Free Product Decomposition of Galois Groups of Number Fields

by Kay Wingberg at Heidelberg

Let G(k(𝔠)|k) be the Galois group of the maximal pro-𝔠 extension k(𝔠) of a number field k, where 𝔠 is a full class of finite groups which is closed under taking subgroups, quotients and extensions. If 𝔭 is a prime of k and 𝔓 an extension of 𝔭 to k(𝔠), then the decomposition group G_{𝔓}(k(𝔠)|k) with respect to 𝔓 is isomorphic to the Galois group G(k_{𝔭}(𝔠)|k_{𝔭}) of the maximal pro-𝔠 extension k_{𝔭}(𝔠) of k_{𝔭}, cf. [4] theorem (9.3.1). In this paper we consider the question whether the decomposition groups G_{𝔓}(k(𝔠)|k) or the inertia groups T_{𝔓}(k(𝔠)|k) for some primes 𝔭 form a free pro-𝔠-product inside G(k(𝔠)|k). More precisely, if S(k) and T_{0}(k) are sets of primes of k, then

k_{S}(𝔠) is the maximal pro-𝔠 extension which is unramified outside S,
k^{T_{0}}(𝔠) is the maximal pro-𝔠 extension which is completely decomposed at T_{0},

and we have canonical homomorphisms

ϕ_{S}(𝔠) : ∗_{𝔭∉S(k_{S}(𝔠))}T_{𝔓}(k(𝔠)|k) ⟶ G(k(𝔠)|k_{S}(𝔠)),

of the free pro-𝔠-product of the groups T_{𝔓}(k(𝔠)|k) into G(k(𝔠)|k_{S}(𝔠)) and, if T_{0} is finite,

ϕ^{T_{0}}(𝔠) : ∗_{𝔭∈T_{0}(k^{T_{0}}(𝔠))}G_{𝔓}(k(𝔠)|k) ⟶ G(k(𝔠)|k^{T_{0}}(𝔠)),

of the free pro-𝔠-products of the groups T_{𝔓}(k(𝔠)|k) and G_{𝔓}(k(𝔠)|k), respectively; here the prime 𝔓 is a fixed extensions of 𝔭 to k(𝔠). It is known that the answer of the questions, whether ϕ_{S}(𝔠) or ϕ^{T_{0}}(𝔠) are isomorphisms, do not depend on the choice of the extensions 𝔓|𝔭 if 𝔠 is the class of finite p-groups, p a prime number, but in general it does, see [4] chap.IV §2. Observe further that in general the sets S(k_{S}(𝔠)) (resp. its complement) and T_{0}(k^{T_{0}}(𝔠)) are infinite, and we recall the definition of free profinite products over profinite spaces in section 1 (see [4]). Let us collect some known results.

Communicated by A.Tamagawa. Received July 22, 2003.

2000 Mathematics Subject Classifications: 11R32, 11R34, 20E18.

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

If 𝔠 is the class of finite p-groups and S(k) is an arbitrary set containing the set S_{p}(k) of primes above p, then ϕ_{S} is an isomorphism, see [4] theorem (10.5.1) for the case that p is odd or S(k) contains the set S_{ℝ} of real places of k and [6] theorem 2 in the remaining case.

If 𝔠 is the class of all finite groups (and T_{0}(k) is finite), then there exist suitable extensions 𝔓|𝔭 such that ϕ^{T_{0}} is an isomorphism of profinite groups. If T_{0} = S_{∞} is the set archimedean primes, this is a result of Fried-Haran-Völklein [1] and in general it is proven by Pop [5].

We also would like to mention a result of Neukirch, [3] Satz (12.2), which says that finitely many decomposition groups G_{𝔓}(k(p)|k) form a free pro-p-product inside G(k(p)|k).

In this paper we will prove that ϕ^{T_{0}}(𝔠) is an isomorphism if 𝔠 is the class of finite p-groups. We get even more: let S, T_{0} be sets of primes of k such that S_{p} ∪ S_{∞} ⊂ S, the density δ(S) of S is equal to 1 and T_{0} is a finite subset of S and let

k_{S}^{T_{0}}(p) be the maximal pro-p extension of k which is unramified outside S

and completely decomposed at T_{0}.

