imply that ϕ_{i} is an isomorphism for all i. Hence condition (iv) of Proposition 2.1 is satisfied for X ∖ S and p.

Let F be a locally constant sheaf on (X ∖ S)_{et}. For each prime 𝔭 the composite map O_{k,S} → k → k_{𝔭} induces natural maps H_{et}^{i}(X ∖ S, F) → H^{i}(k_{𝔭}, F) for all i ≥ 0.

Definition. For any locally constant sheaf F on (X ∖ S)_{et} we put

� ^{i}(k, S, F) := ker (H_{et}^{i}(X ∖ S, F) ⟶ ∏_{𝔭∈S} H^{i}(k_{𝔭}, F)).

Assume a prime number p is fixed. Then we write � ^{i}(k, S) := � ^{i}(k, S, ℤ/pℤ) and, following historical notation, we put � _{S}(k) := V_{S}(k)^{∨}, where ∨ denotes the Pontryagin dual.

The next theorem is sharper than [NSW], Thm. 8.7.4, as the group � ^{2}(G_{S}), which was considered there, is a subgroup of � ^{2}(k, S). If p = 2 and k has real places, then Theorem 3.6 remains true after replacing étale cohomology with its modified version.

Theorem 3.6. Assume p ≠ 2 or that k is totally imaginary. Then there exists a natural isomorphism

� ^{2}(k, S) ⟶^{∼} � _{S}(k).

Proof. The proof of [NSW], Thm. 8.7.4 can be adapted to show also the stronger statement here. However, we decided to take the short cut by using flat duality. For any prime 𝔭 of k one easily computes the local cohomology groups for the flat topology as H_{fl ,𝔭}^{1}(X, μ_{p}) = 0 and H_{fl ,𝔭}^{2}(X, μ_{p}) ≅ k_{𝔭}^{×}/U_{𝔭}k_{𝔭}^{×p}. Therefore excision and Kummer theory imply an exact sequence

0 → H_{fl}^{1} (X, μ_{p}) → k^{×}/k^{×p} → ⨁_{𝔭} k_{𝔭}^{×}/U_{𝔭}k_{𝔭}^{×p}.

As H_{fl}^{1} (X_{𝔭}^{h}, μ_{p}) ≅ U_{𝔭}/p, we obtain the exact sequence

(∗)  0 → V_{S}(k) → H_{fl}^{1} (X, μ_{p}) → ⨁_{𝔭∈S} H_{fl}^{1} (X_{𝔭}^{h}, μ_{p}).

By excision, and noting H_{𝔭}^{3}(X, ℤ/pℤ) ≅ H^{2}(k_{𝔭}, ℤ/pℤ), we have an exact sequence

(∗∗)  ⨁_{𝔭∈S} H_{𝔭}^{2}(X, ℤ/pℤ) → H_{et}^{2}(X, ℤ/pℤ) → � ^{2}(k, S) → 0.

Comparing sequences (∗) and (∗∗) via local and global flat duality, we obtain the asserted isomorphism.

We provide the following lemma for further use.

Lemma 3.7. Let K ⊂ k_{S}(p) be an extension of k inside k_{S}(p) and let (X ∖ S)_{K} be the normalization of X ∖ S in K. If δ = 0, or S ≠ ∅ or K|k is infinite, then

H_{et}^{3}((X ∖ S)_{K}) = 0.

Proof. We denote the normalization of X ∖ S in any algebraic extension field k′ of k by (X ∖ S)_{k′}. Étale cohomology commutes with inverse limits of schemes if the transition maps are affine (see [AGV], VII, 5.8). Therefore

H^{3}((X ∖ S)_{K}) = \underrightarrow{lim}_{k′⊂K} H^{3}((X ∖ S)_{k′}),

where k′ runs through all finite subextensions of k in K. If δ = 0 or S ≠ ∅, then, by Theorem 3.4, H_{et}^{3}((X ∖ S)_{k′}) = 0 for all k′ and the limit is obviously zero. Assume δ = 1 and S = ∅. Then, by Artin-Verdier duality,

H_{et}^{3}(X_{k′}) ≅ μ_{p}(k′)^{∨}.

For k′ ⊂ k′′ ⊂ K, the transition map

H_{et}^{3}(X_{k′}) → H_{et}^{3}(X_{k′′})

is the dual of the norm map N_{k′′|k′}: μ_{p}(k′′) → μ_{p}(k′), hence the zero map if k′ ≠ k′′. As K|k is infinite, the limit vanishes.

4 Removing the h^{2}-defect

We start by extending the notions introduced before to infinite sets of primes S. Let k be a number field and S a set of nonarchimedean primes of k. We set X = Spec(O_{k}) and

X ∖ S = Spec(O_{k,S}),

which makes sense also if S is infinite. Let F be a sheaf on X ∖ S which comes by restriction from X ∖ T for some finite subset T ⊂ S. As each open subscheme of X is affine, we have

H_{et}^{i}(X ∖ S, F) ≅ \underrightarrow{lim}_{\begin{matrix}T ⊂S′⊂S \\ S′ finite \end{matrix}} H_{et}^{i}(X ∖ S′, F)

for all i ≥ 0.

We fix a prime number p and put the running assumption that k is totally imaginary if p = 2. Hence we may ignore archimedean primes for cohomological considerations. The notion of being a K(π, 1) for p extends in an obvious manner to the case when S is infinite. Also the isomorphism

� ^{2}(k, S) ⟶^{∼} � _{S}(k)

generalizes to infinite S by passing to the limit over all finite subsets S′ ⊂ S. In particular, � ^{2}(k, S) is finite.

For the remainder of this paper, we assume that S ∩ S_{p} = ∅. We also keep the running assumption p ≠ 2 or k is totally imaginary.

For shorter notation, we drop p wherever possible. We write G_{S} instead of G_{S}(p), k_{S} for k_{S}(p), and so on. Unless mentioned otherwise, all cohomology groups are taken with values in ℤ/pℤ. We keep this notational convention for the rest of this paper.

If 𝔭 ∤ p is a prime with μ_{p} ⊄ k_{𝔭}, then every p-extension of the local field k_{𝔭} is unramified (see [NSW], Proposition 7.5.1). Therefore primes 𝔭 ∉ S_{p} with N(𝔭) ≢ 1 mod p cannot ramify in a p-extension. Removing all these redundant primes from S, we obtain a subset S_{min} ⊂ S which has the property that G_{S} = G_{S_{min}}. Moreover, we have the

Lemma 4.1. The natural map

(X ∖ S)_{et}^{(p)} ⟶ (X ∖ S_{min})_{et}^{(p)}

is a weak homotopy equivalence.

Proof. By [AM], (4.3), it suffices to show that for every discrete p-primary G_{S}-module M the natural maps H_{et}^{i}(X ∖ S_{min}, M) → H_{et}^{i}(X ∖ S, M) are isomorphisms for all i. By the same argument, as in the proof of Proposition 2.1, (iv)⇒(v), we may suppose that M = ℤ/pℤ. Using the excision sequence, it therefore suffices to show that the group H_{𝔭}^{i}(X ∖ S_{min}, ℤ/pℤ) vanishes for all 𝔭 ∈ S ∖ S_{min}. This follows from Proposition 3.1.

Therefore we can replace S by S_{min} and make the following notational convention for the rest of this paper.

The word ‘prime’ means nonarchimedean prime with norm ≡ 1 mod p.

At this point it is useful to redefine the notion of Dirichlet density.

Definition. Let S be a set of primes of k (of norm ≡ 1 mod p). The p-density Δ^{p}(S) is defined by

Δ^{p}(S) = δ_{k(μ_{p})}(S(k(μ_{p}))),

where S(k(μ_{p})) is the set of prolongations of primes in S to k(μ_{p}) and δ_{k(μ_{p})} denotes the Dirichlet density on the level k(μ_{p}). In other words,

Δ^{p}(S) = d · δ_{k}(S), where d = [k(μ_{p}) : k].

The set of all primes (of norm ≡ 1 mod p) has p-density equal to 1.

Proposition 4.2. Let S be a set of primes of p-density Δ^{p}(S) = 1. Then there exists a finite subset T ⊂ S with � _{T} (k) = 0. In particular, � _{S}(k) = 0 = � ^{2}(k, S).

Proof. By the Hasse principle for the module μ_{p}, see [NSW], Thm. 9.1.3 (ii), and Kummer theory, the natural map

k^{×}/k^{×p} ⟶ ∏_{𝔭∈S} k_{𝔭}^{×}/k_{𝔭}^{×p}

is injective, hence V_{S}(k) = 0. Furthermore V_{∅}(k) is finite. Choosing to each nonzero element α of V_{∅}(k) a prime 𝔭 ∈ S with α ∉ k_{𝔭}^{×p}, we obtain a finite subset T ⊂ S with V_{T} (k) = 0.

