Corestricted Free Products of Profinite Groups

by Jochen Gärtner and Kay Wingberg

Abstract

We introduce the notion of corestricted free products of a family of profinite groups indexed over an arbitrary profinite space. Using arithmetic results of the second author, this enables us to prove an analogue of Riemann’s existence theorem for the decomposition groups of certain infinite sets of primes of a number field.

In 1971, Neukirch [4] introduced the concept of free products of families of pro-𝔠-groups indexed over a discrete set. Since in number theory projective limits of free pro-𝔠-products arise in a natural way, it became necessary to generalize this concept to families varying continuously over a profinite index space. This has been worked out by several authors by introducing the notion of compact bundles G of pro-𝔠-groups over a profinite space T, see [1], [2], [3] and [5]. These bundles are group objects in the category of profinite spaces over T such that every fibre is a pro-𝔠-group. The generalized free products obtained in this way turned out to be the key tool to prove number theoretical analogues of the Riemann existence theorem over large number fields, i.e. inertia groups, indexed by a projective limit of possibly infinite sets of primes of a tower of number fields, form a free product. Furthermore, analogous results hold for the decomposition groups lying over a finite set of primes of a number field, see [5] chap.10 §5.

From the arithmetic point of view, it is interesting whether infinitely many decomposition groups also form a free product. However the notion of compact bundles turns out to be too restrictive. Inspired by the abelian case in number theory, i.e. the idele group of a number field which is a restricted product and topological not compact, Neukirch introduced the concept of corestricted free products of pro-𝔠-groups over discrete sets. Generalizing this concept to a more general profinite space T leads to possibly non-compact bundles, which are group objects in the category of totally disconnected Hausdorff spaces over T.

Of special interest are so-called corestricted bundles G over a profinite space T which are corestricted by a compact subbundle U of G, i.e. G has the final topology with respect to the inclusions U ↪ G and G_{t} ↪ G, t ∈ T, where G_{t} denotes the fibre of G at t. Alexander Schmidt pointed out that a good way to think of this object is as a “hedgehog ” having a compact body U which is surrounded by “spines ” corresponding to the sets G_{t}\U_{t}, t ∈ T.

In the first section we introduce the notion of (not necessarily compact) bundles and their free products. We investigate bundles endowed with an additional action by a pro-𝔠-group G and their corresponding free products, which are pro𝔠-G operator groups. In the second section, we define corestricted bundles and corestricted free products. One main goal of this paper is to study the properties of the functor

F : Bundles ⟶ Pro-𝔠-groups

which associates to a bundle G over a profinite space T the free pro-𝔠-product ∗_{T} G. This functor commutes with projective limits on the subcategory of compact bundles. We prove that this remains true for certain corestricted G-operator bundles. Furthermore, this also holds for projective limits of corestricted bundles over the one-point compactifications of discrete sets.

In the third section, we show that corestricted bundles naturally arise from families of closed subgroups of a pro-𝔠-group. Finally we consider the cohomology of a corestricted free product over the one-point compactification of a discrete set. This yields the number theoretic application we have in mind: Under some conditions there exists a Galois group of a large number field which is a free prop-product of infinity many decomposition groups corestricted by inertia groups.

1 Bundles and free products

Let 𝔠 be a class of finite groups closed under taking subgroups, homomorphic images and finite direct products. A pro-𝔠-group is a projective limit of groups in 𝔠. In [5], chap. IV §3, the notion of a bundle of pro-𝔠-groups is introduced. Here we use the notion bundle in a more general context, but we will follow partly the presentation of [5]; see also [3].

Definition 1.1 Let T be a profinite space, i.e. a topological projective limit of finite discrete spaces. A bundle of pro-𝔠-groups

G = (G, pr, T)

over T is a group object in the category of totally disconnected Hausdorff spaces over T such that the fibre over every point of T is a pro-𝔠-group, i.e. there are continuous maps

pr : G → T (the structure map), m : G ×_{T} G → G (the multiplication), e : T → G a section to pr (the unit), ι : G → G (the inversion),

such that the fibre G_{t} = pr^{−1}(t) together with the induced maps m_{t} : G_{t} × G_{t} → G_{t}, ι_{t} : G_{t} → G_{t} and the unit element e_{t} = e(t) is a pro-𝔠-group for every point t ∈ T.

Here G ×_{T} G = {(x, y) ∈ G × G|pr(x) = pr(y)} denotes the fibre product of G and G over T, which is a totally disconnected Hausdorff space and has a structure map pr: G ×_{T} G → T. Furthermore, the maps m and ι commute with the structure maps.

Of special interest are bundles which are compact, i.e. G is a profinite space. These are the bundles which were considered in [5], and which we now call compact bundles.

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

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 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 for every t ∈ T the associated map ϕ_{t} : G_{t} → H_{ϕ_{T} (t)} is a group homomorphism. We say that (G, pr_{G}, T) is a subbundle of (H, pr_{H}, S) if T = S, ϕ_{T} = id and ϕ_{t} is injective for all t ∈ T. We say that ϕ is fibrewise surjective if ϕ is surjective and ϕ_{t} is surjective for all t ∈ T.

A morphism from a bundle (G, pr, T) to a pro-𝔠-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.

1.1 Free pro-𝔠-products

Let (G, pr, T) be a bundle of pro-𝔠-groups.

Definition 1.2 The free pro-𝔠-product of (G, pr, T) of 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.

As in [5](4.3.6) we have the

Proposition 1.3 The free pro-𝔠-product ∗_{T} G exists and is unique up to unique

isomorphism.