Then we have a canonical isomorphism

ϕ_{S}^{T_{0}} : ∗_{𝔭∈T_{0}(k_{S}^{T_{0}}(p))}G_{𝔭}(k(p)|k) ∗ ∗_{𝔭∉S(k_{S}^{T_{0}}(p))}T_{𝔭}(k(p)|k) ⟶^{∼} G(k(p)|k_{S}^{T_{0}}(p)).

Observe that T_{0} may contain the set S_{p} ∪ S_{∞} (and then the cyclotomic ℤ_{p}- extension of k is not contained in k_{S}^{T_{0}}(p)). Furthermore we would like to mention that this result, in the case where S is the set of all places of k, is not an easy consequence of the theorem of Pop, although the proof of that theorem is much more difficult than this one which is only based on the theorem of Grunwald/Wang. From Pop’s result it only follows that we have a free pro-p product decomposition into decomposition groups over the field k^{T_{0}} (the maximal Galois extension of k which is completely decomposed at all primes of T_{0}) whereas we show that this decomposition already exists over the maximal p-extension of k inside k^{T_{0}}.

Finally, I would like to thank Alexander Schmidt for many helpful discussions on this subject.

1 Free Products

We briefly collect some facts on free products of profinite groups. For a more detailed presentation and proofs we refer the reader to [4], chap. IV and chap. X §1 and [2].

A profinite space is a topological inverse limit of finite discrete spaces and a pro-𝔠-group is a profinite group which is the inverse limit of groups in a full class of finite groups 𝔠 which is closed under taking subgroups, quotients and extensions.

Definition 1.1 Let T be a profinite space. A bundle of pro-𝔠-groups

G = (G, pr, T)

over T is a group object in the category of profinite spaces over T together with a continuous structure map pr : G → T such that the fibre G_{t} of G over every point t ∈ T is a pro-𝔠-group.

Example 1. If G is a pro-𝔠-group and T a profinite space, then we always have the constant bundle (G × T, pr, T), where pr is the projection G × T → T.

Example 2. Let {G_{t}}_{t∈T} be a continuous family of closed subgroups of a pro𝔠-group G indexed by the points of a profinite space T, i.e. it has the property that for every open subgroup U ⊆ G the set T (U) = {t ∈ T | G_{t} ⊆ U} is open in T. Then

G = {(g, t) ∈ G × T | g ∈ G_{t}}

is in a natural way a bundle of pro-𝔠-groups over T.

Example 3. Let T̄ = T ∪ {∗} be the one-point compactification of a discrete set T. Let {G_{i}}_{i∈T} be a (discrete) family of pro-𝔠-groups. Then the bundle G over T̄ is defined by

G := ⋃ ._{i∈T} G_{i} ⋃ . {∗} ,

with the following topology: G_{i} ⊆ G (together with its profinite topology) is open in G for all i, and for every open neighbourhood V ⊆ T̄ of ∗ ∈ T̄, let

⋃ ._{\begin{matrix}i ∈ V \\ i ≠ ∗ \end{matrix}} G_{i} ⋃ . \{∗\}

be an open neighbourhood of ∗ ∈ G. One checks that the map

p : G → T̄; G_{i} ∋ g_{i} ↦ i, ∗ ↦ ∗

is continuous. Viewing {∗} as the group with one element, we see that the group operations on the G_{i}’s induce the structure of a bundle of pro-𝔠-groups on the triple (G, p, T̄).

Definition 1.2 A morphism of bundles

ϕ : (G, pr_{G}, T) → (H, pr_{H}, S)

is a pair ϕ_{G} : G → H, ϕ_{T} : T → S of continuous maps such that

(i) the diagram

\xymatrix{G \ar[r]^{ϕ_{G}} \ar[d]_{pr_{G}} & H \ar[d]^{pr_{H}} \\ T \ar[r]_{ϕ_{T}} & S}

commutes and

(ii) for every t ∈ T the associated map ϕ_{t} : G_{t} → H_{ϕ_{T} (t)} is a group homomor-

phism. We say that ϕ is surjective if ϕ_{G} (and hence also ϕ_{T}) is surjective.