Theorem 4.3. Let k be a number field and let S be a set of primes of k of p-density Δ^{p}(S) = 1. Then X ∖ S is a K(π, 1) for p.

Proof. Let T ⊂ S be a finite subset. By [NSW], Thm. 9.2.2 (ii), the natural map

H_{et}^{1}(X ∖ (S ∪ S_{p})) ⟶ ∏_{𝔭∈T ∪S_{p}} H^{1}(k_{𝔭})

is surjective. A class in H_{et}^{1}(X ∖ (S ∪ S_{p})) which maps to zero in H^{1}(k_{𝔭}) for all 𝔭 | p is contained in H_{et}^{1}(X ∖ S). Therefore, also the map

H_{et}^{1}(X ∖ S) ⟶ ∏_{𝔭∈T} H^{1}(k_{𝔭})

is surjective. Hence the maximal elementary abelian extension of k in k_{S} realizes the maximal elementary abelian extension of k_{𝔭} in k_{𝔭}(p) for all 𝔭 ∈ S. Applying the same argument to each finite subextension of k in k_{S}, we conclude that k_{S} realizes k_{𝔭}(p) for all 𝔭 ∈ S. In particular,

∏_{𝔭∈S(k_{S})} H^{2}((k_{S})_{𝔭}) = 0.

Furthermore, by Proposition 4.2, � ^{2}(K, S(K)) = 0 for all finite subextensions K of k in k_{S}. We obtain

H_{et}^{2}((X ∖ S)_{k_{S}}) = 0.

As there is no cohomology in dimension greater or equal 3, condition (iii) of Proposition 2.1 is satisfied.

In order to proceed, we make the following definitions.

Definition. Let S be a finite set of primes (of norm ≡ 1 mod p).

(i) We say that S is p-large if � _{S}(k, p) = 0.

(ii) We put

δ_{S}^{2}(k) = h^{2}(X ∖ S) − h^{2}(G_{S})

and call this number the h^{2}-defect of S (with respect to p).

(iii) We denote by k_{S}^{el} the maximal elementary abelian p-extension of k inside k_{S}.

If S is p-large, then � ^{2}(k, S) = 0, and so, for any set T ⊃ S, the natural maps H^{2}(G_{S}) → H^{2}(G_{T}) and H_{et}^{2}(X ∖ S) → H_{et}^{2}(X ∖ T) are injective.

Lemma 4.4. Let S be p-large and let 𝔭 be a prime (of norm ≡ 1 mod p) which does not split completely in k_{S}^{el}|k. Then

δ_{S∪{𝔭}}^{2}(k) ≤ δ_{S}^{2}(k).

Furthermore, the natural map H^{2}(G_{S∪{𝔭}}) ⟶ H^{2}(k_{𝔭}) is surjective.

Proof. Put S′ = S ∪ {𝔭}. By Theorem 3.4, the extension k_{S′}^{el}|k is ramified at 𝔭. Therefore k_{S′}^{el} realizes the maximal elementary abelian p-extension k_{𝔭}^{el} of the local field k_{𝔭}, i.e. the map

H^{1}(G_{S′}(p)) ⟶ H^{1}(k_{𝔭})

is surjective. As the cup-product H^{1}(k_{𝔭}) × H^{1}(k_{𝔭}) → H^{2}(k_{𝔭}) is surjective, the natural map

H^{2}(G_{S′}) ⟶ H^{2}(k_{𝔭})

is also surjective. The statement of the lemma now follows from the commutative and exact diagram

0 ⟶ H^{2}(G_{S}) ⟶ H_{et}^{2}(X ∖ S)
[figure: ∩; ∩]
0 ⟶ H^{2}(G_{S′}) ⟶ H_{et}^{2}(X ∖ S′)
↓ ↓
H^{2}(k_{𝔭}).

Lemma 4.5. Let S be p-large and let 𝔭 be a prime. Let T be a set of primes of p-density Δ^{p}(T) = 1. Then there exists a prime 𝔭′ ∈ T such that

(i) 𝔭′ does not split completely in k_{S}^{el}|k.

(ii) 𝔭 does not split completely in k_{S∪{𝔭′}}^{el}|k.

In particular, δ_{S∪{𝔭,𝔭′}}^{2}(k) ≤ δ_{S}^{2}(k).

Proof. If 𝔭 does not split completely in k_{S}^{el}|k, then condition (ii) is void. By assumption, Δ^{p}(T) = δ_{k(μ_{p})}(T (k(μ_{p})) = 1. By Čebotarev’s density theorem, we can find a prime 𝔓′ ∈ T (k(μ_{p})) which does not split completely in k_{S}^{el}(μ_{p})|k(μ_{p}). Then 𝔭′ = 𝔓′|_{k} satisfies (i). Therefore we may assume that 𝔭 splits completely in k_{S}^{el}|k. By class field theory, there exists an s ∈ k^{×} with (a) v_{𝔭}(s) ≡ 1 mod p,

(b) v_{𝔮}(s) ≡ 0 mod p for all 𝔮 ∉ S, 𝔮 ≠ 𝔭, and

(c) s ∈ k_{𝔮}^{×p} for all 𝔮 ∈ S. Since S is p-large, s is well-defined as an element in k^{×}/k^{×p}. Now consider the extensions k(μ_{p}, \sqrt[p]{s}) and k_{S∪\{𝔭\}}^{el}(μ_{p}) of k(μ_{p}). The first one might be contained in the second (only if ζ_{p} ∈ k) but this does not matter. Using Čebotarev’s density theorem, we find 𝔓′ ∈ T (k(μ_{p})) such that Frob_{𝔓′} is nonzero in Gal(k(μ_{p}, \sqrt[p]{s})|k(μ_{p})) and non-zero in Gal(k_{S∪\{𝔭\}}^{el}(μ_{p})|k(μ_{p})). We put 𝔭′ = 𝔓′|_{k}. Then 𝔭′ does not split completely in k_{S}^{el}|k and s ∉ k_{𝔭′}^{×p} = k(μ_{p})_{𝔓′}^{×p}. We claim that 𝔭 does not split completely in k_{S∪{𝔭′}}^{el}|k: Otherwise there would exist a t ∈ k^{×} satisfying conditions (a) – (c) above and with t ∈ k_{𝔭′}^{×p}. Since s/t ∈ � _{S}(k) = 0, we obtain s/t ∈ k^{×p}. Hence s ∈ k_{𝔭′}^{×p} giving a contradiction. Hence condition (i) and (ii) are satisfied.

Lemma 4.6. Let S be a finite set of primes and let T be a set of primes of p-density Δ^{p}(T) = 1. Then there exists a finite subset T_{1} ⊂ T such that S ∪ T_{1} is p-large and such that the natural inclusion

H^{2}(G_{S∪T_{1}}(k)) ⟶ H_{et}^{2}(X ∖ (S ∪ T_{1}))

is an isomorphism.

Proof. We first move finitely many primes from T to S to achieve that S is p-large. If δ_{S}^{2}(k) is zero, we are ready. Otherwise, consider the commutative diagram

H^{2}(G_{S}) ⟶  H^{2}(G_{S∪T})
∩  ↓^{≀}
H_{et}^{2}(X ∖ S) ⟶ H_{et}^{2}(X ∖ (S ∪ T))

in which the right hand isomorphism follows from Theorem 4.3. Let x ∈ H_{et}^{2}(X ∖ S) but x ∉ H^{2}(G_{S}). Then there exists a finite subset T_{0} ⊂ T such that x ∈ H^{2}(G_{S∪T_{0}}). Let T_{0} = {𝔭_{1}, …, 𝔭_{n}}. We choose 𝔭′_{1}, …, 𝔭′_{n} ∈ T according to Lemma 4.5 and put T_{1} = {𝔭_{1}, …, 𝔭_{n}, 𝔭′_{1}, …, 𝔭′_{n}}. Then the natural map

H^{2}(G_{S∪T_{1}}) ⟶^{ϕ} ∏_{i=1}^{n} H^{2}(k_{𝔭_{i}}) × ∏_{i=1}^{n} H^{2}(k_{𝔭′_{i}})

is surjective. We have H^{2}(G_{S}) ⊂ ker(ϕ) and also x ∈ ker(ϕ). Hence δ_{S∪T_{1}}^{2}(k) < δ_{S}^{2}(k). Iterating this process, we obtain a set with trivial h^{2}-defect.

5 Review of mild pro-p-groups

In the following we recall definitions and results from J. Labute’s paper [La]. Only interested in our application, we are slightly less general than Labute.

