Riemann’s existence theorem and the K(π, 1)-property of rings of integers

by Kay Wingberg at Heidelberg

Preliminary version: December 6, 2007

Let k be a number field, S a finite set of nonarchimedean primes of k and p a prime number. We assume that p is odd or that k is totally imaginary. Let k_{S}(p) be the maximal p-extension of k unramified outside S and G_{S}(p) = Gal(k_{S}(p)|k). In geometric terms, we have

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

where (Spec(O_{k})\S)_{et}^{(p)} is the p-completion of the étale homotopy type of the scheme Spec(O_{k})\S. If S contains the set S_{p} of primes dividing p (the wild case), then G_{S}(p) has cohomological dimension less or equal to 2. Furthermore, if T ⊇ S ⊇ S_{p} are sets of primes of k, then the canonical homomorphisms

ϕ_{T,S} : ∗_{𝔭∈(T \S)(k_{S}(p))}T_{𝔭}(k(p)|k) ⟶ G(k_{T} (p)|k_{S}(p))

of the free pro-p product of the groups T_{𝔭}(k(p)|k) into G(k_{T} (p)|k_{S}(p)); here T_{𝔭}(k(p)|k) is the inertia subgroup of the decomposition group G_{𝔭}(k(p)|k) ≅ G(k_{𝔭}(p)|k_{𝔭}), where k_{𝔭} is the completion of k with respect to the prime 𝔭. We say that Riemann’s existence theorem holds for k, S, T.

In the tame case, i.e. S ∩S_{p} = ∅, and in the mixed case, i.e. ∅ ≠ S ∩S_{p} ⊈ S_{p}, until recently not much was known about the group G_{S}(p): In the tame case G_{S}(p) is a finitely presented pro-p-group (Koch), which can be infinite (Golod-Šafarevič), and which is a fab-group, i.e. U^{ab} is finite for each open subgroup U ⊆ G_{S}(p).

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 [S2] A. Schmidt also considered the tame case: he showed that for a number filed k, which does not contain the group of p-th roots of unity and whose p-part of its ideal class group is trivial, there always exists a finite set T of primes with T ∩ S_{p} = ∅, such that (Spec(O_{k})\(S ∪ T))_{et}^{(p)} is a K(π, 1) for p, i.e. the higher étale homotopy groups of (Spec(O_{k})\(S ∪ T))_{et}^{(p)} vanish; in particular, cd_{p} G_{S∪T} (p) ≤ 2.

In this paper we will study the relationship of the K(π, 1)-property of the scheme Spec(O_{k})\S and Riemann’s existence theorem for sets T ⊇ S, where S is an arbitrary finite set of nonarchimedean primes. We extend results of [5] in the following way (see also [6]): Theorem. Let p be a prime number and k a number field where p is odd or k is totally imaginary. Let T ⊇ S be finite sets of nonarchimedean primes of k. Assume that (k_{S}(p))_{𝔭} ≠ k_{𝔭} for all 𝔭 ∈ (T \S) ∩ S_{p}. Then we have the following assertions are equivalent:

(i) Spec(O_{k})\S is a K(π, 1) for p and (k_{S}(p))_{𝔮} ≠ k_{𝔮} for all 𝔮 ∈ (T \(S ∪S_{p}))_{min}.

(ii) Spec(O_{k})\T is a K(π, 1) for p and

∗_{𝔭∈T \S(k_{S}(p))}T_{𝔭}(k) ⟶^{∼} G(k_{T} (p)|k_{S}(p)).

Using this theorem and results of [5], we will show that not only in the tame case but also in the mixed case one can find finite sets S of primes such that cd_{p} G_{S}(p) ≤ 2.

1 Free product decomposition

We introduce some notation. If p is a fixed prime number and G a prop group, then H^{i}(G) denotes the cohomology group H^{i}(G, ℤ/pℤ) and we put h^{i}(G) = dim_{𝔽_{p}} H^{i}(G). Furthermore,

χ(G) = ∑_{i}(−1)^{i}h^{i}(G) and χ_{n}(G) = ∑_{i=0}^{n}(−1)^{i}h^{i}(G)

denotes the Euler-Poincaré characteristic and partial Euler-Poincaré characteristic of G, respectively. If K|k is a Galois p-extension with Galois group G(K|k), we sometimes write H^{i}(K|k) for H^{i}(G(K|k)).

Let k is a number field with absolute Galois group by G_{k}. If p is a prime number, then k(p) is the maximal p-extension of k with Galois group G_{k}(p) = G(k(p)|k). If K|k is a Galois p-extension with Galois group G(K|k), we sometimes write H^{i}(K|k) for H^{i}(G(K|k)).

By S_{∞}, S_{ℝ} and S_{ℂ} we denote the sets of archimedean, real and complex primes of k and put r_{1}(k) = #S_{ℝ} and r_{2}(k) = #S_{ℂ}, respectively. We consider the extension ℂ|ℝ as ramified. If p is a prime number, then S_{p} is the set of all primes of K above p.

If 𝔭 is a prime k, then k_{𝔭} is the completion of k with respect to 𝔭 with absolute Galois group G_{k_{𝔭}}, and U_{𝔭} denotes is group of units.

If K|k is a Galois extension, then we denote the decomposition group and inertia group of the Galois group G(K|k) with respect to 𝔭 by G_{𝔭}(K|k) and T_{𝔭}(K|k), respectively. We write G_{𝔭} = G_{𝔭}(k) = G_{𝔭}(k(p)|k) ≅ G(k_{𝔭}(p)|k_{𝔭}) and T_{𝔭} = T_{𝔭}(k) = T_{𝔭}(k(p)|k) ≅ T (k_{𝔭}(p)|k_{𝔭}); then G_{𝔭}/T_{𝔭} = G(k_{𝔭}^{nr}(p)|k_{𝔭}), where k_{𝔭}^{nr}(p) is the maximal unramified p-extension of k_{𝔭}.