Proof: Let D = ∗_{t∈T}^{d} G_{t} be the abstract free product of the family of groups (G_{t})_{t∈T} and let λ : G → D be the map which is given on every G_{t} as the natural inclusion of G_{t} into D. Then define the free product G = ∗_{T} G as the completion of D with respect to the topology which is given by the family of normal subgroups N ⊆ D of finite index for which

(a) D/N ∈ 𝔠,

(b) the composition of λ with the natural projection D → D/N is continuous. It is easily verified that G has the required universal property. The uniqueness assertion is clear by the universal property. □

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

∗_{t∈T} G_{t} = ∗_{T} G.

Observe that there are different “free pro-𝔠-products” of a family of pro-𝔠-groups {G_{t}}_{t∈T} depending on the topology on G = ⋃ . _{t∈T} G_{t}.

If ϕ : (G, pr_{G}, T) → (H, pr_{H}, S) is a morphism of bundles, then by the universal property of the free product the map G →^{ϕ_{G}} H →^{ω}∗_{S}H induces a homomorphism

ϕ_{∗} : ∗_{T} G → ∗_{S} H,

i.e. there is a functor

B ⟶ Pro-𝔠-groups, G ↦ ∗_{T} G,

between the category B of bundles of pro-𝔠-groups and the category of pro-𝔠-groups.

Proposition 1.4 Let (G, pr, T) be a bundle of pro-𝔠-groups and assume that T = T_{1} ∪. · · · ∪. T_{k} is a finite disjoint decomposition of T into non-empty open subsets. Then the following assertions hold.

(i) There is a canonical isomorphism

∗_{T_{1}} G_{1} ∗ ⋯ ∗ ∗_{T_{k}} G_{k} ≅ ∗_{T} G,

where G_{i} denotes the bundle pr^{−1}(T_{i}) over T_{i} for i = 1, …, k. (ii) The canonical homomorphism ϕ_{∗}: ∗_{T_{i}}G_{i} →∗_{T} G, which is induced by the

bundle morphism ϕ: G_{i} → G, has a splitting; in particular, ϕ_{∗} is injective.

Proof: The first assertion follows from the universal property. For the second consider the map φ_{G}: G ↠ G_{i} which maps G_{j}, j ≠ i, to the unit element of G_{t_{0}} for some fixed t_{0} ∈ T_{i} and which is the identity on G_{i}, and the map φ_{T} : T ↠ T_{i} which maps T_{j}, j ≠ i, to t_{0} and which is the identity on T_{i}. Then (φ_{G}, φ_{T}) is a morphism of bundles which induces a splitting of ϕ_{∗} □ Corollary 1.5 Let (G, pr, T) be a bundle, where T = T_{0} ∪. {∗} is the one-point compactification of the discrete set T_{0} and G_{∗} = {∗}. If t_{0} ∈ T, then the canonical

map ω_{G_{t_{0}}} : G_{t_{0}} ⟶ ∗_{t∈T} G_{t}

has a splitting. In particular, it is injective.

We do not know whether the map ω_{G_{t_{0}}} is injective for bundles over arbitrary profinite spaces (but see remark 5 in section 3).

1.2 Compact bundles

First we prove a criterion for subsets of a compact bundle for being open.

Lemma 1.6 Let G be a compact bundle of pro-𝔠-groups over a profinite space T. For every t ∈ T let an open subset W_{t} of G_{t} be given. Then the following is equivalent.

(i) For every closed subset Y of G the set {t ∈ T | Y_{t} ⊆ W_{t}} is open in T.

(ii) W = ⋃ ._{t∈T} W_{t} is open in G.

Proof: Assume that (i) holds. Let

W̄ = G\W = ⋃ ._{t∈T} W̄_{t}, W̄_{t} = G_{t}\W_{t}.

Let g ∈ W_{t_{0}} ⊆ W. Since W̄_{t_{0}} is closed in G_{t_{0}} (and so in G), there are open subsets R, Q of G such that W̄_{t_{0}} ⊆ Q, g ∈ R and R ∩ Q = ∅. Since Q̄ = G\Q is closed in G, the set

S := {t ∈ T |W̄_{t} ⊆ Q} = {t ∈ T |Q̄_{t} ⊆ W_{t}}

is open in T by assumption and t_{0} ∈ S. Thus V = pr_{U}^{−1}(S) ∩ R is an open neighborhood of g. Furthermore, if t ∈ S, then V_{t} ⊆ R_{t} ⊆ Q̄_{t} ⊆ W_{t}. Thus V is contained in W. It follows that W is open.

Now assume that (ii) holds. Then {t ∈ T | Y_{t} ⊆ W_{t}} = T \pr(Y ∩ W̄).

Since (Y ∩ W̄) is closed in G, hence compact, pr(Y ∩ W̄) is a compact subset of T, and so T \pr(Y ∩ W̄) is open. □

Let G be a pro-𝔠-group and let (G_{t})_{t∈T} be a family of closed subgroups of G indexed by the points of a profinite space T. We say that (G_{t})_{t∈T} is a continuous family, if and only if for every open neighborhood V of the identity of G the set T (V) := {t ∈ T | G_{t} ⊆ V } is open in T.

The following proposition shows that the concepts of compact bundles and continuous families coincide.

Proposition 1.7

(i) The set

G = {(g, t) ∈ G × T | g ∈ G_{t}}

equipped with the induced topology of the constant bundle (G × T, pr, T) is a compact bundle over T.