Let R be a principal ideal domain and let L be the free R-Lie algebra over ξ_{1}, …, ξ_{n}, n ≥ 1. We view L as graded algebra where the degree of ξ_{i} is 1. Let ρ_{1}, …, ρ_{m}, m ≥ 1, be homogeneous elements in L with ρ_{i} of degree h_{i} and let 𝔯 = (ρ_{1}, …, ρ_{m}) be the ideal of L generated by ρ_{1}, …, ρ_{m}. Let 𝔤 = L/𝔯 and U_{𝔤} be the universal enveloping algebra of 𝔤. Then M = 𝔯/[𝔯, 𝔯] is a U_{𝔤}-module via the adjoint representation.

Definition. The sequence ρ_{1}, …, ρ_{m} is called strongly free if U_{𝔤} is a free R-module and M = 𝔯/[𝔯, 𝔯] is the free U_{𝔤}-module on the images of ρ_{1}, …, ρ_{m} in M.

Let us consider the special case when R = k[π] is the polynomial ring in one variable π over a field k. Then L̄ = L/π is a free k-Lie algebra and the next theorem shows that we can detect strong freeness by reduction. We denote the image in L̄ of an element ρ ∈ L by ρ̄.

Theorem 5.1. ([La], Th. 3.10) The sequence ρ_{1}, …, ρ_{m} in L is strongly free if and only if the sequence ρ̄_{1}, …, ρ̄_{m} is strongly free in L̄.

Over fields, we have the following criterion for strong freeness. Let R = k be a field, X a finite set and S ⊂ X a subset. Let L(X) be the free Lie algebra over X and let 𝔞 be the ideal of L(X) generated by the elements ξ ∈ X ∖ S. Put

T = { [ξ, ξ′] | ξ ∈ X ∖ S, ξ′ ∈ S} ⊂ 𝔞.

Theorem 5.2. ([La], Th. 3.3, Cor. 3.5) If ρ_{1}, …, ρ_{m} are homogeneous elements of 𝔞 which lie in the k-span of T modulo [𝔞, 𝔞] and which are linearly independent over k modulo [𝔞, 𝔞] then the sequence ρ_{1}, …, ρ_{m} is strongly free in L.

Let p be an odd prime number and let G be a pro-p-group. We consider the descending p-central series (G_{n})_{n≥1}, which is defined recursively by

G_{1} = G, G_{n+1} = G_{n}^{p}[G, G_{n}].

The quotients gr_{n}(G) = G_{n}/G_{n+1}, denoted additively, are 𝔽_{p}-vector spaces. The graded vector space

gr(G) = ⨁_{n≥1} gr_{n}(G)

has a Lie algebra structure over the polynomial ring 𝔽_{p}[π], where multiplication by π is induced by x ↦ x^{p} and the bracket operation for homogeneous elements is induced by the commutator operation in G, see [NSW], III, §8. For g ∈ G, g ≠ 1, let the natural number h(g) be defined by

g ∈ G_{h(G)}, g ∉ G_{h(G)+1}.

This definition makes sense because ⋂_{n} G_{n} = {1}, see [NSW], Prop. 3.8.2. The image ω(g) of g in gr_{h(g)}(G) is called the initial form of g.

Let F be the free pro-p-group over elements x_{1}, …, x_{n}, n ≥ 1. Then h(x_{i}) = 1, i = 1, …, n, and

L = gr(F)

is the free 𝔽_{p}[π]-Lie algebra over ξ_{1}, …, ξ_{n}, where ξ_{i} = ω(x_{i}), i = 1, …, n, see [Lz]. Let r_{1}, …, r_{m}, m ≥ 1, be a sequence of elements in F_{2} = F^{p}[F, F] ⊂ F and let R = (r_{1}, …, r_{m})_{F} be the closed normal subgroup of F generated by r_{1}, …, r_{m}. Put ρ_{i} = ω(r_{i}) ∈ L.

Definition. A pro-p-group G is called mild if there exists a presentation

1 ⟶ R ⟶ F ⟶ G ⟶ 1

with F a free pro-p-group on generators x_{1}, …, x_{n} and R = (r_{1}, …, r_{m})_{F} such that the associated sequence ρ_{1}, …, ρ_{m} is strongly free in L = gr(F).

Essential for our application is the following property of mild pro-p-groups.

Theorem 5.3. ([La], Th. 1.2(c)) If G is a mild pro-p-group, then cd G = 2.

Now let G be a finitely presented pro-p-group and let

1 ⟶ R ⟶ F ⟶ G ⟶ 1

be a minimal presentation, i.e. F is the free pro-p-group on generators x_{1}, …, x_{n} where n = dim_{𝔽_{p}} H^{1}(G) and R = (r_{1}, …, r_{m})_{F} with m = dim_{𝔽_{p}} H^{2}(G), cf. [NSW], (3.9.5). Then the images ξ_{i} = ω(x_{i}), i = 1, …, n, of x_{1}, …, x_{n} are a basis of the 𝔽_{p}-vector space F/F_{2} = H_{1}(F) = H_{1}(G) = G/G_{2}. For y ∈ F_{n} and a ∈ ℤ_{p} the class of y^{a} modulo F_{n+1} only depends on the residue class ā ∈ 𝔽_{p} of a. Every element r ∈ R ⊂ F_{2} has a representation

r ≡ ∏_{j=1}^{n}(x_{j}^{p})^{a_{j}} · ∏_{1≤k<l≤n}[x_{k}, x_{l}]^{a_{kl}} mod F_{3},

where a_{j}, a_{k,l} ∈ 𝔽_{p}. These coefficients are uniquely defined and can be calculated as follows. As F is free, we have an isomorphism H_{2}(G) = R_{G}^{ab}/p. Let r̄ ∈ H_{2}(G) be the image of r and let χ_{1}, …, χ_{n} ∈ H^{1}(G) be the dual 𝔽_{p}-basis of ξ_{1}, …, ξ_{n}.

Theorem 5.4. a_{kl} = −r̄(χ_{k} ∪ χ_{l}) for k < l.

For a proof see [NSW], Prop. 3.9.13, which also gives a description of the a_{j} using the Bockstein operator.

Using the results above, we obtain a criterion for mildness.

Theorem 5.5. Let G be a finitely presented pro-p-group. Assume there exists a basis χ_{1}, …, χ_{n} of H^{1}(G), a basis r̄_{1}, …, r̄_{m} of H_{2}(G) and a natural number a, 1 ≤ a < n, such that the following conditions are satisfied

(i) r̄_{i}(χ_{k} ∪ χ_{l}) = 0 for a < k < l ≤ n and i = 1, …, m.

(ii) The m×a(n − a) matrix

(r̄_{i}(χ_{k} ∪ χ_{l}))_{i,(k,l)}, 1 ≤ i ≤ m, 1 ≤ k ≤ a < l ≤ n

has rank m.

Then G is a mild pro-p-group.

Proof. Let ξ_{1}, …, ξ_{n} ∈ H_{1}(G) be the dual basis of χ_{1}, …, χ_{n}. We choose a minimal presentation

(∗)  1 ⟶ R ⟶ F ⟶ G ⟶ 1

and generators x_{1}, …, x_{n} ∈ F mapping to ξ_{1}, …, ξ_{n} ∈ H_{1}(F) = H_{1}(G). Then we choose elements r_{1}, …, r_{m} ∈ R mapping to r̄_{1}, …, r̄_{m} ∈ R_{G}^{ab}/p = H_{2}(G). Let X = {ξ_{1}, …, ξ_{n}}. Then L = gr(F) is the free 𝔽_{p}[π]-Lie algebra over X and L̄ = L/π is the free 𝔽_{p}-Lie algebra over X. In order to show that G is mild, we have to show that the initial forms ρ_{1}, …, ρ_{m} of r_{1}, …, r_{m} are a strongly free sequence in L. By Theorem 5.1 it suffices to show that ρ̄_{1}, …, ρ̄_{m} ⊂ L̄ are a strongly free sequence. By condition (ii) and Theorem 5.4, we have ρ̄_{1}, …, ρ̄_{m} ∈ gr_{2}(L̄) = F_{2}/F_{3}F^{p}. Now we apply Theorem 5.2 with S = {ξ_{a+1}, …, ξ_{n}} ⊂ X. In the notation of this theorem, 𝔞 is the ideal generated by ξ_{1}, …, ξ_{a} in L̄ and

T = {[ξ_{i}, ξ_{j}] | 1 ≤ i ≤ a, a + 1 ≤ j ≤ n}.