If S = S(k) is a set of primes and k′|k an algebraic extension of k, then we denote the set of primes of k′ consisting of all prolongations of S by S(k′). Furthermore,

k_{S}  is the maximal extension of k which is unramified outside S,
k_{S}(p) is the maximal p-extension of k which is unramified outside S,

and by G_{S} = G_{S}(k) and G_{S}(p) = G_{S}(k)(p) we denote the Galois groups G(k_{S}|k) and G(k_{S}(p)|k), respectively.

For an arbitrary set S of primes of k we define the Šafarevič-Tate groups � ^{i}(G_{S}(p)) = � ^{i}(G_{S}(p), ℤ/pℤ) and the groups coker^{i}(G_{S}(p)) by the exactness of the sequences

0 ⟶ � ^{i}(G_{S}(p)) ⟶ H^{i}(G_{S}(p)) ⟶ ∏_{𝔭∈S}H^{i}(G_{𝔭}) ⟶ coker^{i}(G_{S}(p)) ⟶ 0 .

Let

V_{S}(k) = ker (k^{×}/k^{×p} ⟶ ∏_{𝔭∈S}k_{𝔭}^{×}/k_{𝔭}^{×p} ×∏_{𝔭∉S}k_{𝔭}^{×}/U_{𝔭}k_{𝔭}^{×p}),

and � _{S}(k) = V_{S}(k)^{∨}. Observe that when μ_{p} ⊆ k

\begin{aligned}� _{S}(k) & = & ker(H^{1}(G_{S}(p), μ_{p}) → ∏_{𝔭∈S} H^{1}(G_{𝔭}, μ_{p}))^{∨} \\ & = & (Cl_{S}(k)/p)(−1) . \end{aligned}

Furthermore, we set

δ = \{ \begin{matrix}1, & μ_{p} ⊆ k, \\ 0, & μ_{p} ⊆ / k, \end{matrix} and δ_{𝔭} = \{ \begin{matrix}1, & μ_{p} ⊆ k_{𝔭}, \\ 0, & μ_{p} ⊆ / k_{𝔭}. \end{matrix}

The following primes cannot ramify in a p-extension, and are therefore redundant in S:

1. Complex primes.

2. Real primes if p ≠ 2.

3. Primes 𝔭 ∤ p with N(𝔭) ≢ 1 mod p.

Removing all these redundant places from S, we obtain a subset S_{min} ⊆ S which has the property that

G_{S}(p) = G_{S_{min}}(p).

We need some results on the cohomology of a free product in the following case, see [3] chap.IV: Let T = lim_{⟵}_{λ} T̄_{λ}, where the sets T̄_{λ} = T_{λ} ∪ {∗_{λ}} are the one-point compactifications of discrete sets T_{λ}. Let G = lim_{⟵}_{λ} G_{λ} be the projective

limit of bundles G_{λ} = ⋃ . _{t_{λ}∈T_{λ}} G_{t_{λ}} ⋃ . {∗_{λ}}, and let G_{t} = lim_{⟵}_{λ} G_{t_{λ}}. Let A be an abelian torsion group considered as a trivial G-module where G = ∗_{T} G. Then

there are isomorphisms

H^{i}(G, A) = \underrightarrow{lim}_{λ} ⨁_{T_{λ}} H^{i}(G_{t_{λ}}, A), i ≥ 0.

We will use the notation

\begin{matrix}⨁ & ′ \\ T \end{matrix} H^{i}(G_{t}, A) := \underrightarrow{lim}_{λ} ⨁_{T_{λ}} H^{i}(G_{t_{λ}}, A).

We need the following

Lemma 1.1 Let

[figure: 1; ℋ_{λ}; 𝒢_{λ}; G_{λ}; 1; 1; ℋ; 𝒢; G; 1]

be an exact and commutative diagram of pro-p-groups and assume that ℋ is a free pro-p-product of the form

∗_{λ∈S} ∗_{σ∈G|G_{λ}} ℋ_{λ}^{σ} ⟶^{∼} ℋ ,

where S is a profinite set, ℋ_{λ}^{σ} is a closed subgroup of ℋ , which is conjugated to ℋ_{λ} under an arbitrary extension of σ to 𝒢, and G|G_{λ} is a complete system of representatives of G_{λ} in G. Assume that cd_{p} ℋ_{λ} ≤ 1 and cd_{p} G_{λ} ≤ 1 for all λ ∈ S. Then there is an exact sequence

0 → H^{1}(G, A) → H^{1}(𝒢 , A) → \begin{matrix}⨁ & ′ \\ λ \end{matrix} H^{1}(ℋ_{λ}, A)^{G_{λ}}

→ H^{2}(G, A) → H^{2}(𝒢 , A) → \begin{matrix}⨁ & ′ \\ λ \end{matrix} H^{2}(𝒢_{λ}, A) → H^{3}(G, A) → H^{3}(𝒢 , A) → 0,

where A is a torsion group (considered as a 𝒢 -module with trivial action), and (i) cd_{p} 𝒢 ≤ 2 implies cd_{p} G ≤ 3, (ii) cd_{p} G ≤ 2 implies cd_{p} 𝒢 ≤ 2.

Proof: Using the results on the cohomology of free products, see [3] chap.IV, we obtain

H^{i}(G, H^{j}(ℋ , A)) ≅ \begin{matrix}⨁ & ′ \\ λ∈S \end{matrix} H^{i}(G_{λ}, H^{j}(ℋ_{λ}, A)), j ≥ 1.

These groups can be non-trivial only for i = 0, 1 and j = 1. Furthermore, we have

H^{1}(G_{λ}, H^{1}(ℋ_{λ}, A)) ≅ H^{2}(𝒢_{λ}, A).

Since cd_{p} ℋ ≤ 1, the Hochschild-Serre spectral sequence gives the result. □

Corollary 1.2 Let k be number field and p prime number. Assume that k is totally imaginary if p = 2. Let T ⊇ S be non-empty sets of primes of k. Assume that S_{p} ⊆ T. Assume further that we have a free product decomposition

∗_{𝔭∈(T \S)(k_{S}(p))}T_{𝔭}(k) ⟶^{∼} G(k_{T} (p)|k_{S}(p)),

and that (k_{S}(p))_{𝔭} = k_{𝔭}^{nr}(p) for all 𝔭 ∈ (T \S)_{min}. Then

cd_{p} G(k_{S}(p)|k) ≤ 2.