We will not distinguish between the pro-𝔠-group G and the bundle (G, pr, {∗}) over the one point space {∗}. In particular, a morphism from a bundle (G, pr, T) to a group G is a continuous map ϕ : G → G such that the induced maps ϕ_{t} : G_{t} → G are group homomorphisms for every t ∈ T.

Definition 1.3 The free pro-𝔠-product of a bundle (G, pr, T) of pro-𝔠-groups is a pro-𝔠-group

G = ∗_{T} G

together with a morphism ω : G → G, which has the following universal property: for every morphism f : G → H from G to a pro-𝔠-group H there exists a unique homomorphism of pro-𝔠-groups ϕ : G → H with f = ϕ ∘ ω.

Proposition 1.4 The free pro-𝔠-product of the bundle G of pro-𝔠-groups over T exists and is unique up to unique isomorphism.

Let G be a bundle of pro-𝔠-groups over a profinite space T and let {G_{t}}_{t∈T} be its fibers, i.e.

G = ⋃ ._{t∈T} G_{t}.

For the free pro-𝔠-product of the bundle (G, pr, T) we often write

∗_{T} G = ∗_{t∈T} G_{t}.

Proposition 1.5 Let

(G, p, T) = \underleftarrow{lim}_{i∈I}(G_{i}, p_{i}, T_{i})

be the inverse limit of the pro-𝔠-group bundles G_{i}. Then

∗_{T} G = \underleftarrow{lim}_{i∈I} ∗_{T_{i}} G_{i}.

Let G be a pro-𝔠-group and let {G_{t}}_{t∈T} be a continuous family of closed subgroups indexed by the points of a profinite space T, and so G = {(g, t) ∈ G × T | g ∈ G_{t}} is a bundle over T. We have a canonical homomorphism

ϕ : ∗_{T} G ⟶ G.

Definition 1.6 A pro-𝔠-group G is the free product of the continuous family {G_{t}}_{t∈T} if ϕ is an isomorphism.

Now we consider the cohomology of a free product in the following case:

Let T = \underleftarrow{lim}_{λ} T̄_{λ}, where the sets T̄_{λ} = T_{λ} ∪ \{∗_{λ}\} are the one-point compactifications of discrete sets T_{λ}. Let G = \underleftarrow{lim}_{λ} G_{λ} be the projective limit of bundles G_{λ} = ⋃ . _{t_{λ}∈T_{λ}} G_{t_{λ}} ⋃ . {∗_{λ}}.

Let A be an abelian torsion group on which the groups ∗T̄_{λ} G_{λ} act and assume that these actions are compatible with respect to the transition maps. Then A is also a G-module where G = ∗_{T} G.

Proposition 1.7 With the notation and assumptions as above there is an exact sequence

0 → A/A^{G} → \underrightarrow{lim}_{λ} ⨁_{T_{λ}} A/A^{G_{t_{λ}}} → H^{1}(G, A) → \underrightarrow{lim}_{λ} ⨁_{T_{λ}} H^{1}(G_{t_{λ}}, A) → 0

and isomorphisms

H^{i}(G, A) = \underrightarrow{lim}_{λ} ⨁_{T_{λ}} H^{i}(G_{t_{λ}}, A), i ≥ 2.

Proof: Using proposition 1.5, we have

H^{i}(G, A) = \underrightarrow{lim}_{λ} H^{i}(∗_{T̄_{λ}} G_{λ}, A), i ≥ 0,

and so the result follows from [4], theorem (4.1.4). □

If k is a number field, then we denote the one-point compactification of the set of all places of k by Sp(k). The compactifying point will be denoted by η_{k}. For an infinite extension K|k, we set

Sp(K) = \underleftarrow{lim}_{k′} Sp(k′),

where k′|k runs through all finite subextensions of K. Let S be a set of primes of k and S̄ its closure in Sp(k). The pre-image S̄(K) of S̄ under the natural projection Sp(K) → Sp(k) is the closure of the set S(K) of all prolongations of primes in S to K in Sp(K).