By condition (i) and Theorem 5.4, we have ρ̄_{i} in the 𝔽_{p}-span H of T modulo [𝔞, 𝔞]. The elements of T are a basis of H and the coefficient matrix of ρ̄_{1}, …, ρ̄_{m} is up to sign the transpose of the matrix written in condition (ii). Hence ρ̄_{1}, …, ρ̄_{m} are linearly independent and, by Theorem 5.2, a strongly free sequence. This concludes the proof.

6 Existence of K(π, 1)’s

Let k be a number field and let p be a prime number with μ_{p} ⊄ k and assume that Cl(k)(p) = 0. The exact sequence

0 ⟶ O_{k}^{×} ⟶ k^{×} ⟶^{(v_{𝔮})_{𝔮}} ⨁_{𝔮} ℤ ⟶ Cl(k) ⟶ 0

implies the exactness of

0 ⟶ O_{k}^{×}/p ⟶ k^{×}/k^{×p} ⟶ ⨁_{𝔮} ℤ/pℤ ⟶ 0.

Let S = {𝔭_{1}, …, 𝔭_{m}} be a finite set of primes of norm ≡ 1 mod p. We choose for i = 1, …, m elements s_{i} ∈ k^{×}/k^{×p} with v_{𝔭_{i}}(s_{i}) ≡ 1 mod p and v_{𝔮}(s_{i}) ≡ 0 mod p for all primes 𝔮 ≠ 𝔭_{i} of k. Let furthermore, e_{1}, …, e_{r}, r = r_{1} + r_{2} − 1, be a basis of O_{k}^{×}/p.

Consider the field

K = k(μ_{p}, \sqrt[p]{s_{1}}, …, \sqrt[p]{s_{m}}, \sqrt[p]{e_{1}}, …, \sqrt[p]{e_{r}}).

An inspection of the ramification behaviour shows that Gal(K|k(μ_{p})) has the Galois group (ℤ/pℤ)^{m+r}: Indeed, k(μ_{p}, \sqrt[p]{e_{1}}, …, \sqrt[p]{e_{r}})|k(μ_{p}) is unramified outside S_{p} and has Galois group (ℤ/pℤ)^{r} by Kummer theory. Adjoining \sqrt[p]{s_{i}}, i = 1, …, m, yields a cyclic extension of degree p which is unramified outside S_{p} ∪ {𝔭_{i}} and ramified at 𝔭_{i}. Since μ_{p} ⊄ k, the extensions k_{S}^{el}(μ_{p})|k(μ_{p}) and K|k(μ_{p}) lie in different eigenspaces for the action of Gal(k(μ_{p})|k). Therefore Kk_{S}^{el}|k(μ_{p}) has Galois group (ℤ/pℤ)^{m+r+n}, with n = dim_{𝔽_{p}} Gal(k_{S}^{el}|k) = dim_{𝔽_{p}} H^{1}(G_{S}).

Assume now that we are given

• a set of primes T of k with T ∩ S = ∅ and with p-density Δ_{k}^{p}(T) = 1,

• a nonzero element F ∈ Gal(k_{S}^{el}|k) = Gal(k_{S}^{el}(μ_{p})|k(μ_{p})),

• to each 𝔭_{i}, i = 1, …, m, a condition C_{i} which says “split” or “inert”.

By Čebotarev’s density theorem applied to the extension Kk_{S}^{el}(μ_{p})|k(μ_{p}), we find a prime 𝔓 ∈ T (Kk_{S}^{el}(μ_{p})) such that

• the image of Frob_{𝔓} in Gal(k(μ_{p}, \sqrt[p]{e_{1}}, …, \sqrt[p]{e_{r}})|k(μ_{p})) is trivial,

• the image of Frob_{𝔓} in Gal(k(μ_{p}, \sqrt[p]{s_{i}})|k(μ_{p})) is trivial if C_{i} is “split” and nontrivial otherwise, and

• the image of Frob_{𝔓} in Gal(k_{S}^{el}(μ_{p})|k(μ_{p})) is equal to F.

Let 𝔭 ∈ T be the restriction of 𝔓 to k. Then the natural map O_{k}^{×}/p → k_{𝔭}^{×}/k_{𝔭}^{×p} is the zero map. Since _{p}Cl(k) = 0, we obtain O_{k}^{×}/p →^{∼} V_{∅}(k) = V_{{𝔭}}(k). By Theorem 3.4, k_{{𝔭}}^{el}|k is cyclic of order p and 𝔭 is ramified in this extension. Recall that H_{nr}^{1}(G_{𝔭}) is defined as the exact annihilator of the inertia group T_{𝔭}(k_{𝔭}^{el}|k_{𝔭}) ⊂ H_{1}(G_{𝔭}) in the natural pairing

H_{1}(G_{𝔭}) × H^{1}(G_{𝔭}) ⟶ 𝔽_{p}.

Dually, T_{𝔭}(k_{𝔭}^{el}|k_{𝔭}) is the exact annihilator of H_{nr}^{1}(G_{𝔭}). The equation T_{𝔭}(k_{{𝔭}}^{el}|k) = Gal(k_{{𝔭}}^{el}|k) yields an isomorphism

H^{1}(G_{{𝔭}}) ⟶^{∼} H^{1}(G_{𝔭})/H_{nr}^{1}(G_{𝔭}).

By class field theory, 𝔭_{i} splits in k_{{𝔭}}^{el}|k if and only if there exists an element s′_{i} ∈ k^{×}/k^{×p} with v_{𝔭_{i}}(s′_{i}) ≡ 1 mod p, v_{𝔮}(s′_{i}) ≡ 0 mod p for all 𝔮 ≠ 𝔭_{i} and s′_{i} ∈ k_{𝔭}^{×p}. Then s′_{i}/s_{i} lies in O_{k}^{×}/p, and therefore s_{i} ∈ k_{𝔭}^{×p}. Hence 𝔭_{i} splits in k_{{𝔭}}^{el}|k if and only if s_{i} is a p-th power in k_{𝔭}. On the other hand, by our choice of 𝔓, s_{i} is a p-th power in k_{𝔭} if and only if C_{i} is “split”. Therefore the following holds:

• the natural map O_{k}^{×}/p → k_{𝔭}^{×}/k_{𝔭}^{×p} is the zero map,

• Frob_{𝔭} = F ∈ Gal(k_{S}^{el}|k),

• k_{{𝔭}}^{el}|k is cyclic of order p,

• each 𝔭_{i}, i = 1, …, m, satisfies condition C_{i} in k_{{𝔭}}^{el}|k.

Now assume that � _{S ∖ {𝔮}}(k) = 0 for all 𝔮 ∈ S, in particular, S is p-large. Then all 𝔭_{i} ∈ S ramify in k_{S}^{el}|k and the 1-dimensional subspaces T_{𝔭_{i}}(k_{S}^{el}|k), i = 1, …, m, in H_{1}(G_{S}) are pairwise different and generate H_{1}(G_{S}). Furthermore assume that δ_{S}^{2}(k) = 0. As 𝔭 does not split completely in k_{S}^{el}|k, Lemma 4.4 implies δ_{S∪{𝔭}}^{2}(k) = 0. Since μ_{p} ⊄ k and by Theorem 3.4, the natural maps H^{2}(G_{S}) → ∏_{𝔮∈S} H^{2}(G_{𝔮}) and H^{2}(G_{S∪{𝔭}}) → ∏_{𝔮∈S∪{𝔭}} H^{2}(G_{𝔮}) are isomorphisms. We denote the 𝔮-component of a global cohomology class α by α_{𝔮}.

Next we fix a primitive p-th root of unity in k(μ_{p}) and to each 𝔭_{1}, …, 𝔭_{m} a prolongation to k(μ_{p}). After this choice we have identifications μ_{p}(k_{𝔭}) = μ_{p}((Kk_{S}^{el})_{𝔓}) ≅ 𝔽_{p} and μ_{p}(k_{𝔭_{i}}) ≅ 𝔽_{p}, i = 1, …, m. In particular, we have an isomorphism H^{2}(G_{𝔭}) = H^{2}(G_{𝔭}, μ_{p}) = 𝔽_{p}, and similarly for the 𝔭_{i}. Via these isomorphisms we consider the 𝔮-component α_{𝔮} of a class α ∈ H^{2}(G_{S∪{𝔭}}) as an element in 𝔽_{p}. Let

π_{𝔭} ∈ H^{1}(G_{𝔭})/H_{nr}^{1}(G_{𝔭}) = H^{1}(G_{𝔭}, μ_{p})/H_{nr}^{1}(G_{𝔭}, μ_{p}) = k_{𝔭}^{×}/U_{𝔭}k_{𝔭}^{×p}

be the image of a uniformizer and let χ_{𝔭} ∈ H^{1}(G_{{𝔭}}) be the unique pre-image. We denote the image of χ_{𝔭} in H^{1}(G_{S∪{𝔭}}) by the same letter. Thus χ_{𝔭} maps to π_{𝔭} under the natural map H^{1}(G_{S∪{𝔭}}) → H^{1}(G_{𝔭})/H_{nr}^{1}(G_{𝔭}). Consider the exact pairing