Proof: Since

cd_{p} T_{𝔭}(k) = 1, cd_{p} G_{𝔭}(k)/T_{𝔭}(k) = 1, cd_{p} G(k_{T} (p)|k) ≤ 2,

we obtain from lemma (1.1), that the vertical left sequence in the commutative diagram

[figure: ⨁_{𝔭∈S}H^{2}(G_{𝔭}(k)); H^{2}(G(k_{T} (p)|k)); ⨁_{𝔭∈T}H^{2}(G_{𝔭}(k)); Σ; H^{0}(G(k_{T} |k), μ_{p})^{∨}; ⨁_{𝔭∈T \S} H^{2}(G_{𝔭}(k)); ⨁_{𝔭∈T \S} H^{2}(G_{𝔭}(k)); H^{3}(G(k_{S}(p)|k)).]

is exact. By the theorem of Poitou-Tate, see [3] (8.6.13), the horizontal sequence is exact. We obtain H^{3}(G(k_{S}(p)|k)) = 0, hence cd_{p} G(k_{S}(p)|k) ≤ 2. □

Proposition 1.3 Let p be a prime number and let k be the number field.

(i) For an arbitrary set S of primes of k there is a canonical exact and commutative diagram

[figure: H^{1}(G(k(p)|k)); ⨁_{𝔭∉S}H^{1}(T_{𝔭}(k))^{G_{𝔭}(k)}; � _{S}(k); 0; H^{1}(G(k(p)|k)); H^{1}(G(k(p)|k_{S}(p)))^{G_{S}(p)}; � ^{2}(G_{S}(p)); 0.]

(ii) Let T ⊇ S be sets of primes of k. Assume that

\underrightarrow{lim}_{k′⊆k_{S}(p)} � _{S}(k′) = 0 ,

where k′ runs through the finite extensions of k inside k_{S}(p). Then the canonical map

H^{1}(G(k_{T} (p)|k_{S}(p))) ⟶^{∼} \begin{matrix}⨁ & ′ \\ 𝔭∈T \S(k_{S}(p)) \end{matrix}H^{1}(T_{𝔭}(k))^{G_{𝔭}(k_{S}(p))}

is an isomorphism.

(iii) Let T ⊇ S ⊇ S_{p} ∪ S_{∞} be sets of primes of k. Then the canonical map

H^{1}(G(k_{T} (p)|k_{S}(p))) ⟶^{∼} \begin{matrix}⨁ & ′ \\ 𝔭∈T \S(k_{S}(p)) \end{matrix}H^{1}(T_{𝔭}(k))

is an isomorphism.

Proof: Let T_{S} = G(k(p)|k_{S}(p)). We consider the group extension

1 ⟶ T_{S} ⟶ G_{k}(p) ⟶ G_{S}(p) ⟶ 1 .

From the commutative exact diagram

\xymatrix{H^{1}(G_{k}(p)) \ar[r] & H^{1}(T_{S})^{G_{S}(p)} \ar[r] & H^{2}(G_{S}(p)) \ar[r] \ar[d] & H^{2}(G_{k}(p)) \ar@{^{(}->}[d] \\  &  & ⨁_{𝔭∈S}H^{2}(G_{𝔭}(k)) \ar@{^{(}->}[r] & ⨁_{𝔭}H^{2}(G_{𝔭}(k)),}

where the right-hand vertical map is injective by [3](9.1.10) and (10.4.8), we obtain the exact sequence

H^{1}(G_{k}(p)) ⟶ H^{1}(T_{S})^{G_{S}(p)} ⟶ � ^{2}(G_{S}(p)) ⟶ 0 .

Furthermore, we consider the commutative exact diagram

[figure: H^{1}(T_{S})^{G_{S}(p)}; ⨁_{𝔭∉S}H^{1}(T_{𝔭})^{G_{𝔭}}; H^{1}(G_{k}(p)); \underline{∏}_{𝔭}H^{1}(G_{𝔭}); H^{1}(G_{k}, μ_{p})^{∨}; \underline{∏}_{𝔭∈S}H^{1}(G_{𝔭}) × ∏_{𝔭∉S}H_{nr}^{1}(G_{𝔭}); H^{1}(G_{k}, μ_{p})^{∨}; � _{S}(k) .]

The row in the middle is exact by the Poitou-Tate theorem, see [3] (8.6.10) and (9.1.10), and the upper map is injective by definition of the group T_{S}. The exactness of the bottom row follows from the definition of � _{S}(k) = (V_{S}(k))^{∨} and from H_{nr}^{1}(G_{𝔭})^{∨} = k_{𝔭}^{×}/U_{𝔭}k_{𝔭}^{×p}. This diagram and the exact sequence above imply that the commutative diagram

[figure: H^{1}(G_{k}(p)); ⨁_{𝔭∉S}H^{1}(T_{𝔭}(k))^{G_{𝔭}(k)}; � _{S}(k); 0; H^{1}(G_{k}(p)); H^{1}(T_{S})^{G_{S}(p)}; � ^{2}(G_{S}(p)); 0]

is exact. This finishes the proof of (i).