(ii) If H is a compact bundle, then H is the bundle associated to the family (H_{t})_{t∈T} of pro-𝔠-groups given by the fibres of H, where each fibre is considered as closed subgroup of ∗_{T} H.

Proof: For the first assertion see [5] (4.3.3). In order to prove the second, we first remark that a fibre H_{t} is a closed subgroup of the free pro-𝔠-product ∗_{T} H, see [5] (4.3.12)(i), thus

H = {(h, t) ∈ ∗_{T} H × T | h ∈ H_{t}}.

Now it is easy to see that the family (H_{t})_{t∈T} is continuous: Let W be an open neighborhood of the unit of ∗_{T} H. Then, using lemma (1.6), the set T (W) = {t ∈ T | H_{t} ⊆ W } = {t ∈ T | H_{t} ⊆ (W ∩ H)_{t}} is open in T. □

1.3 Bundles with operators

Definition 1.8 Let G be a pro-𝔠-group. A bundle (G, pr, T) of pro-𝔠-groups is called G-bundle if G acts continuously on G, i.e. there is a commutative diagram

[figure: G × G; pr; G; pr; (σ, x) ↦ σx,; G × T; T; (σ, t) ↦ σt,]

of continuous maps such that

(i) for all σ ∈ G the map G → G, x ↦ σx, is a morphism of bundles, (ii) (στ)(x) = σ(τ(x)) and ex = x for all σ, τ ∈ G and all x ∈ G.

Definition 1.9 Let (G, pr, T) and (H, pr, S) be G-bundles of pro-𝔠-groups and let ϕ: G → H be a morphism of bundles. Then ϕ is called G-invariant, if the diagrams

[figure: G × G; ϕ_{G}; G; ϕ_{G}; G × T; ϕ_{T}; T; ϕ_{T}; G × H; H; G × S; S]

commute. The morphism ϕ is called G-transitive, if ϕ is G-invariant and for t, t′ ∈ ϕ_{T}^{−1}(s), s ∈ S, there exists σ ∈ G such that σt = t′.

Let (G, pr, T) be a G-bundle and let σ ∈ G. Then

σG_{t} = G_{σt} for t ∈ T .

Furthermore, the homeomorphism σ: G ⟶^{∼} G induces an automorphism

σ : ∗_{T} G ⟶^{∼} ∗_{T} G

of pro-𝔠-groups, i.e. ∗_{T} G is a pro-𝔠-G operator group, see [5] (4.3.8). Thus ∗_{T} G possesses a system of neighborhoods of the identity consisting of open G-invariant normal subgroups, see [6] theorem 17:

If U is an open subgroup of ∗_{T} G, then Ũ := ⋂_{σ∈G} σU is a G-invariant subgroup of ∗_{T} G. We will show that Ũ is open. Since σe = e ∈ U, there exist open neighborhoods V_{σ} and W_{σ} of e ∈ U and σ ∈ G, respectively, such that W_{σ}V_{σ} ⊆ U. Since ⋃_{σ} W_{σ} = G and G is compact, we find a finite covering {W_{σ_{1}}, …, W_{σ_{n}}} of G. Let V = ⋂_{i=1}^{n} V_{σ_{i}}, then σV ⊆ Ũ for every σ ∈ G and consequently for every x ∈ Ũ we have V x ⊆ Ũ.

Since the map λ: G →∗_{T}^{d} G is G-invariant, where the action of G on ∗_{T}^{d} G is defined in the obvious way, it follows that

∗_{T} G = lim_{⟵}_{N} (∗_{T}^{d}G)/N,

where N runs through all G-invariant normal subgroups of ∗_{T}^{d} G such that (∗_{T}^{d} G)/N ∈ 𝔠 and the map G →∗_{T}^{d} G ↠ (∗_{T}^{d} G)/N is continuous.

1.4 Projective limits of bundles

Let I be a directed set and let {G_{i}, T_{i}, ϕ_{ij}}_{I} be a projective system of bundles of pro-𝔠-groups with transition morphisms

ϕ_{ij} : G_{i} → G_{j}, i ≥ j.

A bundle (G, pr, T) of pro-𝔠-groups together with morphisms of bundles ϕ_{i}: G → G_{i} compatible with ϕ_{ij} is called projective limit of {G_{i}, T_{i}, ϕ_{ij}}_{I}, if it satisfies the usual universal property. The projective limit exists and is unique up to isomorphism and we write

(G, pr, T) = lim_{⟵}_{i∈I} (G_{i}, pr_{i}, T_{i}).

Indeed, the uniqueness follows from the universal property and for the existence we consider the Cartesian product G̃ := ∏_{i∈I} G_{i} which, being endowed with the product topology, is a bundle over T = lim_{⟵} T_{i}. Then the subbundle

G := {(g_{i}) ∈ G̃ | ϕ_{ij}(g_{i}) = g_{j}}

satisfies the universal property of the projective limit and its fibres are given by

G_{t} = lim_{⟵}_{i∈I} G_{i,t_{i}}

for t = (t_{i})_{i∈I} ∈ lim_{⟵} T_{i}. Clearly, the transition morphisms ϕ_{ij} give rise to homomorphisms ∗_{T_{i}} G_{i} → ∗_{T_{j}} G_{j}, i ≥ j. It is a natural question whether the induced

homomorphism

∗_{T} G ⟶ lim_{⟵}_{i∈I} ∗_{T_{i}} G_{i}

is an isomorphism, i.e. whether free products commute with projective limits. This holds if the bundles G_{i} and hence G are compact, see [5](4.3.6). Unfortunately, the argument cannot be carried over to arbitrary bundles. However, under additional conditions for the maps G → G_{i}, we can prove the following criterion:

Proposition 1.10 Let G be a pro-𝔠-group and let {G_{i}, T_{i}, ϕ_{ij}}_{I} be a projective system of G-bundles of pro-𝔠-groups with G-invariant transition maps ϕ_{ij}. Then the bundle

G = lim_{⟵}_{i∈I} G_{i}

is a G-bundle over T = lim_{⟵} T_{i}. Assume further that the following holds:

(i) For all i ∈ I, the canonical map ϕ_{i}: G → G_{i} is surjective. (ii) Let W be a closed resp. open subset of G which is invariant under an open

subgroup M of G. Then ϕ_{i}(W) is closed resp. open in G_{i} for all i ∈ I. Then

∗_{T} G = lim_{⟵}_{i∈I} ∗_{T_{i}} G_{i}.

Proof: The fact that G is a G-bundle over T is obvious. Let D = ∗_{t∈T}^{d} G_{t} be the abstract free product of the family of groups G_{t} and D_{i} = ∗_{t_{i}∈T_{i}}^{d}G_{i,t_{i}} for i ∈ I. Using (i), we see that the canonical maps

φ_{i}: D −↠ D_{i}, i ∈ I,

are surjective. Furthermore the action of G on G induces a G-action on D_{i} and D, and φ_{i} is G-invariant. For a fixed i ∈ I, the correspondence N ↦ N_{i} := φ_{i}(N) induces a bijection between the set of G-invariant normal subgroups N of D such that N ⊇ ker φ_{i} and D/N ∈ 𝔠, and the set of normal subgroups N_{i} of D_{i} such that D_{i}/N_{i} ∈ 𝔠. For all such N and N_{i}, we have the isomorphism

D/N ⟶^{∼} D_{i}/N_{i}

and the commutative diagram

\xymatrix{G_{i} \ar[r]^{λ_{i}} & D_{i} \ar@{->>}[r]^{f_{i}} & D_{i}/N_{i} \\ G \ar[r]^{λ} \ar@{->>}[u]^{ϕ_{i}} & D \ar@{->>}[u]^{φ_{i}} \ar@{->>}[r]^{f} & D/N . \ar[u]^{∼}}

Obviously, if f_{i}λ_{i} is continuous, then fλ is continuous. Conversely, assume that fλ is continuous and let a ∈ D_{i}/N_{i}. Then W = (fλ)^{−1}(a) is open and closed in G and invariant under any open subgroup of G acting trivially on the finite group D/N. Furthermore, since ϕ_{i} is surjective, ϕ_{i}(W) = (f_{i}λ_{i})^{−1}(a). Using (ii), ϕ_{i}(W) is closed resp. open in G_{i} . This shows that f_{i}λ_{i} is continuous.

Now by the general theory of pro-𝔠-G operator groups presented in the previous section, we have ∗_{T} G = lim_{⟵}_{N} D/N where N runs through the G-invariant

normal subgroups of D such that D/N ∈ 𝔠 and fλ is continuous. Analogously, ∗_{T} G_{i} = lim_{⟵}_{N_{i}} D_{i}/N_{i} where N_{i} runs through the G-invariant normal subgroups of

D_{i} such that D_{i}/N_{i} ∈ 𝔠 and f_{i}λ_{i} is continuous. Noting that any normal subgroup N of D of finite index contains ker φ_{i} for some i ∈ I, the above considerations imply

∗_{T} G = lim_{⟵}_{N} D/N = lim_{⟵}_{i∈I} ( lim_{⟵}_{N⊇ker(φ_{i})} D/N) = lim_{⟵}_{i∈I} ( lim_{⟵}_{N_{i}} D_{i}/N_{i}) = lim_{⟵}_{i∈I} ∗_{T_{i}} G_{i}.

□

Remark: In section 2 we will work with projective limits which are formed in the subcategory of corestricted bundles. As a set, the projective limit G = lim_{⟵} G_{i} is given as above by G := {(g_{i}) ∈ ∏ G_{i} | ϕ_{ij}(g_{i}) = g_{j}}, however it has to be endowed with a topology which is finer than the product topology. It is important to remark that (1.10) remains valid as long as the projections G → G_{i} are continuous and condition (ii) holds with respect to this finer topology.

2 Corestricted bundles and their free products

Now we introduce the notion of a corestricted bundle of pro-𝔠-groups with respect to a compact bundle U of pro-𝔠-groups.

Definition 2.1 Let (G, pr_{G}, T) be a bundle of pro-𝔠-groups. We say that G is corestricted with respect to a compact bundle (U, pr_{U}, T) over T if there exists an injective morphism of bundles ϕ: U ↪ G over T, i.e. the diagram

\xymatrix{U \ar@{^{(}->}[rr]^{ϕ} \ar[dr]_{pr_{U}} &  & G \ar[dl]^{pr_{G}} \\  & T}

commutes, and G is equipped with the final topology with respect to the family of inclusions {ϕ: U ↪ G, G_{t} ↪ G, t ∈ T }. We denote the bundle G by (G, U) if it is corestricted with respect to U.

The corestricted free pro-𝔠-product of the corestricted bundle (G, U) is the free pro-𝔠-product ∗_{T} (G, U), and we write

∗_{t∈T}(G_{t}, U_{t}) = ∗_{T}(G, U).