H_{nr}^{1}(G_{𝔭}) × H^{1}(G_{𝔭})/H_{nr}^{1}(G_{𝔭}) → H^{2}(G_{𝔭}) = 𝔽_{p},

which is induced by local Tate duality, see [NSW], Thm. 7.2.15. Let δ : k_{𝔭}^{×}/k_{𝔭}^{×p} →^{∼} H^{1}(G_{𝔭}) be the boundary isomorphism of the Kummer sequence and let rec : k_{𝔭}^{×}/k_{𝔭}^{×p} →^{∼} H_{1}(G_{𝔭}) be the mod-p reciprocity map. Put φ = rec ∘ δ^{−1}. Then the image of χ_{𝔭} under the composition

H^{1}(G_{S}) ⟶ H^{1}(G_{𝔭}) ⟶^{ϕ} H_{1}(G_{𝔭}) ⟶ H_{1}(G_{S})

is Frob_{𝔭}, the Frobenius automorphism of the unramified prime 𝔭 in k_{S}^{el}|k. By [NSW], Prop. 7.2.13 ^{2}), the diagram

H^{1}(G_{𝔭}) × H^{1}(G_{𝔭}) ⟶^{∪} H^{2}(G_{𝔭})
∥  ↓_{≀ φ}  ↓_{≀ inv}
H^{1}(G_{𝔭}) × H_{1}(G_{𝔭}) ⟶^{can}  𝔽_{p}

2This proposition contains a sign error, see the errata file on the author’s homepage.

Documenta Mathematica 12 (2007) 441–471

commutes. We obtain for any χ ∈ H^{1}(G_{S}) ⊂ H^{1}(G_{S∪{𝔭}}) the following formula for the 𝔭-component of χ ∪ χ_{𝔭} ∈ H^{2}(G_{S∪{𝔭}}):

(χ ∪ χ_{𝔭})_{𝔭} = χ(Frob_{𝔭}).

The image of χ_{𝔭} in H^{1}(G_{𝔭_{i}}) obviously lies in the subgroup H_{nr}^{1}(G_{𝔭_{i}}). By the same argument, noting that the cup-product is anti-symmetric, we obtain the equality

(χ ∪ χ_{𝔭})_{𝔭_{i}} = −χ_{𝔭}(Frob_{𝔭_{i}}),

for any χ ∈ H^{1}(G_{S}) mapping to π_{𝔭_{i}} ∈ H^{1}(G_{𝔭_{i}})/H_{nr}^{1}(G_{𝔭_{i}}), where Frob_{𝔭_{i}} is the Frobenius automorphism of the unramified prime 𝔭_{i} in k_{{𝔭}}^{el}|k. As χ_{𝔭} is unramified at 𝔭_{i}, the element (χ ∪ χ_{𝔭})_{𝔭_{i}} depends only on the image of χ in the one-dimensional 𝔽_{p}-vector space H^{1}(G_{𝔭_{i}})/H_{nr}^{1}(G_{𝔭_{i}}). Since 𝔭_{i} ramifies in k_{S}^{el}|k, the map H^{1}(G_{S}) → 𝔽_{p}, χ ↦ (χ ∪ χ_{𝔭})_{𝔭_{i}} is the linear form associated to an element t_{i} ∈ T_{𝔭_{i}}(k_{S}^{el}|k) ⊂ H_{1}(G_{S}).

Summing up and using the notation and choices above, we obtain the

Lemma 6.1. Let k be a number field and let p be a prime number with μ_{p} ⊄ k and Cl(k)(p) = 0. Let S = {𝔭_{1}, …, 𝔭_{m}} be a finite p-large set of primes and assume δ_{S}^{2}(k) = 0. Let for i = 1, …, m elements ε_{i} ∈ {0, 1} and for i = 1, …, n elements d_{i} ∈ 𝔽_{p} be given, where not all d_{i} are zero. Let χ_{1}, …, χ_{n} be a basis of H^{1}(G_{S}). Furthermore, let T be a set of primes of p-density Δ_{p}(T) = 1 and with T ∩ S = ∅. Then there exists a prime 𝔭 ∈ T such that the following conditions hold with respect to the identifications H^{2}(G_{𝔭_{i}}) = 𝔽_{p}, i = 1, …, m, and H^{2}(G_{𝔭}) = 𝔽_{p}.

• 𝔭 does not split completely in k_{S}^{el}|k,

• k_{{𝔭}}^{el}|k is cyclic of order p,

• χ_{1}, …, χ_{n}, χ_{𝔭} is a basis of H^{1}(G_{S∪{𝔭}}),

• (χ_{i} ∪ χ_{𝔭})_{𝔭} = d_{i} for i = 1, …, n,

• For i = 1, …, m we have c_{i} = 0 if and only if ε_{i} = 0, where c_{i} ∈ T_{𝔭_{i}}(k_{S}^{el}|k) ⊂ H_{1}(G_{S}) represents the map H^{1}(G_{S}) → 𝔽_{p}, χ ↦ (χ ∪ χ_{𝔭})_{𝔭_{i}}.

Now we are able to prove the following result, which is unessentially sharper than Theorem 1 of the introduction.

Theorem 6.2. Let k be a number field and let p be a prime number such that

μ_{p} ⊄ k and Cl(k)(p) = 0.

Let S be a finite set of primes of k and let T be a set of primes of p-density Δ^{p}(T) = 1. Then there exists a finite subset T_{1} ⊂ T such that Spec(O_{k}) ∖ (S ∪ T_{1}) is a K(π, 1) for p.

Proof. We may suppose that T ∩ S = ∅. After moving finitely many primes of T to S, we may assume that the following conditions hold:

• � _{S ∖ {𝔭}}(k) = 0 for all 𝔭 ∈ S,

• δ_{S}^{2}(k) = 0.

Now let S = {𝔭_{1}, …, 𝔭_{m}}. Then m = h^{2}(G_{S}). Let n = m − r = h^{1}(G_{S}). We will achieve the K(π, 1)-situation by adding m further primes to S. We choose any basis χ_{1}, …, χ_{n} of H^{1}(G_{S}). Let t_{1}, …, t_{m} be generators of the inertia groups T_{𝔭_{i}}(k_{S}^{el}|k) ⊂ H_{1}(G_{S}). Now we add a prime 𝔭_{m+1} in the following way: Let i_{1} ∈ {1, …, n} be an index such that χ_{i_{1}}(t_{1}) ≠ 0, and let i′_{1} ∈ {1, …, n}, i′_{1} ≠ i_{1}, be any other index. Now, according to Lemma 6.1, we put the conditions

ε_{1} = 1 and ε_{i} = 0 for i ∈ {2, …, m},
d_{i′_{1}} = 1 and d_{i} = 0 for i ∈ {1, …, n}, i ≠ i′_{1}

to choose a prime 𝔭_{m+1} ∈ T such that for i = 1, …, n

(χ_{i} ∪ χ_{𝔭_{m+1}})_{𝔭_{1}} = λ_{1}χ_{i}(t_{1}), λ_{1} ∈ 𝔽_{p}^{×}, (χ_{i} ∪ χ_{𝔭_{m+1}})_{𝔭_{j}} = 0, j = 2, …, m
and (χ_{i} ∪ χ_{𝔭_{m+1}})_{𝔭_{m+1}} = d_{i}.

Then in the matrix

((χ_{i} ∪ χ_{𝔭_{m+1}})_{𝔭_{j}})\begin{matrix}i=1,...,n \\ j=1,...,m+1 \end{matrix}

the i_{1}-line has entry ≠ 0 at (i_{1}, 1) and all other entries zero, while the i′_{1}-line has some entry at (i′_{1}, 1), the entry 1 at (i′_{1}, m + 1) and all other entries zero. In order to proceed, we put χ_{n+1} = χ_{𝔭_{m+1}} and choose an index i_{2} ∈ {1, …, n} with χ_{i_{2}}(t_{2}) ≠ 0 and any i′_{2} ∈ {1, …, n} with i′_{2} ≠ i_{2}. We choose conditions as before, completed by ε_{m+1} = 0 and d_{n+1} = 0. Then we choose 𝔭_{m+2} according to Lemma 6.1 and such that in the matrix

((χ_{i} ∪ χ_{𝔭_{m+2}})_{𝔭_{j}})\begin{matrix}i=1,...,n \\ j=1,...,m+2 \end{matrix}

the i_{2}-line has entry ≠ 0 at (i_{2}, 2) and all other entries zero, while the i′_{2}-line has some entry at (i′_{2}, 2), the entry 1 at (i′_{2}, m + 2) and all other entries zero. In addition, our choice implies

(χ_{𝔭_{m+1}} ∪ χ_{𝔭_{m+2}})_{𝔭_{m+1}} = 0 = (χ_{𝔭_{m+1}} ∪ χ_{𝔭_{m+2}})_{𝔭_{m+2}}

As χ_{𝔭_{m+1}} and χ_{𝔭_{m+2}} are unramified at 𝔭_{1}, …, 𝔭_{m} by construction, we have furthermore (χ_{𝔭_{m+1}} ∪ χ_{𝔭_{m+2}})_{𝔭_{i}} = 0 for i = 1, …, m.