Now let T ⊇ S be sets of primes of k. Using (i) and passing to limit, we obtain

H^{1}(G(k(p)|k_{S}(p))) ⟶^{∼} \begin{matrix}⨁ & ′ \\ 𝔭∉S(k_{S}(p)) \end{matrix}H^{1}(T_{𝔭}(k))^{G_{𝔭}(k_{S}(p))},

as \underrightarrow{lim}_{k′⊆k_{S}(p)} � _{S}(k′) = 0 by assumption. From this assumption follows that \underrightarrow{lim}_{k′⊆k_{S}(p)} � _{T} (k′) = 0, as � _{S}(k′) surjects onto � _{T} (k′). Thus we also obtain an isomorphism

H^{1}(G(k(p)|k_{T} (p)))^{G(k_{T} (p)|k_{S}(p))} ⟶^{∼} \begin{matrix}⨁ & ′ \\ 𝔭∉T (k_{S}(p)) \end{matrix}H^{1}(T_{𝔭}(k))^{G_{𝔭}(k_{S}(p))}

Now the the exact sequence

0 → H^{1}(G(k_{T} (p)|k_{S}(p))) → H^{1}(G(k(p)|k_{S}(p))) → H^{1}(G(k(p)|k_{T} (p)))^{G(k_{T} (p)|k_{S}(p))}.

implies assertion (ii).

If S_{∞} ∪ S_{p} ⊆ S, then we have an isomorphism of finite groups

� ^{2}(G_{S}(p)) ≅ � _{S}(k)

by [3] (10.4.8) and (8.6.9). Therefore the map

H^{1}(G(k(p)|k_{S}(p)))^{G_{S}(k)(p)} ⟶^{∼} ⨁_{𝔭∉S(k)}H^{1}(T_{𝔭}(k))^{G_{𝔭}(k)},

is an isomorphism. Passing to the limit and observing that G_{𝔭}(k_{S}(p)) = T_{𝔭}(k) for 𝔭 ∉ S as k_{S}(p) contains the cyclotomic ℤ_{p}-extension, we obtain

H^{1}(G(k(p)|k_{S}(p))) ⟶^{∼} \begin{matrix}⨁ & ′ \\ 𝔭∉S(k_{S}(p)) \end{matrix}H^{1}(T_{𝔭}(k)).

By the same argument as in (ii), the last assertion follows. □

Proposition 1.4 Let p be a prime number, k a the number field and T ⊇ S sets of primes of k. Assume that

(i) \underrightarrow{lim}_{k′⊆k_{S}(p)} � _{S}(k′) = 0,

(ii) the local extensions (k_{S}(p))_{𝔭}|k_{𝔭} are infinite for all 𝔭 ∈ T_{min}\S_{∞}, and, if p = 2, then (k_{S}(2))_{𝔭} = ℂ for all 𝔭 ∈ S ∩ S_{∞}.

Then there is a free product decomposition

∗_{𝔭∈T \S(k_{S}(p))}T_{𝔭}(k) ⟶^{∼} G(k_{T} (p)|k_{S}(p)).

Proof: We may assume that T = T_{min}. Since (k_{S}(p))_{𝔭}|k_{𝔭} is infinite for a prime 𝔭 ∈ T \(S ∪ S_{∞}), the field k_{S}(p)_{𝔭} is the maximal unramified p-extension of k_{𝔭}. Using proposition (1.3)(ii), it follows that

H^{1}(G(k_{T} (p)|k_{S}(p))) ⟶^{∼} \begin{matrix}⨁ & ′ \\ 𝔭∈T \S(k_{S}(p)) \end{matrix}H^{1}(T_{𝔭}(k)).

Now we consider the exact sequence

0 ⟶ � ^{2}(G_{T} (k′)(p)) ⟶ H^{2}(G(k_{T} (p)|k′)) ⟶ ⨁_{𝔭∈T (k′)}H^{2}(G_{𝔭}(k′)),

where k′ is a finite extension of k inside k_{S}(p). Passing to the limit, we obtain

0 ⟶ \underrightarrow{lim}_{k′⊆k_{S}(p)} � ^{2}(G_{T} (k′)(p)) ⟶ H^{2}(G(k_{T} (p)|k_{S}(p))) ⟶ \begin{matrix}⨁ & ′ \\ 𝔭∈T (k_{S}(p)) \end{matrix}H^{2}(G_{𝔭}(k_{S}(p))).

By proposition (1.3)(i), we have an injection

� ^{2}(G_{T} (k′)(p)) ↪ � _{T} (k′),

and the group on the right-hand side is an homomorphic image of � _{S}(k′). Since \underrightarrow{lim}_{k′⊆k_{S}(p)} � _{S}(k′) is trivial by assumption, it follows that

\underrightarrow{lim}_{k′⊆k_{S}(p)} � ^{2}(G_{T} (k′)(p)) = 0.

Furthermore, H^{2}(G_{𝔭}(k_{S}(p))) ≅ H^{2}(G(k_{𝔭}(p)|k_{S}(p)_{𝔭})) = 0 for all 𝔭 ∈ T \S_{∞} as k_{S}(p)_{𝔭}|k_{𝔭} is infinite, see [3] (7.1.8)(i), (7.5.8). It follows that

H^{2}(G(k_{T} (p)|k_{S}(p))) ⟶ \begin{matrix}⨁ & ′ \\ 𝔭∈(S_{∞}∩(T \S))(k_{S}(p)) \end{matrix}H^{2}(T_{𝔭}(k)) = \begin{matrix}⨁ & ′ \\ 𝔭∈T \S(k_{S}(p) \end{matrix}H^{2}(T_{𝔭}(k))

is injective. Thus we proved that

H^{i}(G(k_{T} (p)|k_{S}(p))) ⟶ H^{i}( ∗_{𝔭∈T \S(k_{S}(p))}T_{𝔭}(k))

is an isomorphism for i = 1 and injective for i = 2. By [3](1.6.15), the desired result follows. □

2 The K(π, 1)-property

A locally noetherian scheme Y is called a K(π, 1) for a prime number p if the higher homotopy groups of the p-completion Y_{et}^{(p)} of its etale homotopy type Y_{et} vanish, see [5] §2.

Let p a fixed prime number. Let k be a number field and S a finite set of nonarchimedean primes of k. We assume that k is totally imaginary if p = 2. For the scheme X = Spec(O_{k})\S we have

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

where we omit the base point. We consider the property

K(O_{k}, S) : Spec(O_{k})\S is a K(π, 1) for p.