Let M ⊇ K ⊇ k be (possibly infinite) extensions of k such that M|K is a Galois extension and G(M|K) is a pro-𝔠-group. Let S be a set of primes of k. For a fixed section s : S̄(K) → S̄(M) of the natural projection S̄(M) → S̄(K) we consider the family {G_{s(𝔭)}(M|K)}_{𝔭∈S̄(K)} of decomposition groups and the family {T_{s(𝔭)}(M|K)}_{𝔭∈S̄(K)} of inertia groups, where by convention G_{η_{M}} = {1}. Since a finite extension of number fields is ramified only at finitely many primes, the later is a continuous family of subgroups of G(M|K) indexed by S̄(K). We obtain a natural homomorphism

φ : ∗_{S(K)} T_{s(𝔭)}(M|K) → G(M|K)

and, if S(k) is finite, a homomorphism

ϕ : ∗_{S(K)} G_{s(𝔭)}(M|K) → G(M|K).

If 𝔠 is the class of p-groups, where p is a prime number, we have the following well-known

Lemma 1.8 Let f : G′ ⟶ G be a homomorphism of pro-p-groups. Then f is an isomorphism if and only if the induced homomorphism

H^{i}(f) : H^{i}(G, ℤ/pℤ) ⟶ H^{i}(G′, ℤ/pℤ)

is an isomorphism for i = 1 and injective for i = 2.

In particular, it follows that the questions whether the homomorphisms ϕ or φ are isomorphisms (in the case of pro-p-groups) do not depend on the section s. In the following we will omit the chosen section s in the notation.

For the cohomology of a free product with values in a torsion group A (considered as a module with trivial action) it follows from proposition 1.7 the

Proposition 1.9 With the notation as above let S_{1} and S_{2} be sets of primes of k where S_{1} is finite. Then, for i ≥ 1, the following holds:

H^{i}( ∗_{𝔭∈S_{1}(K)}G_{𝔭}(M|K) ∗ ∗_{𝔭∈S_{2}(K)}T_{𝔭}(M|K), A)

= \underrightarrow{lim}_{k′} ⨁_{𝔭∈S_{1}(k′)}H^{i}(G_{𝔭}(M′|k′), A) ⊕ \underrightarrow{lim}_{k′} ⨁_{𝔭∈S_{2}(k′)}H^{i}(T_{𝔭}(M′|k′), A).

where k′ runs through all finite subextensions of k in K and M′ is the maximal pro-𝔠 Galois subextension of M|k′ (so M = \underrightarrow{lim} M′).

The limits on the right-hand side depend on K and not on k and we denote them by

\begin{matrix}⨁ & ′ \\ 𝔭∈S_{1}(K) \end{matrix}H^{i}(G_{𝔭}(M|K), A) and _{\begin{matrix}⨁ & ′ \\ 𝔭∈S_{2}(K) \end{matrix}}H^{i}(T_{𝔭}(M|K), A).

2 Galois Groups of large Number Fields

Let k be a number field and P the set of all primes of k. Let p be a prime number and S ⊇ T sets of primes of k, where S contains the set S_{p} ∪ S_{∞} of archimedean primes and primes above p, and let k_{S} be the maximal extension of k which is unramified outside S.

We denote the density of a set S of primes of k by δ(S) and the inertia subgroup of the decomposition group G_{𝔭}(k) by T_{𝔭}(k). In the following we write H^{i}(−) for H^{i}(−, ℤ/pℤ).

One version of the theorem of Grunwald/Wang says (see [4], theorem (9.2.2))

Theorem 2.1 Let k be a number field and let S ⊇ T be sets of primes of k, where S ⊇ S_{p} ∪ S_{∞}, δ(S) = 1 and T is finite. Then the canonical homomorphism

H^{1}(k_{S}|k) ⟶ ⨁_{𝔭∈T}H^{1}(G_{𝔭}(k))

is surjective.