Now we proceed to construct 𝔭_{m+3}, …, 𝔭_{2m} in a similar way, and apply Theorem 5.5 with a = m. For each j, the j-th of the m-steps in the construction produced the two lines ((i_{j}, j), −) and ((i′_{j}, j), −) in the nm×2m-matrix

((χ_{i} ∪ χ_{𝔭_{j}})_{𝔭_{k}})\begin{matrix}i=1,...,n,j=m+1,...,2m \\ k=1,...,2m \end{matrix}

According to our choices these 2m lines are linearly independent, hence the matrix has rank 2m. Putting T_{1} = {𝔭_{m+1}, …, 𝔭_{2m}}, we conclude by Theorem 5.5 that G_{S∪T_{1}} is a mild pro-p-group. Hence cd G_{S∪T_{1}} = 2 by Theorem 5.3. By Lemma 4.4, we didn’t produce new h^{2}-defect during our construction, hence δ_{S∪T_{1}}^{2}(k) = 0. As the étale cohomology is trivial in dimension ≥ 3, we conclude that the homomorphisms

ϕ_{i} : H^{i}(G_{S∪T_{1}}, ℤ/pℤ) ⟶ H_{et}^{i}(Spec(O_{k}) ∖ (S ∪ T_{1}), ℤ/pℤ)

are isomorphisms for all i ≥ 0. Hence condition (v) of Proposition 2.1 is satisfied.

7 Consequences of the K(π, 1)-property

In this section we assume that S is finite and we exclude the case S = ∅ from our considerations. Keeping all conventions made before, we assume

p ≠ 2 or k is totally imaginary and S is a non-empty finite set of nonarchimedean primes 𝔭 with norm N(𝔭) ≡ 1 mod p.

Lemma 7.1. G_{S} is a fab-group, i.e. the abelianization U^{ab} of every open subgroup U of G_{S} is finite.

Proof. Let U ⊂ G_{S} be an open subgroup. The abelianization U^{ab} of U is a finitely generated abelian pro-p-group. If U^{ab} were infinite, it would have a quotient isomorphic to ℤ_{p}, which by Galois theory corresponds to a ℤ_{p}- extension K_{∞} of the number field K = k_{S}^{U} inside k_{S}. By [NSW], Theorem 10.3.20 (ii), a ℤ_{p}-extension of a number field is ramified at at least one prime dividing p. This contradicts K_{∞} ⊂ k_{S} and we conclude that U^{ab} is finite.

The group theoretical structure of the local Galois groups is well known.

Proposition 7.2. Let 𝔭 ∈ S. Then Gal(k_{𝔭}(p)|k_{𝔭}) is the pro-p-group on two generators σ, τ subject to the relation στσ^{−1} = τ^{q}. The element τ is a generator of the inertia group, σ is a Frobenius lift and q = N(𝔭).

Proof. This follows from [NSW], Thm. 7.5.2 by passing to the maximal pro-p-factor group.

We obtain the following corollary.

Corollary 7.3. Assume that G_{S} is infinite. Then, for each 𝔭 ∈ S, the decomposition group G_{𝔭} of 𝔭 in G_{S} has infinite index.

Proof. The decomposition group G_{𝔭} is a quotient of the local Galois group Gal(k_{𝔭}(p)|k_{𝔭}). If G_{𝔭} ⊂ G_{S} would have finite index, it would be an infinite fabgroup by Lemma 7.1. By Proposition 7.2, each infinite quotient of Gal(k_{𝔭}(p)|k_{𝔭}) has a surjection to ℤ_{p} and is therefore not a fab-group. This contradiction shows that G_{𝔭} has infinite index in G_{S}.

The next proposition classifies the degenerate K(π, 1)-case.

Proposition 7.4. X ∖ S is a K(π, 1) and G_{S} = 1 if and only if S = {𝔭} consists of a single prime and one of the following cases occurs.

(a) p = 2, k ≠ ℚ(\sqrt{−1}) is imaginary quadratic, 2 ∤ h_{k} and N(𝔭) ≢ 1 mod 4,

(b) p = 2, k = ℚ(\sqrt{−1}) and N(𝔭) ≢ 1 mod 8,

(c) p = 3, k = (ℚ\sqrt{−3}) and N(𝔭) ≢ 1 mod 9.

Proof. Assume G_{S} = 1 and that X ∖ S is a K(π, 1). Then H_{et}^{i}(X ∖ S) = 0 for all i ≥ 1. In particular, p ∤ h_{k}. By Theorem 3.4, h^{2}(X ∖ S) = 0 implies δ = 1, #S = 1 and V_{S} = 0. Then, using h^{1}(X ∖ S) = 0, we obtain r = 1. As δ = 1, the following possibilities remain

(a) p = 2, k ≠ ℚ(\sqrt{−1}) is imaginary quadratic and 2 ∤ h_{k},

(b) p = 2, k = ℚ(\sqrt{−1}),

(c) p = 3, k = (ℚ\sqrt{−3}).

In all cases, Proposition 3.3 yields an isomorphism O_{k}^{×}/p →^{∼} V_{∅}. The second exact sequence of Proposition 3.3 and the isomorphism U_{𝔭}/p ≅ U_{𝔭}k_{𝔭}^{×p}/k_{𝔭}^{×p} imply

0 = V_{S} = ker (O_{k}^{×}/p → U_{𝔭}/p).

Note that O_{k}^{×}/p is one-dimensional. In case (a), the unit −1 is a generator of O_{k}^{×}/2 which must not be a square in U_{𝔭}, implying N(𝔭) ≢ 1 mod 4. In case (b), \sqrt{−1} is a generator, and in case (c), a generator is given by ζ_{3} = \frac{1}{2}(−1 + \sqrt{−3}). The assertions in the cases (b) and (c) follow similarly. Conversely, assume we are in case (a), (b) or (c). Then we can reverse the given arguments and obtain h^{i}(X ∖ S) = 0 for all i ≥ 1.

Theorem 7.5. Assume G_{S} ≠ 1 and that X ∖ S is a K(π, 1). Then the following hold.

(i) cd G_{S} = 2, scd G_{S} = 3.

(ii) G_{S} is a duality group (of dimension 2).

Proof. By Lemma 7.1 and Corollary 3.5, G_{S} is a fab-group and cd G_{S} ≤ 2. Now the assertions follow in a purely group-theoretical way: As G_{S} ≠ 1 and G_{S}^{ab} is finite, G_{S} is not free, and we obtain cd G_{S} = 2. By [NSW], Proposition 3.3.3, it follows that scd G_{S} ∈ {2, 3}. Assume scd G = 2. We consider the G_{S}-module

D_{2}(ℤ) := \underrightarrow{lim}_{U} U^{ab},

where the limit runs over all open normal subgroups U ⊲G_{S} and for V ⊂ U the transition map is the transfer Ver: U^{ab} → V^{ab}, i.e. the dual of the corestriction map cor: H^{2}(V, ℤ) → H^{2}(U, ℤ) (see [NSW], I, §5). By [NSW], Theorem 3.6.4 (iv), we obtain G_{S}^{ab} = D_{2}(ℤ)^{G_{S}}. On the other hand, U^{ab} is finite for all U and the group theoretical version of the Principal Ideal Theorem (see [Ne], VI, Theorem 7.6) implies D_{2}(ℤ) = 0. Hence G_{S}^{ab} = 0 which implies G_{S} = 1 producing a contradiction. Hence scd G_{S} = 3. It remains to show that G_{S} is a duality group. By [NSW], Theorem 3.4.6, it suffices to show that the terms

D_{i}(G_{S}, ℤ/pℤ) := \underrightarrow{lim}_{U} H^{i}(U, ℤ/pℤ)^{∨}

are zero for i = 0, 1. Here U runs through the open subgroups of G_{S}, ∨ denotes the Pontryagin dual and the transition maps are the duals of the corestriction maps. For i = 0, and V ⫋ U, the transition map

cor^{∨}: ℤ/pℤ = H^{0}(V, ℤ/pℤ)^{∨} → H^{0}(U, ℤ/pℤ)^{∨} = ℤ/pℤ

is multiplication by (U : V), hence zero. Therefore D_{0}(G_{S}, ℤ/pℤ) = 0, as G_{S} is infinite. Furthermore,

D_{1}(G_{S}, ℤ/pℤ) = \underrightarrow{lim}_{U} U^{ab}/p = 0

by the Principal Ideal Theorem. This finishes the proof.