If S is infinite, one can extend the notion of being a K(π, 1) for p in an obvious manner, see [5] §4. In the following we write H_{et}^{i}(Spec(O_{k})\S) for the group H_{et}^{i}(Spec(O_{k})\S, ℤ/pℤ) and h^{i}(Spec(O_{k})\S) = dim_{𝔽_{p}} H_{et}^{i}(Spec(O_{k})\S)

Proposition 2.1 Let p be a prime number and k a number field where p is odd or k is totally imaginary. Let S be a non-empty set of non-archimedean primes of k. Then the following assertions are equivalent:

(i) Spec(O_{k})\S is a K(π, 1) for p. (ii) cd_{p} G_{S}(p) ≤ 2 and the canonical map

H^{2}(G_{S}(p)) ↪ H_{et}^{2}(Spec(O_{k})\S)

is surjective. (iii) cd_{p} G_{S}(p) ≤ 2, � ^{2}(G_{S}(p)) →^{∼} � _{S}(k) and dim_{𝔽_{p}} coker^{2}(G_{S}(p)) = δ.

(iv) cd_{p} G_{S}(p) ≤ 2, H^{1}(G(k_{T} (p)|k_{S}(p)))^{G_{S}(p)} ⟶^{∼} ⨁_{𝔭∈T \S} H^{1}(T_{𝔭}(k))^{G_{𝔭}(k)}

for some set T containing S ∪ S_{p} and dim_{𝔽_{p}} coker^{2}(G_{S}(p)) = δ. If S is finite, then these assertions are equivalent to

(v) cd_{p} G_{S}(p) ≤ 2 and χ(G_{S}(p)) = r_{1}(k) + r_{2}(k) − ∑_{𝔭∈S∩S_{p}}[k_{𝔭} : ℚ_{p}].

Proof: For the equivalence (i)⇔ (ii) see [5] cor. 3.5. In order to show (ii)⇔ (iii) we only have to consider the commutative and exact diagram

[figure: � ^{2}(G_{S}(p)); H^{2}(G_{S}(p)); ∏_{𝔭∈S}H^{2}(G_{𝔭}(k)); � _{S}(k); H_{et}^{2}(Spec(O_{k})\S); ∏_{𝔭∈S}H^{2}(G_{𝔭}(k)); H_{et}^{3}(Spec(O_{k})),]

where dim_{𝔽_{p}} H_{et}^{3}(Spec(O_{k}))) = δ, see [5] thm.3.4 and thm.3.6.

By (1.3)(i), the surjectivity of the map � ^{2}(G_{S}(p)) → � _{S}(k) is equivalent to

H^{1}(G(k(p)|k_{S}(p)))^{G_{S}(p)} ⟶^{∼} ⨁_{𝔭∉S}H^{1}(T_{𝔭}(k))^{G_{𝔭}(k)}.

Using T ⊇ S_{p} and (1.3)(iii), we obtain

H^{1}(G(k(p)|k_{T} (p)))^{G_{T} (p)} ⟶^{∼} ⨁_{𝔭∉T}H^{1}(T_{𝔭}(k))^{G_{𝔭}(k)}.

Therefore the commutative and exact diagram

\xymatrix{⨁_{𝔭∈T \S} H^{1}(T_{𝔭}(k))^{G_{𝔭}(k)} \ar@{^{(}->}[r] & ⨁_{𝔭∉S}H^{1}(T_{𝔭}(k))^{G_{𝔭}(k)} \ar@{->>}[r] & ⨁_{𝔭∉T}H^{1}(T_{𝔭}(k))^{G_{𝔭}(k)} \\ H^{1}(k_{T} (p)|k_{S}(p))^{G_{S}(p)} \ar@{^{(}->}[r] \ar@{^{(}->}[u] & H^{1}(k(p)|k_{S}(p))^{G_{S}(p)} \ar[r] \ar@{^{(}->}[u] & H^{1}(k(p)|k_{T} (p))^{G_{T} (p)} \ar[u]_{≅}}

shows (iii)⇔ (iv).

Now let S be finite. By [5] prop.3.2,

\begin{aligned}χ(Spec(O_{k})\S) & := & ∑_{i}(−1)^{i}h^{i}(Spec(O_{k})\S) \\ & = & r_{1}(k) + r_{2}(k) − ∑_{𝔭∈S∩S_{p}}[k_{𝔭} : ℚ_{p}]. \end{aligned}

Since cd_{p} G_{S}(p) ≤ 2, we have

\begin{aligned}χ(G_{S}(p)) & = & ∑_{i=0}^{2}(−1)^{i}h^{i}(G_{S}(p)) \\ & = & χ(Spec(O_{k})\S) + h^{2}(G_{S}(p)) − h^{2}(Spec(O_{k})\S). \end{aligned}

This shows (ii)⇔(v).

□

Remarks:

(i) If S contains S_{p}, then Spec(O_{k})\S is a K(π, 1) for p. This follows from the equivalence (i)⇔(iv) of proposition (2.1) and [3] (8.3.18),(10.4.9), see also [5] prop.2.3 (ii) Let p be a prime number and k a number field where p is odd or k is totally imaginary. Let S be a non-empty finite set of non-archimedean primes of k. Assume that Spec(O_{k})\S is a K(π, 1) for p. Then the sequence

0 ⟶ � _{S}(k) ⟶ H^{2}(G_{S}(p)) ⟶ ∏_{𝔭∈S}H^{2}(G_{𝔭}) \underrightarrow{∑} H^{0}(G_{k}, μ_{p})^{∨} ⟶ 0

is exact, where Σ is the dual map of the diagonal embedding

H^{0}(G_{k}, μ_{p}) → ∏_{𝔭∈S}H^{0}(G_{k_{𝔭}}, μ_{p}) ≅ ∏_{𝔭∈S}H^{2}(G_{𝔭})^{∨}.