Remarks: 1. The topology on G_{t} induced by the topology of (G, U) is the pro-𝔠 topology of G_{t}: if V_{0} is an open subgroup of the pro-𝔠-group G_{t_{0}}, then G_{t_{0}}\V_{0} is closed in G_{t_{0}} and so in G since every fiber of G is closed. It follows that V = ⋃ . _{t≠t_{0}} G_{t} ∪. V_{0} is open in G, and V ∩ G_{t_{0}} = V_{0}.

2. If T is the one-point compactification of a discrete set T_{0}, then the definition of the corestricted free pro-𝔠-product of the corestricted bundle (G, U) over the profinite space T coincides with that given in [4] §2. If T is finite, then (G, U) is a compact bundle independent of U.

3. We have two extreme cases. If U_{t} = G_{t} for all t ∈ T, then (G, U) is the equal to the compact bundle U. If U_{t} = {1} for all t ∈ T, i.e. U is the trivial bundle 1 , then the topology of (G, 1 ) is given by the sets V ⊆ G such that V_{t} = V ∩ G_{t} is an open subgroup of G_{t} for all t ∈ T and S = {t ∈ T |1 ∈ V_{t}} is open in T. The corestricted free pro-𝔠-product

∗_{T}(G, 1 )

of the corestricted bundle (G, 1 ) is sometimes called the unrestricted free pro-𝔠-product of G.

Definition 2.2 Let T, S be profinite spaces and (G, U), (H, V) be corestricted bundles of pro-𝔠-groups over T and S, respectively. A morphism of bundles

ϕ : (G, U) ⟶(H, V)

is called morphism of corestricted bundles if ϕ(U) ⊆ V. Furthermore, we say that ϕ is strict if ϕ^{−1}(V) = U.

As a special case of morphisms of corestricted bundle, we consider morphisms (G, U ′) → (G, U) where U ′ ⊆ U are compact subbundles. In this situation, we have the following

Proposition 2.3 Let (G, U) and (G, U ′) be corestricted bundles of pro-𝔠-groups with respect to compact subbundles U and U ′, respectively, with continuous inclusion U ′ ⊆ U. Then the morphism of corestricted bundles id : (G, U ′) → (G, U) induces a canonical surjection

∗_{T}(G, U ′) −↠ ∗_{T}(G, U)

and there is an isomorphism

lim_{⟵}_{N} (∗_{T}(G, U ′))/N ⟶^{∼} ∗_{T}(G, U).

Here N runs through the open normal subgroups of ∗_{T} (G, U ′) such that for every a ∈ ∗_{T} (G, U ′) the preimage of aN under the map

(G, U) ⟶^{id} (G, U ′) ⟶^{ω} ∗_{T}(G, U ′)

is open in (G, U), i.e.

(i) ω^{−1}(aN) ∩ U is open in U, (ii) ω^{−1}(aN) ∩ G_{t} is open in G_{t} for all t ∈ T.

Proof: This follows directly from the universal property of the free product. In fact, ∗_{T} (G, U) satisfies the property of the completion of ∗_{T} (G, U ′) with respect to the family of open normal subgroups N satisfying (i) and (ii). Hence the homomorphism ∗_{T} (G, U ′) −↠ ∗_{T} (G, U) induces the isomorphism

lim_{⟵}_{N}(∗_{T} (G, U ′))/N ⟶^{∼} ∗_{T} (G, U). □

In particular, it follows that any corestricted free product can be recovered from the unrestricted free product ∗_{T} (G, 1 ).

2.1 Quotients of corestricted bundles

Let (G, U) be a corestricted bundle of pro-𝔠-groups and let V be a closed subbundle of G such that V_{t} is a normal subgroup of G_{t} for all t ∈ T. Set \overline{V} := V ∩U and let Ũ := ⋃ . _{t∈T} U_{t}/\overline{V}_{t} be endowed with the quotient topology with respect to the canonical surjection U ↠ Ũ. Then Ũ is a compact bundle over T and we define the quotient bundle (G̃, Ũ) := (G/V, U/\overline{V}) of (G, U) with respect to V as the set

(G̃, Ũ) = ⋃ ._{t∈T} G_{t}/V_{t}

endowed with the final topology with respect to the inclusions G_{t}/V_{t} ↪ (G̃, Ũ), t ∈ T, and Ũ ↪ (G̃, Ũ). Thus (G̃, Ũ) is a corestricted bundle of pro-𝔠-groups and there is a canonical surjective morphism of corestricted bundles

ϕ : (G, U) −↠ (G̃, Ũ)

which induces a surjection

∗_{T}(G, U) −↠ ∗_{T}(G̃, Ũ).

Indeed, ϕ is continuous: Let Ṽ be open in (G̃, Ũ), and so Ṽ ∩Ũ is open in Ũ. Then the equality ϕ^{−1}(Ṽ) ∩ U = ϕ^{−1}(Ṽ ∩ Ũ) ∩ U = ϕ_{|U}^{−1}(Ṽ ∩ Ũ) shows that ϕ^{−1}(Ṽ) ∩ U is open in U. With an analogous argument for the fibres it follows that ϕ^{−1}(Ṽ) is open in (G, U).