In order to proceed, we introduce some notation in order to deal with the case of infinite extensions. For a (possibly infinite) algebraic extension K of k we denote by S(K) the set of prolongations of primes in S to K. The set S(K) carries a profinite topology in a natural way. Now assume that M|K|k is a tower of Galois extensions. We denote the inertia group of a prime 𝔭 ∈ S(K) in the extension M|K by T_{𝔭}(M|K). For i ≥ 0 we write

\begin{matrix}⨁ & ′ \\ 𝔭∈S(K) \end{matrix} H^{i}(T_{𝔭}(M|K), ℤ/pℤ) =^{df} \underrightarrow{lim}_{k′⊂K} ⨁_{𝔭∈S(k′)} H^{i}(T_{𝔭}(M|k′), ℤ/pℤ),

where the limit on the right hand side runs through all finite subextensions k′ of k in K. The Gal(K|k)-module ^{⨁′}_{𝔭∈S(K)} H^{i}(T_{𝔭}(M|K), ℤ/pℤ) is the maximal discrete submodule of the product ∏_{𝔭∈S(K)} H^{i}(T_{𝔭}(M|K), ℤ/pℤ).

Whenever we deal with local terms associated to the elements of S(K) (e.g. étale cohomology groups) we use restricted sums, which are, in the same manner as above, defined as the inductive limit over the similar terms associated to all finite subextensions of k in K.

A natural question is how far we get locally at the primes in S when going up to k_{S}.

Proposition 7.6. Assume that X ∖ S is a K(π, 1) and that G_{S} ≠ 1. Then k_{S} realizes the maximal unramified p-extension of k_{𝔭} for all 𝔭 ∈ S, i.e.

k_{𝔭}^{nr}(p) ⊂ (k_{S})_{𝔭} for all 𝔭 ∈ S.

If 𝔭 ∈ S ramifies in k_{S}, then (k_{S})_{𝔭} = k_{𝔭}(p), i.e. k_{S} realizes the maximal p-extension of k_{𝔭}.

Proof. For an integral normal scheme Y we write Y_{L} for the normalization of Y in an algebraic extension L of its function field. Then (X ∖ S)_{k_{S}} is the universal pro-p covering of X ∖ S. We consider the following part of the excision sequence for the pair (X_{k_{S}}, (X ∖ S)_{k_{S}})

H_{et}^{2}((X ∖ S)_{k_{S}}) → \begin{matrix}⨁ & ′ \\ 𝔭∈S(k_{S}) \end{matrix} H_{𝔭}^{3}((X_{k_{S}})_{𝔭}) → H_{et}^{3}(X_{k_{S}}).

As G_{S} is infinite, Lemma 3.7 implies H_{et}^{3}(X_{k_{S}}) = 0. By condition (iii) of Proposition 2.1 we have H_{et}^{2}((X ∖ S)_{k_{S}}) = 0. Hence H_{𝔭}^{3}((X_{k_{S}})_{𝔭}) = 0 for all 𝔭 ∈ S(k_{S}). As H_{et}^{i}((X_{k_{S}})_{𝔭}) = 0 for i ≥ 2, we obtain

H_{𝔭}^{3}((X_{k_{S}})_{𝔭}) ≅ H^{2}((k_{S})_{𝔭}),

where the group on the right hand side is Galois cohomology with values in ℤ/pℤ. As μ_{p} ⊂ k_{𝔭} by assumption, the vanishing of H^{2}((k_{S})_{𝔭}) implies p^{∞} | [(k_{S})_{𝔭} : k_{𝔭}]. In other words, the decomposition group G_{𝔭}(k_{S}|k) of each 𝔭 ∈ S is infinite. As a subgroup of G_{S}, it has cohomological dimension ≤ 2. Furthermore, G_{𝔭}(k_{S}|k) is a factor group of the local Galois group Gal(k_{𝔭}(p)|k_{𝔭}), which, by Proposition 7.2, has only three quotients of cohomological dimension less or equal to 2: itself, the trivial group and the Galois group of the maximal unramified p-extension of k_{𝔭}. Hence k_{𝔭}^{nr}(p) ⊆ (k_{S})_{𝔭} and (k_{S})_{𝔭} = k_{𝔭}(p) if 𝔭 ramifies in k_{S}.

In order to deduce Theorem 3, it remains to show that each 𝔭 ∈ S ramifies in k_{S}. The following lemma provides a first step.

Lemma 7.7. Let 𝔭 ∈ S be a prime and let S′ = S ∖ {𝔭}. Assume that the natural injection V_{S} ↪ V_{S′} is an isomorphism. Then 𝔭 ramifies in k_{S}.

Proof. Since the map H^{1}(G_{S}) → H_{et}^{1}(X ∖ S) is an isomorphism, Theorem 3.4 implies

dim_{𝔽_{p}} H^{1}(G_{S}) = 1 + #S − δ + dim_{𝔽_{p}} V_{S} − r,

and the same formula holds with S replaced by S′. Hence

dim_{𝔽_{p}} H^{1}(G_{S}) = dim_{𝔽_{p}} H^{1}(G_{S′}) + 1 .

In particular, G_{S′} is a proper quotient of G_{S} and therefore 𝔭 ramifies in k_{S}.

Corollary 7.8. Assume that X ∖ S is a K(π, 1) and that G_{S} ≠ 1. Let 𝔭 ∈ S be a prime and let S′ = S ∖ {𝔭}. Assume that V_{S′} = 0. Then (k_{S})_{𝔭} = k_{𝔭}(p).

Remark: If V_{∅} = 0, then the given criterion applies to any set S and each 𝔭 ∈ S. This was used in [S1] for k = ℚ and in [Vo] for imaginary quadratic number fields. If the unit rank of k is non-zero, then V_{∅} ≠ 0 and the criterion applies only to sufficiently large sets S.

8 Enlarging the set of primes

Next we consider the problem of enlarging the set S.

Proposition 8.1. Let S ⊂ S′ be finite sets of primes of norm congruent to 1 modulo p. Assume that X ∖ S is a K(π, 1) and that G_{S} ≠ 1. Further assume that each 𝔮 ∈ S′ ∖ S does not split completely in k_{S}. Then the following holds.

(i) X ∖ S′ is a K(π, 1).

(ii) (k_{S′})_{𝔮} = k_{𝔮}(p) for all 𝔮 ∈ S′ ∖ S.

Furthermore, H^{i}(Gal(k_{S′}|k_{S})) = 0 for i ≥ 2. For i = 1 we have a natural isomorphism

H^{1}(Gal(k_{S′}|k_{S})) ≅ \begin{matrix}⨁ & ′ \\ 𝔭∈S′ ∖ S(k_{S})) \end{matrix}H^{1}(T_{𝔭}(k_{S′}|k_{S})),

In particular, Gal(k_{S′}|k_{S}) is a free pro-p-group.

Proof. Let 𝔮 ∈ S′ ∖ S. Since 𝔮 does not split completely in k_{S} and since cd G_{S} = 2, the decomposition group of 𝔮 in k_{S}|k is a non-trivial and torsionfree quotient of ℤ_{p} ≅ G(k_{𝔮}^{nr}(p)|k_{𝔮}). Therefore (k_{S})_{𝔮} is the maximal unramified p-extension of k_{𝔮}. We denote the normalization of an integral normal scheme Y in an algebraic extension L of its function field by Y_{L}. Then (X ∖ S)_{k_{S}} is the universal pro-p covering of X ∖ S. We consider the étale excision sequence for the pair ((X ∖ S)_{k_{S}}, (X ∖ S′)_{k_{S}}). By assumption, X ∖ S is a K(π, 1), hence H_{et}^{i}((X ∖ S)_{k_{S}}) = 0 for i ≥ 1 by condition (iii) of Proposition 2.1. This implies isomorphisms

H_{et}^{i}((X ∖ S′)_{k_{S}}) →^{∼} \begin{matrix}⨁ & ′ \\ 𝔭∈S′ ∖ S(k_{S}) \end{matrix} H_{𝔭}^{i+1}(((X ∖ S)_{k_{S}})_{𝔭})

for i ≥ 1. As k_{S} realizes the maximal unramified p-extension of k_{𝔮} for all 𝔮 ∈ S′ ∖ S, the schemes ((X ∖ S)_{k_{S}})_{𝔭}, 𝔭 ∈ S′ ∖ S(k_{S}) have trivial cohomology with values in ℤ/pℤ and we obtain isomorphisms

H^{i}((k_{S})_{𝔭}) →^{∼} H_{𝔭}^{i+1}(((X ∖ S)_{k_{S}})_{𝔭})

for i ≥ 1. These groups vanish for i ≥ 2. This implies

H_{et}^{i}((X ∖ S′)_{k_{S}}) = 0

for i ≥ 2. The scheme (X ∖ S′)_{k_{S′}} is the universal pro-p covering of (X ∖ S′)_{k_{S}}. The Hochschild-Serre spectral sequence yields an inclusion

H^{2}(Gal(k_{S′}|k_{S})) ↪ H_{et}^{2}((X ∖ S′)_{k_{S}}) = 0.