This follows from (2.1)(i)⇔(iii) and the commutative and exact diagram

\xymatrix{H^{2}(G_{S}(p)) \ar[r] \ar[d] & ∏_{𝔭∈S}H^{2}(G_{𝔭}(k)) \ar@{^{(}->}[d] \ar[r] \ar@{->>}[dr] & coker^{2}(G_{S}(p)) \ar@{->>}[d] \ar[r] & 0 \\ H^{2}(G_{S∪S_{p}}(p)) \ar[r] & ∏_{𝔭∈S∪S_{p}} H^{2}(G_{𝔭}(k)) \ar[r]^{∑} & H^{0}(G_{k}, μ_{p})^{∨} \ar[r] & 0,}

where the lower exact sequence is part of the 9-term exact sequence of the theorem of Poitou-Tate.

The following proposition is taken from [5] cor.2.2, and the proof presented here from [1].

Proposition 2.2 Let p be a prime number and k a number field where p is odd or k is totally imaginary. Let S be a non-empty finite set of non-archimedean primes of k. Let k′|k be a finite extension inside k_{S}(p). Then the following assertions are equivalent:

(i) Spec(O_{k})\S is a K(π, 1) for p.

(ii) Spec(O_{k′})\S is a K(π, 1) for p.

Proof: Let R(k, S) = r_{1}(k) + r_{2}(k) − ∑_{𝔭∈S∩S_{p}}[k_{𝔭} : ℚ_{p}]. Since p is odd or k is totally imaginary, we have R(k′, S) = [k′ : k]R(k, S). Therefore, using the equivalence (i)⇔(v) of proposition (2.1) and χ(G_{S}(k′)(p)) = χ(G_{S}(k)(p))[k′ : k], assertion (i) implies (ii). Conversely, let k′′|k be a finite extension inside k_{S}(p) containing k′. Then, using the implication (i)⇒(ii), we obtain

χ_{2}(G_{S}(k′′)(p)) = χ(G_{S}(k′′)(p))
= χ(Spec(O_{k′′})\S)
= [k′′ : k]χ(Spec(O_{k})\S)
= [k′′ : k](χ_{2}(G_{S}(k)(p)) + h^{2}(G_{S}(k)(p)) − h^{2}(Spec(O_{k})\S))
≥ [k′′ : k]χ_{2}(G_{S}(k)(p))

Using [3] (3.3.15) equality follows, and so h^{2}(G_{S}(k)(p)) = h^{2}(Spec(O_{k})\S), and by [3] (3.3.16), cd_{p} G_{S}(k)(p) ≤ 2. □

The following proposition is taken from [5] thm.9.1.

Proposition 2.3 Let p be a prime number and k a number field where p is odd or k is totally imaginary. Assume that Spec(O_{k})\S is a K(π, 1) for p and G_{S}(p) ≠ 1. Then k_{S}(p) realizes the maximal p-extension k_{𝔮}(p) of k_{𝔮} where 𝔮 ∈ S_{min}\S_{p}.

Proof: We have only to show that 𝔮 ramifies in k_{S}(p)|k. Suppose not, then k_{S}(p) = k_{S′}(p), where S′ = S\{𝔮}. By proposition (2.1)(i)⇔(v), it follows that Spec(O_{k})\S′ is a K(π, 1) for p, and so � ^{2}(G_{S′}(p)) →^{∼} � _{S′}(k). The commutative and exact diagram

[figure: 0; � _{S′}(k); H^{2}(G_{S′}(p)); ∏_{𝔭∈S′} H^{2}(G_{𝔭}(k)); 0; � _{S}(k); H^{2}(G_{S}(p)); ∏_{𝔭∈S}H^{2}(G_{𝔭}(k)),]

shows that � _{S′}(k) →^{∼} � _{S}(k). Using [3] (10.7.12), it follows that h^{1}(G_{S}(p)) = h^{1}(G_{S′}(p)) + 1 which is a contradiction. □

Let T ⊇ S be sets of nonarchimedean primes of k. We consider the properties

L_{0}(k, S, T) : (k_{S}(p))_{𝔭} ≠ k_{𝔭} for all 𝔭 ∈ (T \S) ∩ S_{p} ,
L_{1}(k, S, T) : (k_{S}(p))_{𝔮} ≠ k_{𝔮} for all 𝔮 ∈ (T \(S ∪ S_{p}))_{min} ,
R(k, S, T) : ∗_{𝔭∈T \S(k_{S}(p))}T_{𝔭}(k) ⟶^{∼} G(k_{T} (p)|k_{S}(p)).

Using the subgroup theorem for free products, see [3](4.2.1), one has

R(k, S, T) ⇒ (R(k, U, T) and R(k, S, U)),

where T ⊇ U ⊇ S.

If T ∩ S_{p} = ∅, then one part of the following theorem is also proved in [5] prop.8.1 and cor.8.2.

Theorem 2.4 (Reducing and enlarging the set of primes) Let p be a prime number and k a number field where p is odd or k is totally imaginary. Let T ⊇ S be finite sets of nonarchimedean primes of k. Assume that L_{0}(k, S, T) holds. Then we have the following assertions are equivalent:

(i) Spec(O_{k})\S is a K(π, 1) for p and (k_{S}(p))_{𝔮} ≠ k_{𝔮} for all 𝔮 ∈ (T \(S ∪S_{p}))_{min}.

(ii) Spec(O_{k})\T is a K(π, 1) for p and

∗_{𝔭∈T \S(k_{S}(p))}T_{𝔭}(k) ⟶^{∼} G(k_{T} (p)|k_{S}(p)).

The implication (i) ⇒(ii) also holds when S or T is infinite.

Proof: Assume that L_{1}(k, S, T) and K(O_{k}, S) holds. We may further assume that (T \S)_{min} ≠ ∅; in particular, G_{S}(p) ≠ 1. By proposition (2.2), it follows that K(O_{k′}, S) for all finite extensions k′|k inside k_{S}(p). Thus, using proposition (2.1) (i)⇔(iii),

\underrightarrow{lim}_{k′⊆k_{S}(p)} � _{S}(k′) = \underrightarrow{lim}_{k′⊆k_{S}(p)} � ^{2}(G_{S}(k′)(p)) ⊆ \underrightarrow{lim}_{k′⊆k_{S}(p)} H^{2}(G_{S}(k′)(p)) = 0.