Remark: Let (G, U) be a corestricted bundle of pro-𝔠-groups such that U_{t} is a normal subgroup of G_{t} for all t ∈ T. Then the topology of the quotient bundle (G/U, 1 ), where 1 is the compact unit bundle ⋃ . _{t∈T} {1_{t}}, can be described as follows: A subset V ⊆ (G/U, 1 ) is open if and only if

(i) V ∩ G_{t}/U_{t} is open in G_{t}/U_{t} for all t ∈ T, (ii) T (V) = {t ∈ T | 1_{t} ∈ V } is open in T where 1_{t} denotes the unit in G_{t}/U_{t}. In fact, a subset V of (G/U, 1 ) is open if and only if π^{−1}(V) is open in (G, U), i.e. if and only if π^{−1}(V) ∩ U is open in U and π^{−1}(V) ∩ G_{t} is open in G_{t} for any t ∈ T. The last statement is equivalent to (i), and since

π^{−1}(V) ∩ U = ⋃ ._{t∈T (V)} U_{t}

the first statement is equivalent to the assertion that T (V) is open in T.

Of particular interest are quotient bundles which occur when passing to 𝔠̃- completions, where 𝔠̃ is a class of finite groups which is closed under taking subgroups, homomorphic images and finite direct products and contained in the class 𝔠. If G is a pro-𝔠-group, then G(𝔠̃) denotes its maximal pro-𝔠̃ factor group.

Proposition 2.4 Let (G, U) be a corestricted bundle of pro-𝔠-groups over T. Assume that either

G = U is a compact bundle or

T = T_{0} ∪ {∗} is the one-point compactification of a discrete set T_{0} (and G_{∗} = U_{∗} = {∗}). Then, with the notion as above, we have a surjective morphism of corestricted bundles

ϕ : (G, U) ↠ (G̃, Ũ).

This morphism induces an isomorphism

( ∗_{T}(G, U))(𝔠̃) ≅ ∗_{T}^{}˜ (G̃, Ũ),

where ∗^{}˜ denotes the free product the category of pro-𝔠̃-groups.

Proof: Denote by V_{t} the kernel of the homomorphism ρ_{t}: G_{t} → G̃_{t}, t ∈ T. Once we have shown that V := ⋃ . _{t∈T} V_{t} is a closed subbundle of (G, U), we proceed as follows: The bundle U ∩ V is closed in (G, U) and therefore (G̃, Ũ) = (G/V, U/(V ∩ U)) is the quotient bundle of (G, U) with respect to V and the canonical surjection ϕ: (G, U) ↠ (G̃, Ũ) is a morphism of corestricted bundles. Furthermore, ϕ induces a surjection

ϕ_{∗} : ( ∗_{T}(G, U))(𝔠̃) ↠ ∗_{T}^{}˜ (G̃, Ũ).

On the other hand, the morphism φ: (G, U) →^{ω}∗_{T} (G, U) ↠ ( ∗_{T}(G, U))(𝔠̃) induces a map

φ̃: (G̃, Ũ) → (∗_{T} (G, U))(𝔠̃).

We will show that φ̃ is continuous. Let W be open in (∗_{T} (G, U))(𝔠̃) and let Ṽ = φ̃^{−1}(W). Then V := φ^{−1}(W) is open in (G, U), and so V ∩ U is open in U. Since Ũ has the quotient topology with respect to the surjection U ↠ Ũ, the equality

V ∩ U = ϕ^{−1}(Ṽ ∩ Ũ) ∩ U = ϕ_{|U}^{−1}(Ṽ ∩ Ũ),

shows that Ṽ ∩ Ũ is open in Ũ. Using an analogous argument for the intersection of Ṽ with a fibre, it follows that Ṽ is open in (G̃, Ũ).

Thus we get a homomorphism ∗^{}_{˜ T} (G̃, Ũ) → (∗_{T} (G, U))(𝔠̃) which is inverse to ϕ_{∗}. Now we will show that under our assumptions V is a closed subbundle.

If G = U is a compact bundle, then G is projective limit of finite bundles G_{λ}, see [5] (4.3.10). Since G̃_{λ} is obviously a (compact) bundle of pro-𝔠̃-groups, it follows that G̃ = lim_{⟵}_{λ} G̃_{λ} is a compact bundle of pro-𝔠̃-groups and V is a closed subbundle of G.

If T = T_{0} ∪ {∗} is the one-point compactification of a discrete set T_{0}, then the fibres U_{t}, t ∈ T_{0}, are open in U. Thus the set ⋃ . _{t∈T_{0}} U_{t}\V_{t} is open in U, hence V = ⋃ . _{t∈T} V_{t} is closed. □

If 𝔞 is the class of finite abelian groups, then the free pro-𝔞-product and the free pro-(𝔠 ∩ 𝔞)-product of abelian pro-𝔠-groups coincide. Thus we obtain the following Corollary 2.5 Let (G, U) be a corestricted bundle of pro-𝔠-groups over T, where is the one-point compactification of a discrete set T_{0} and G_{∗} = U_{∗} = {∗}. Then there is a canonical isomorphism

( ∗_{t∈T}(G_{t}, U_{t}))^{ab} ≅ ∗_{t∈T}^{}˜ (G_{t}^{ab}, Ũ_{t}),

where Ũ_{t} = U_{t}[G_{t}, G_{t}]/[G_{t}, G_{t}] and ∗^{}˜ denotes the free product in the category of pro-abelian-groups.