Corollary 2.2 Let k be a number field, and let T and T_{0} be sets of primes such that T_{0} is finite, δ(T) = 0 and T ∩ T_{0} = ∅. Then the canonical homomorphism

H^{1}(k̄|k) ⟶ ⨁_{𝔭∈T_{0}} H^{1}(G_{𝔭}(k)) ⊕ ⨁_{𝔭∈T}H^{1}(T_{𝔭}(k))^{G_{𝔭}(k)}

is surjective.

Proof: Let T_{1} = T_{0} ∪ S_{p} ∪ S_{∞} and S = (P\T) ∪ S_{p} ∪ S_{∞}. It follows that δ(S) = 1, P\S = T \(S_{p} ∪ S_{∞}) and T_{1} ⊂ S.

By theorem 2.1, the canonical map

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

is surjective. Furthermore, by [4] corollary (10.5.3), we have a canonical isomorphism

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

and so an isomorphism

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

Consider the commutative and exact diagram

\xymatrix{H^{1}(k̄|k_{S})^{G(k_{S}|k)} \ar[r]^{∼} & ⨁_{T \(S_{p}∪S_{∞})} H^{1}(T_{𝔭}(k))^{G_{𝔭}(k)} \\ H^{1}(k̄|k) \ar[r] \ar[u] & ⨁_{T_{1}} H^{1}(G_{𝔭}(k)) ⊕ ⨁_{T \(S_{p}∪S_{∞})} H^{1}(T_{𝔭}(k))^{G_{𝔭}(k)} \ar@{->>}[u] \\ H^{1}(k_{S}|k) \ar@{->>}[r] \ar@{^{(}->}[u] & ⨁_{T_{1}} H^{1}(G_{𝔭}(k)) \ar@{^{(}->}[u]}

We will show that the vertical upper map on the left-hand side is surjective. Using the Hochschild-Serre spectral sequence, we have to show that the inflation map

H^{2}(k_{S}|k) ⟶ H^{2}(k̄|k)

is injective. But this follows from the commutative diagram

\xymatrix{H^{2}(k̄|k) \ar@{^{(}->}[r] & ⨁_{P} H^{2}(G_{𝔭}(k)) \\ H^{2}(k_{S}|k) \ar@{^{(}->}[r] \ar[u] & ⨁_{S} H^{2}(G_{𝔭}(k)) \ar@{^{(}->}[u]}

where the lower horizontal map is injective since δ(S) = 1, see [4] theorem (9.1.8). It follows that the map

H^{1}(k̄|k) ⟶ ⨁_{T_{1}} H^{1}(G_{𝔭}(k)) ⊕ ⨁_{T \(S_{p}∪S_{∞})} H^{1}(T_{𝔭}(k))^{G_{𝔭}(k)}

is surjective. Since the maps H^{1}(G_{𝔭}(k)) → H^{1}(T_{𝔭}(k))^{G_{𝔭}(k)} are surjective and the set T is contained in (T_{1}\T_{0}) ∪ (T \(S_{p} ∪ S_{∞})), the desired result follows. □

For a number field k and sets S, T_{0} of primes of k recall that

k_{S}^{T_{0}}(p) is the maximal p-extension of k which is unramified outside S

and completely decomposed at T_{0}.

For a prime 𝔭 ∈ S and a fixed extension 𝔓 of 𝔭 to a number field K|k we denote the completion of K with respect to 𝔓 by K_{𝔭}.