Using proposition (2.3) and L_{i}(k, S, T), i = 0, 1, we see that (k_{S}(p))_{𝔮} ≠ k_{𝔮} for all 𝔮 ∈ T_{min}. By proposition (1.4), it follows that R(k, S, T) holds. The spectral sequence

H^{i}(G_{S}(p), H^{j}(G(k_{T} (p)|k_{S}(p))) ⇒ H^{i+j}(G_{T} ((p)))

now shows that cd_{𝔭} G_{T} (p) ≤ 2. Consider the commutative and exact diagram

[figure: ⨁_{𝔭∈T \S} H^{1}(T_{𝔭}(k))^{G_{𝔭}(k)}; ⨁_{𝔭∉S}H^{1}(T_{𝔭}(k))^{G_{𝔭}(k)}; ⨁_{𝔭∉T}H^{1}(T_{𝔭}(k))^{G_{𝔭}(k)}; H^{1}(k_{T} (p)|k_{S}(p))^{G_{S}(p)}; ≅; H^{1}(k(p)|k_{S}(p))^{G_{S}(p)}; res; H^{1}(k(p)|k_{T} (p))^{G_{T} (p)},]

Since K(O_{k}, S) holds, the map res is an isomorphism, and we obtain

H^{1}(k(p)|k_{T} (p))^{G_{T} (p)} ⟶^{∼} ⨁_{𝔭∉T}H^{1}(T_{𝔭}(k))^{G_{𝔭}(k)}.

Using proposition (2.1)(i)⇔(iv), it follows that K(O_{k}, T) holds.

Conversely, assume that K(O_{k}, T) and R(k, S, T) hold. Then, by lemma (1.1), we obtain cd_{p} G_{S}(p) ≤ 3. It follows exactly in the same way as in the proof of corollary (1.2), using the remark (ii), that cd_{p} G_{S}(p) ≤ 2. Furthermore, since cd_{p} G_{S}(p) ≤ 2, cd_{p} G_{T} (p) ≤ 2 and R(k, S, T) holds, we can apply lemma (1.1) and obtain

χ(G_{T} (p)) − χ(G_{S}(p)) = ∑_{𝔭∈(T \S)_{min}}( dim_{𝔽_{p}} H^{1}(T_{𝔭}(k))^{G_{𝔭}(k)} − dim_{𝔽_{p}} H^{2}(G_{𝔭}(k)))
= ∑_{𝔭∈(T \S)∩S_{p}}( dim_{𝔽_{p}} H^{1}(T_{𝔭}(k))^{G_{𝔭}(k)} − dim_{𝔽_{p}} H^{2}(G_{𝔭}(k)))
= ∑_{𝔭∈(T \S)∩S_{p}}[k_{𝔭} : ℚ_{p}]

Using proposition (2.1)(i)⇔(v), we see that K(O_{k}, S) holds.

Let 𝔮 ∈ (T \(S ∪ S_{p}))_{min}. By proposition (2.3), G_{𝔮}(k) is a subgroup of G(k_{T} (p)|k). Since cd_{p} G_{𝔮}(k) = 2, it can not be a subgroup of the free pro-p group G(k_{T} (p)|k_{S}(p)). Therefore G_{𝔮}(k_{S}(p)|k) is non-trivial, and so L_{1}(k, S, T) holds. □

Using remark (i), theorem (2.4) in the case T = S ∪ S_{p}, and

R(k, S, S ∪ S_{p}) ⇒ R(k, S ∪ W, S ∪ S_{p})

for W ⊆ S_{p}, we obtain

Corollary 2.5 Let p be a prime number and k a number field where p is odd or k is totally imaginary. Let S be a finite set of nonarchimedean primes of k with S ∩ S_{p} = ∅ and W ⊆ S_{p}. Assume that (k_{S}(p))_{𝔭} ≠ k_{𝔭} for 𝔭 ∈ S_{p}. Then

(i)

K(O_{k}, S) ⇔ R(k, S, S ∪ S_{p}).

(ii) Assume that K(O_{k}, S) holds. Then also K(O_{k}, S ∪ W) holds, and in partic-

ular,

cd_{p} G(k_{W ∪S}(p)|k) = 2.

Corollary 2.6 Let p be a prime number and k a number field where p is odd or k is totally imaginary. Let S be a finite set of nonarchimedean primes of k with Spec(O_{k})\S is a K(π, 1) for p. Then there exists a set T of nonarchimedean primes with T ∩ S = ∅ and δ(T) = 1, such that there are free product decompositions

(i)

∗_{𝔭∈T (k_{S}(p))}T_{𝔭}(k) ⟶^{∼} G(k_{T ∪S}(p)|k_{S}(p)),

(ii)

∗_{𝔭∉(T ∪S)(k_{T ∪S}(p))}T_{𝔭}(k) ⟶^{∼} G(k(p)|k_{T ∪S}(p)).

Proof: Since k_{S}(p)|k is infinite, it follows from Čebotarev density theorem that the set

V = {𝔮 a prime of k | 𝔮 is completely decomposed in k_{S}(p)|k}

has density zero. Let T be the complement of the set S_{∞} ∪S ∪V, hence δ(T) = 1. By theorem (2.4), we obtain that Spec(O_{k})\(T ∪ S) is a K(π, 1) for p and that there is an isomorphism

∗_{𝔭∈(T ∪S)(k_{S}(p))}T_{𝔭}(k) ⟶^{∼} G(k_{T ∪S}(p)|k_{S}(p)).

Since δ(T ∪ S) = 1, it follows from [3] (10.5.9) that we have the desired decomposition (ii). □

Remarks: (1) If S contains S_{p}, then the corollary above is well-known, see [3] (10.5.1): one can take for T all primes not in S.

(2) It is easy to see, that the corollary above implies that the pro-p-group G(k(p)|k_{S}(p)) is minimal generated by a system of minimal generators of the inertia groups T_{𝔭}(k), 𝔭 ∉ S, with defining relations given by the local relations of the groups G_{𝔭}(k), 𝔭 ∈ V.