2.2 Projective limits of corestricted bundles

If {(G_{i}, U_{i}), T_{i}, ϕ_{ij}}_{I} is a projective system of corestricted bundles of pro-𝔠-groups, i.e. the transition morphisms ϕ_{ij} are morphisms of corestricted bundles, then the projective limit of this system as defined in section 1 need not to be a corestricted bundle. The reason is that the topology on the limit is too coarse. Nevertheless, the category of corestricted bundles is closed under projective limits: We endow the Cartesian product G̃ = ∏_{i∈I} G_{i} with the final topology with respect to the inclusions ∏_{i∈I} G_{i,t_{i}} ↪ G̃ for any (t_{i}) ∈ ∏ T_{i} and Ũ ↪ G̃ where Ũ := ∏_{i∈I} U_{i} is endowed with the product topology. Then G̃ is a corestricted bundle with respect to the compact subbundle Ũ. Furthermore,

G := {(g_{i}) ∈ G̃ | ϕ_{ij}(g_{i}) = g_{j}}

is a corestricted bundle with respect to the compact subbundle

U := {(u_{i}) ∈ G̃ | ϕ_{ij}(u_{i}) = u_{j}}

satisfies the universal property of the projective limit. We write

(G, U) = lim_{⟵}_{i∈I} (G_{i}, U_{i}).

Theorem 2.6 Let G be a pro-𝔠-group and let {(G_{i}, U_{i}), T_{i}, ϕ_{ij}}_{I} be a projective system of corestricted G-bundles of pro-𝔠-groups with G-invariant transition maps ϕ_{ij}. Then the corestricted bundle

(G, U) = lim_{⟵}_{i∈I} (G_{i}, U_{i})

over T = lim_{⟵} T_{i} is a G-bundle. Assume that

(i) the action of G on each T_{i} factors through some finite quotient of G, (ii) the transition maps ϕ_{ij} are surjective strict morphisms and (iii) the maps ϕ_{i}: (G, U) ↠ (G_{i}, U_{i}) are G-transitive. Then

∗_{T}(G, U) = lim_{⟵}_{i∈I} ∗_{T_{i}}(G_{i}, U_{i}).

Using the following lemma, the above result follows from (1.10). Note that (1.10) holds with respect to the topology on the corestricted projective limit bundle (G, U), cf. the remark at the end of section 1.

Lemma 2.7 Let G be a pro-𝔠-group acting on the corestricted bundles (G, U) and (H, V) of pro-𝔠 groups over profinite spaces T and S, respectively. Assume that the action of G on S factors through a finite factor group G/N. Let

ϕ: (G, U) −↠ (H, V)

be a G-transitive, surjective, strict morphism. Then the following holds.

(i) Every fibre H_{s}, s ∈ S, is a finite union of images ϕ(G_{t}) of fibres of G. (ii) Let W be a closed subset of (G, U) which is invariant under an open subgroup

M of G. Then ϕ(W) is closed in (H, V).

Proof: For s ∈ S we have H_{s} = ϕ(⋃_{t∈ϕ_{T}^{−1}(s)} G_{t}) and ϕ_{T}^{−1}(H_{s}) = ⋃_{t∈ϕ_{T}^{−1}(s)} G_{t}, hence

ϕ(W) ∩ H_{s} = ⋃_{t∈ϕ_{T}^{−1}(s)} ϕ(G_{t} ∩ W).

If σ ∈ N ∩ M and t ∈ ϕ_{T}^{−1}(s), then

ϕ(G_{t} ∩ W) = σϕ(G_{t} ∩ W) = ϕ(σG_{t} ∩ σW) = ϕ(G_{σt} ∩ W).

It follows that

⋃_{t∈ϕ_{T}^{−1}(s)} ϕ(G_{t} ∩ W) = ⋃_{σ∈R} ϕ(G_{σt_{0}} ∩ W), t_{0} ∈ ϕ_{T}^{−1}(s),

where R is a (finite) system of representatives of G/(N ∩ M) in G. In particular,

H_{s} = ⋃_{σ∈R′} ϕ(G_{σt_{0}}) ,

R′ a system of representatives of G/N in G, hence we proved (i).

Since for any t ∈ ϕ_{T}^{−1}(s) the surjective homomorphism of pro-𝔠-groups G_{t} −↠ H_{s} is closed, we see that ϕ(W) ∩ H_{s} is closed in H_{s}. Furthermore, since ϕ is strict, we have

ϕ(W) ∩ V = ϕ(U ∩ W)

and ϕ_{|U} : U → V is a surjection. Since U and V are compact, ϕ_{|U} is closed. It follows that ϕ(W) ∩ V is closed in V. Thus ϕ(W) is a closed subset of (H, V).

□

2.3 Corestricted bundles over a one-point compactification of a discrete set

Let T = T_{0} ∪. {∗} be the one-point compactification of a discrete set T_{0}. Let (G_{t})_{t∈T_{0}} be a family of pro-𝔠-groups indexed by the elements of the discrete set T_{0}. Consider the set

G := ⋃ ._{t∈T_{0}} G_{t} ∪. {∗} ,

where {∗} is the group with one element, equipped with the following topology: G_{t} ⊆ G (together with its profinite topology) is open in G for all t, and for every open neighborhood V ⊆ T of ∗ ∈ T, let

⋃ ._{t∈V} G_{t} ∪. {∗}

be an open neighborhood of ∗ ∈ G. Then (G, pr, T), where pr is the continuous map

pr : G → T; G_{t} ∋ g_{t} ↦ t, ∗ ↦ ∗

is a compact bundle of pro-𝔠-groups.

Now we consider the topology of a corestricted bundle over T. Let

(G, U) = ( ⋃ ._{t∈T_{0}} G_{t} ∪. {∗}, ⋃ ._{t∈T_{0}} U_{t} ∪. {∗})

be a corestricted bundle of pro-𝔠-groups over T with respect to the compact subbundle U. Then a subset V of (G, U) is open if and only if the following holds:

(i) If ∗ ∈ V, then U_{t} ⊆ V for almost all t ∈ T_{0}, (ii) V ∩ G_{t} is open in G_{t} for all t ∈ T_{0}.