Hence Gal(k_{S′}|k_{S}) is a free pro-p-group and

H^{1}(Gal(k_{S′}|k_{S})) →^{∼} H_{et}^{1}((X ∖ S′)_{k_{S}}) ≅ \begin{matrix}⨁ & ′ \\ 𝔭∈S′ ∖ S(k_{S}) \end{matrix} H^{1}((k_{S})_{𝔭}).

This shows that each 𝔭 ∈ S′ ∖ S(k_{S}) ramifies in k_{S′}|k_{S}, and since the Galois group is free, k_{S′} realizes the maximal p-extension of (k_{S})_{𝔭}. In particular,

H^{1}(T_{𝔭}(k_{S′}|k_{S})) ≅ H^{1}((k_{S})_{𝔭})

for all 𝔭 ∈ S′ ∖ S(k_{S}). Using that Gal(k_{S′}|k_{S}) is free, the Hochschild-Serre spectral sequence induces an isomorphism

0 = H_{et}^{2}((X ∖ S′)_{k_{S}}) ⟶^{∼} H_{et}^{2}((X ∖ S′)_{k_{S′}})^{Gal(k_{S′} |k_{S})}.

Hence H_{et}^{2}((X ∖ S′)_{k_{S′}}) = 0, since Gal(k_{S′}|k_{S}) is a pro-p-group. Condition (iii) of Proposition 2.1 implies that X ∖ S′ is a K(π, 1).

Corollary 8.2. Assume that X ∖ S is a K(π, 1), and let S ⊂ S′ be a finite set of primes of norm ≡ 1 mod p. Assume that each 𝔮 ∈ S′ ∖ S does not split completely in k_{S}. Then the arithmetic form of Riemann’s existence theorem holds, i.e. the natural homomorphism

∗_{𝔭∈S′\S(k_{S})} T_{𝔭}(k_{S′}|k_{S}) ⟶ Gal(k_{S′}|k_{S})

is an isomorphism. Here T_{𝔭} is the inertia group and ∗ denotes the free pro-p-product of a bundle of pro-p-groups, cf. [NSW], Ch. IV, §3.

Proof. By Proposition 8.1 and by the calculation of the cohomology of a free product ([NSW], 4.3.10 and 4.1.4), ϕ is a homomorphism between free pro-p-groups which induces an isomorphism on mod p cohomology. Therefore ϕ is an isomorphism.

9 Proof of Theorems 3 and 5

Theorem 9.1. Assume that X ∖ S is a K(π, 1) and G_{S} ≠ 1. Then k_{S} realizes the maximal p-extension k_{𝔭}(p) of the local field k_{𝔭} for all 𝔭 ∈ S.

Proof. The decomposition groups of primes in S have infinite index by Corollary 7.3. By Corollary 2.2, we may replace k by a finite subextension in k_{S}, and therefore assume that #S ≥ 2. By Proposition 7.6, it suffices to show that each 𝔭 ∈ S ramifies in k_{S}. Let 𝔭 ∈ S be a prime which does not ramify in k_{S} and put S′ = S ∖ {𝔭}. By Lemma 7.7, the natural injection ϕ: V_{S} ↪ V_{S′} is not an isomorphism. By Proposition 3.3, the cokernel of ϕ is one-dimensional. By Theorem 3.4, we obtain

h^{2}(X ∖ S′) = h^{2}(X ∖ S).

As G_{S} = G_{S′}, we have cd G_{S′} = 2 and

h^{2}(G_{S}) = h^{2}(G_{S′}) ≤ h^{2}(X ∖ S′) = h^{2}(X ∖ S).

As X ∖ S is a K(π, 1), equality holds. Therefore the injection H^{2}(G_{S′}) ↪ H_{et}^{2}(X ∖ S′) is an isomorphism. By Corollary 3.5, X ∖ S′ is a K(π, 1). By Proposition 7.6, 𝔭 does not split completely in k_{S′} = k_{S}. By Proposition 8.1, k_{S} realizes the maximal p-extension of k_{𝔭}. This yields a contradiction.

Now we are in the position to show Theorem 5.

Proof of Theorem 5. We have H_{et}^{2}((X ∖ S)_{k_{S}}) = 0 by condition (iii) of Proposition 2.1. By Theorem 9.1, the local cohomology groups H_{𝔭}^{2}((X_{k_{S}})_{𝔭}) vanish for all 𝔭 ∈ S(k_{S}). Therefore the excision sequence yields H_{et}^{2}(X_{k_{S}}) = 0. By the flat duality theorem of Artin-Mazur ([Mi], III Corollary 3.2) we have H_{et}^{2}(X_{K})^{∨} ≅ H_{fl}^{1} (X_{K}, μ_{p}) for each finite subextension K of k in k_{S}. Hence

\underleftarrow{lim}_{K⊂k_{S}} H_{fl}^{1} (X_{K}, μ_{p}) = 0.

The flat Kummer sequence 0 → μ_{p} → 𝔾_{m} →^{·p} 𝔾_{m} → 0 implies compatible exact sequences

0 → O_{K}^{×}/p → H_{fl}^{1} (X_{K}, μ_{p}) → _{p}H_{fl}^{1} (X_{K}, 𝔾_{m})

for all K. We obtain

\underleftarrow{lim}_{K⊂k_{S}} O_{K}^{×}/p = 0 .

The topological Nakayama-Lemma (see [NSW], Corollary 5.2.8) for the compact ℤ_{p}-module \underleftarrow{lim} O_{K}^{×} ⊗ ℤ_{p} therefore implies

\underleftarrow{lim}_{K⊂k_{S}} O_{K}^{×} ⊗ ℤ_{p} = 0 .

Tensoring the exact sequences (cf. [NSW], Lemma 10.3.11)

0 → O_{K}^{×} → O_{K,S}^{×} → ⨁_{𝔭∈S(K)}(K_{𝔭}^{×}/U_{𝔭}) → Cl(K) → Cl_{S}(K) → 0

by (the flat ℤ-algebra) ℤ_{p}, we obtain exact sequences of finitely generated, hence compact, ℤ_{p}-modules. The field k_{S} admits no unramified p-extensions. Therefore class field theory implies \underleftarrow{lim}_{K} Cl(K)(p) = 0, where K runs through all finite subextensions of k in k_{S}. Thus, passing to the projective limit over K, we obtain the exact sequence

0 → \underleftarrow{lim}_{K⊂k_{S}} O_{K}^{×} ⊗ ℤ_{p} → \underleftarrow{lim}_{K⊂k_{S}} O_{K,S}^{×} ⊗ ℤ_{p} → \underleftarrow{lim}_{K⊂k_{S}} ⨁_{𝔭∈S(K)}(K_{𝔭}^{×}/U_{𝔭}) ⊗ ℤ_{p} → 0.

As k_{S} realizes the maximal unramified p-extension of k_{𝔭} for all 𝔭 ∈ S, local class field theory implies the vanishing of the right hand limit. Therefore the result for the S-units follows from the corresponding result for the units.

We have proven all assertions but the statement on the dualizing module in Theorem 2. In [S1], Th. 5.2 we showed this statement under the assumption that k_{S} realizes the maximal p-extension k_{𝔭}(p) of k_{𝔭} for all 𝔭 ∈ S. This assumption has been shown above, hence the result follows.

Added in proof. As pointed out by K. Wingberg, the proof of Proposition 8.1 does not use that S and S′ are disjoint from S_{p}. Hence Theorem 1 also holds if S ∩ S_{p} ≠ ∅, i.e. the result extends to the mixed case.

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http://www.mathematik.uni-regensburg.de/FGAlgZyk

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Alexander Schmidt NWF I - Mathematik Universität Regensburg D-93040 Regensburg Deutschland 472

Documenta Mathematica 12 (2007)