Using a result of A.Schmidt we will give another application of theorem (2.4). We start with a lemma and introduce the following notation: For a prime number q with q ≡ 1 mod p let L_{q,p} be the maximal p-extension of ℚ inside ℚ(ζ_{q}), where ζ_{q} is a primitive q-th root of unity.

Lemma 2.7 Let p be a prime number and k a number field.

(i) Let r ∈ ℕ. Then the set M_{r}(k) of prime numbers q which are completely decomposed in k and for which the congruences

q ≡ 1 mod p^{2r} and p^{\frac{q−1}{p^{r}}} ≢ 1 mod q

hold has density 1/[k(ζ_{p^{2r}}) : ℚ] − 1/[k(ζ_{p^{2r}}, \sqrt[p^{r}]{p}) : ℚ]. (ii) The set of prime numbers q ≡ 1 mod p which are completely decomposed in

k and which have the property that (L_{q,p} k)_{𝔭} ≠ k_{𝔭} for all 𝔭 ∈ S_{p} has positive density.

Proof: (i) Let q be a prime number which is completely decomposed in k(ζ_{p^{2r}}); in particular, we have q ≡ 1 mod p^{2r}. Let 𝔮 be a prime of k(ζ_{p^{2r}}) above q. Then

p^{\frac{N(𝔮)−1}{p^{r}}} ≡ 1 mod 𝔮 , i.e. ( \sqrt[p^{r}]{p})^{N(𝔮)} ≡ ( \sqrt[p^{r}]{p}) mod 𝔮 ,

if and only if 𝔮 is completely decomposed in k(ζ_{p^{2r}}, \sqrt[p^{r}]{p})). Therefore the density of the set

\{q is completely decomposed in k, q ≡ 1 mod p^{2r} , p^{\frac{q−1}{p^{r}}} ≡ 1 mod q\}

is equal to 1/[k(ζ_{p^{2r}}) : ℚ]·1/[k(ζ_{p^{2r}}, \sqrt[p^{r}]{p}) : k(ζ_{p^{2r}})], and the set M_{r}(k) has density 1/[k(ζ_{p^{2r}}) : ℚ] · (1 − 1/[k(ζ_{p^{2r}}, \sqrt[p^{r}]{p}) : k(ζ_{p^{2r}})]).

(ii) Let r ∈ ℕ be big enough such that \sqrt[p^{r}]{p} ∉ k(ζ_{p^{2r}}) and p^{r} > [k : ℚ]. Then, by (i), the set M_{r}(k) has positive density. Obviously, if q ≡ 1 mod p^{2r} and p^{\frac{q−1}{p^{r}}} ≢ 1 mod q, then the local unramified extension (L_{q,p})_{p}|ℚ_{p} has degree at least p^{r}. Therefore (L_{q,p} k)_{𝔭} is a non-trivial unramified extension of k_{𝔭} for 𝔭|p.

□

Proposition 2.8 Let p be a prime number and k a number field where p is odd or k is totally imaginary. Assume that μ_{p} ⊈ k and Cl_{k}(p) = 0. Let S be a finite set of nonarchimedean primes of k with S ∩ S_{p} = ∅ and W ⊆ S_{p}. Let, in addition, T be a set of primes of Dirichlet density δ(T) = 1. Then there exists a finite subset T_{1} ⊆ T such that K(O_{k}, W ∪ S ∪ T_{1}) holds and

∗_{𝔭∈(S_{p}\W)(k_{W ∪S∪T_{1}}(p))}T_{𝔭}(k) ⟶^{∼} G(k_{S_{p}∪S∪T_{1}}(p)|k_{W ∪S∪T_{1}}(p)).

In particular,

cd_{p} G(k_{W ∪S∪T_{1}}(p)|k) = 2.

Proof: Obviously we may assume that T ∩ (S_{p} ∪ S_{∞}) = ∅ and that the underlying prime numbers of the primes of T are completely decomposed in k. We have to show that there exists a finite subset T_{1} ⊆ T such that (k_{S∪T_{1}}(p))_{𝔭} ≠ k_{𝔭} for 𝔭 ∈ S_{p} and that K(O_{k}, S ∪ T_{1}) holds.

Using lemma (2.7)(ii), there is a prime number q such that S_{q} ⊆ T and (k_{S∪S_{q}}(p))_{𝔭} ≠ k_{𝔭} for 𝔭 ∈ S_{p}. By a result of A. Schmidt, [5] thm.6.2, we obtain a finite subset T_{1} ⊆ T containing S_{q} with the desired properties. □

Remark: The proposition above shows that besides the tame case (W = ∅) also in the “mixed case” (W ≠ ∅ and W ≠ S_{p}) we have examples of Galois groups with cohomological dimension equal to 2.

References

[1] Forré, P. Über pro-p-Erweiterungen algebraischer Zahlkörper mit zahmer Verzweigung. Diplomarbeit, Heidelberg 2008 [2] Labute, J. Mild Pro-p-Groups and Galois Groups of p-Extensions of ℚ.

J. Reine u. Angew. Math. 596 (2006), 155-182

[3] Neukirch, J., Schmidt, A., Wingberg, K. Cohomology of Number Fields. 2nd edition, Springer 2008

[4] Schmidt, A. Circular sets of prime numbers and p-extensions of the rationals. J. Reine u. Angew. Math. 596 (2006),115-130

[5] Schmidt, A. Rings of integers of type K(π, 1). Doc. Math. 12 (2007) 441-471

[6] Schmidt, A. On the K(π, 1)-property of rings of integers in the mixed case. Preprint

[7] Vogel, D. Circular sets of primes of imaginary quadratic number fields. Preprints der Forschergruppe Algebraische Zykel und L-Funktionen. Regensburg/Leipzig Nr.5, 2006. http://www.mathematik.uniregensburg.de/FGAlgZyk

Mathematisches Institut der Universität Heidelberg Im Neuenheimer Feld 288 69120 Heidelberg Germany

e-mail: wingberg@mathi.uni-heidelberg.de