Proposition 2.3 Let S, T_{0} be sets of primes of k with T_{0} ∪ S_{p} ∪ S_{∞} ⊆ S such that T_{0} is finite and δ(S) = 1. Then

(k_{S}^{T_{0}}(p))_{𝔭} = \{ \begin{matrix}k_{𝔭}(p), & if 𝔭 ∈ S\T_{0}, \\ k_{𝔭}^{nr}(p), & if 𝔭 ∉ S, \\ k_{𝔭}, & if 𝔭 ∈ T_{0}. \end{matrix}

Proof: Let T = P\S. For a prime 𝔭_{0} of k let 𝔓_{0} be a fixed extension to k_{S}^{T_{0}}(p). Let K|k be a finite Galois extension inside k_{S}^{T_{0}}(p) and let \overline{𝔓}_{0} be the restriction of 𝔓_{0} to K.

Let 𝔭_{0} ∈ S\T_{0}. By theorem 2.1, the canonical homomorphism

H^{1}(k_{S}|K) ⟶ H^{1}(G_{\overline{𝔓}_{0}}(K)) ⊕ ⨁_{T_{0}(K)} H^{1}(G_{𝔭}(k))

is surjective. In particular, for every α_{\overline{𝔓}_{0}} ∈ H^{1}(G_{\overline{𝔓}_{0}}(K)) there exists an element β ∈ H^{1}(k_{S}|K) which is mapped to (α_{\overline{𝔓}_{0}}, 0, …, 0). But β lies in the subgroup H^{1}(k_{S}^{T_{0}}(p)|K) of H^{1}(k_{S}|K). Therefore

H^{1}(k_{S}^{T_{0}}(p)|K) ⟶ H^{1}(G_{\overline{𝔓}_{0}}(K))

is surjective, and so we proved the first assertion.

Now let 𝔭_{0} ∈ T. Again by theorem 2.1, the canonical homomorphism

H^{1}(k_{S∪\{𝔭_{0}\}}|K) ⟶ H^{1}(G_{\overline{𝔓}_{0}}(K)) ⊕ ⨁_{T_{0}(K)} H^{1}(G_{𝔭}(k))

is surjective. Therefore, for every

α_{\overline{𝔓}_{0}} ∈ H^{1}(G_{\overline{𝔓}_{0}}(K)/T_{\overline{𝔓}_{0}}(K)) ⊆ H^{1}(G_{\overline{𝔓}_{0}}(K))

there exists β ∈ H^{1}(k_{S∪\{𝔭_{0}\}}|K) which is mapped to (α_{\overline{𝔓}_{0}}, 0, …, 0) and is therefore contained in H^{1}(k_{S}^{T_{0}}(p)|K). Thus

H^{1}(k_{S}^{T_{0}}(p)|K) ⟶ H^{1}(G_{\overline{𝔓}_{0}}(K)/T_{\overline{𝔓}_{0}}(K))

is surjective. This proves the second assertion and the third is trivial. □

Corollary 2.4 Let k be a number field and let S, T_{0} be sets of primes of k such that T_{0} ∪ S_{p} ∪ S_{∞} ⊆ S, δ(S) = 1 and T_{0} is finite. Then the canonical homomorphism

H^{1}(k(p)|k_{S}^{T_{0}}(p)) ⟶ \begin{matrix}⨁ & ′ \\ 𝔭∈T_{0}(k_{S}^{T_{0}}(p)) \end{matrix}H^{1}(G_{𝔭}(k(p)|k)) ⊕ \begin{matrix}⨁ & ′ \\ 𝔭∉S(k_{S}^{T_{0}}(p)) \end{matrix}H^{1}(T_{𝔭}(k(p)|k))

is bijective and

H^{2}(k(p)|k_{S}^{T_{0}}(p)) ⟶ \begin{matrix}⨁ & ′ \\ 𝔭∈T_{0}(k_{S}^{T_{0}}(p)) \end{matrix}H^{2}(G_{𝔭}(k(p)|k))

is injective.

Proof: Using corollary 2.2 and taking the inductive limit over all finite subextensions of k_{S}^{T_{0}}(p)|k, we obtain a surjective homomorphism

H^{1}(k̄|k_{S}^{T_{0}}(p)) ↠ \begin{matrix}⨁ & ′ \\ 𝔭∈T_{0}(k_{S}^{T_{0}}(p)) \end{matrix}H^{1}(G_{𝔭}(k)) ⊕ \begin{matrix}⨁ & ′ \\ 𝔭∉S(k_{S}^{T_{0}}(p)) \end{matrix}H^{1}(T_{𝔭}(k))^{G_{𝔭}(k_{S}^{T_{0}}(p))}.

By proposition 2.3, we have

G_{𝔭}(k(p)|k_{S}^{T_{0}}(p)) = T_{𝔭}(k(p)|k) for 𝔭 ∉ S,

and so we obtain a surjection

H^{1}(k(p)|k_{S}^{T_{0}}(p)) ↠ \begin{matrix}⨁ & ′ \\ 𝔭∈T_{0}(k_{S}^{T_{0}}(p)) \end{matrix}H^{1}(G_{𝔭}(k(p)|k)) ⊕ \begin{matrix}⨁ & ′ \\ 𝔭∉S(k_{S}^{T_{0}}(p)) \end{matrix}H^{1}(T_{𝔭}(k(p)|k)).

Since the groups G_{𝔭}(k(p)|k) for 𝔭 ∈ T_{0} and T_{𝔭}(k(p)|k) for 𝔭 ∉ S generate the group G(k(p)|k_{S}^{T_{0}}(p)), the map above is injective. Thus we proved the first assertion.

By [4] theorem (9.1.8), we have an injection

H^{2}(k̄|k_{S}^{T_{0}}(p)) ↪ \begin{matrix}⨁ & ′ \\ 𝔭∈T_{0}(k_{S}^{T_{0}}(p)) \end{matrix}H^{2}(G_{𝔭}(k)) ⊕ \begin{matrix}⨁ & ′ \\ 𝔭∉T_{0}(k_{S}^{T_{0}}(p)) \end{matrix}H^{2}(G_{𝔭}(k_{S}^{T_{0}}(p))).

Since H^{2}(k̄|K) = H^{2}(k(p)|K) and H^{2}(G_{𝔭}(K)) = H^{2}(G_{𝔭}(k(p)|K)), see [4] theorem (10.4.8) and theorem (7.5.7), we get an injection

H^{2}(k(p)|k_{S}^{T_{0}}(p)) ↪ \begin{matrix}⨁ & ′ \\ 𝔭∈T_{0}(k_{S}^{T_{0}}(p)) \end{matrix}H^{2}(G_{𝔭}(k(p)|k)) ⊕ \begin{matrix}⨁ & ′ \\ 𝔭∉T_{0}(k_{S}^{T_{0}}(p)) \end{matrix}H^{2}(G_{𝔭}(k(p)|k_{S}^{T_{0}}(p))).

From proposition 2.3 it follows that

H^{2}(G_{𝔭}(k(p)|k_{S}^{T_{0}}(p))) = 0 for 𝔭 ∉ T_{0},

since G_{𝔭}(k(p)|k_{S}^{T_{0}}(p)) = 1 for 𝔭 ∈ S\T_{0} and cd_{p} G_{𝔭}(k(p)|k_{S}^{T_{0}}(p)) ≤ 1 for 𝔭 ∉ S. This proves the second assertion. □

Using lemma 1.8 and corollary 2.4, we obtain the following theorem.

Theorem 2.5 Let k be a number field and let S, T_{0} be sets of primes of k such that δ(S) = 1, S_{p} ∪ S_{∞} ⊂ S and T_{0} is a finite subset of S. Then there is a canonical isomorphism

ϕ_{S}^{T_{0}} : ∗_{𝔭∈T_{0}(k_{S}^{T_{0}}(p))}G_{𝔭}(k(p)|k) ∗ ∗_{𝔭∉S(k_{S}^{T_{0}}(p))}T_{𝔭}(k(p)|k) ⟶^{∼} G(k(p)|k_{S}^{T_{0}}(p)).

In particular, if S = P, then

∗_{𝔭∈T_{0}(k^{T_{0}}(p))}G_{𝔭}(k(p)|k) ⟶^{∼} G(k(p)|k^{T_{0}}(p)).

References

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