Indeed, assume V is open in (G, U), and so (ii) holds, and W = V ∩ U is open in U. Let S = {t ∈ T |U_{t} ⊆ W }. Then

T \S = {t ∈ T | U_{t} ⊈ W } = pr_{U}(U\W).

Since U is compact, and so U\W is compact, it follows that T \S is compact, hence S is open in T. If ∗ ∈ V, then S contains ∗, thus T \S is finite, i.e. (i) holds. Conversely, assume that (i) and (ii) holds. If ∗ ∈ V, then T \S is finite. It follows that W = U\ ⋃ . _{T \S}(U_{t}\W_{t}) is open in U. If t ≠ ∗, then U_{t} is open in U, and so every open subset of U_{t} is open in U, thus W = V ∩ U is open in U, if ∗ ∉ V.

We will prove that under certain conditions a fibrewise surjective strict morphism ϕ : (G, U) →(H, V) of corestricted bundles is open, where (H, V) is a corestricted bundle over the one-point compactification of a discrete set. We need the following Lemma 2.8 Let (U, pr_{U}, T), (V, pr_{V}, S) be compact bundles of pro-𝔠-groups where T is a profinite space and S = S_{0} ∪. {∗} is the one-point compactification of the discrete set S_{0} such that V_{∗} = {∗_{V}} is the group with one element. Assume that

(i) ϕ : U → V is a fibrewise surjective morphism, (ii) ϕ_{T} : T → S is an open map and (iii) #ϕ^{−1}(∗_{V}) = 1, i.e. ∗_{V} has a unique preimage. Then ϕ is open.

Proof: Let W ⊆ U be an open subset, and so W_{t} = W ∩ U_{t} is open in U_{t} for all t ∈ T. Since ϕ is fibrewise surjective, the homomorphisms ϕ_{t} : U_{t} −↠ U_{ϕ_{T} (t)} are open. Hence it follows that for all s ∈ S the set

ϕ(W)_{s} = ϕ(W) ∩ V_{s} = ⋃_{t∈ϕ_{T}^{−1}(s)} ϕ_{t}(W_{t})

is open in V_{s}. In order to prove that ϕ(W) is open in V, by lemma (1.6) it remains to show that for a closed subset X ⊆ V, the set

S_{X} = {s ∈ S | X_{s} ⊆ ϕ(W)_{s}}

is open in S. Again by lemma (1.6), the set

T_{X} = {t ∈ T | ϕ^{−1}(X)_{t} ⊆ W_{t}}

is open in T. Hence, since ϕ_{T} is open, it follows that ϕ_{T} (T_{X}) is open in S. Obviously ϕ_{T} (T_{X}) ⊆ S_{X}. Since S = S_{0} ∪. {∗} is the one-point compactification of a discrete space, it remains to note that ∗ ∈ ϕ_{T} (T_{X}) if ∗ ∈ S_{X}. In fact, this is guaranteed by the assumption #ϕ^{−1}(∗_{V}) = 1. □

We are now able to prove the following

Proposition 2.9 Let (G, U) be a corestricted bundle of pro-𝔠-groups over a profinite space T and (H, V) be a corestricted bundle over S where S = S_{0} ∪. {∗} is the one-point compactification of the discrete set S_{0} such that H_{∗} = {∗_{H}} is the group with one element. Let

ϕ : (G, U) →(H, V)

be a fibrewise surjective strict morphism of corestricted bundles such that

(i) ϕ_{T} : T → S is an open map and (ii) #ϕ^{−1}(∗_{H}) = 1, i.e. ∗_{H} has a unique preimage. Then ϕ is open.

Proof: Let W ⊆ G be an open subset. For all t ∈ T, the set W_{t} ⊆ G_{t} is open. Since ϕ is fibrewise surjective, the homomorphisms ϕ_{t} : G_{t} −↠ H_{ϕ_{T} (t)} are open and hence for all s ∈ S the set

ϕ(W)_{s} = ϕ(W) ∩ H_{s} = ⋃_{t∈ϕ_{T}^{−1}(s)} ϕ_{t}(W_{t})

is open in H_{s}. It remains to show that ϕ(W) ∩ V is open in V. Note that since ϕ is strict and surjective, we have

ϕ(W) ∩ V = ϕ(W ∩ U).

By lemma (2.8), the induced map ϕ_{|_{U}} : U −↠ V is open which implies the claim. □

One should remark that the assumption (i) in the above “open mapping result” is necessary by the following general observation: If ϕ: (G, U) → (H, V) is an open morphism of corestricted bundles over T and S, respectively, then ϕ_{T} : T → S is open. In fact, let T′ ⊆ T be an open subset and let W = ϕ(⋃ . _{t∈T′} G_{t}) ∩ V which is an open subset of V by assumption. Now lemma (1.6) implies that the set ϕ_{T} (T′) = {s ∈ S | {1_{t}} ⊆ W_{t}}, where 1_{t} denotes the unit element in V_{t}, is open in S.

The following lemma shows that the canonical projections of projective limits of profinite spaces are open provided the transition maps are. More precisely, we have the following

Lemma 2.10 Let {S_{i}, ρ_{ij}}_{I} be a projective system of profinite spaces such that all transition maps ρ_{ij} : S_{i} → S_{j}, i ≥ j are surjective. Let S = lim_{⟵}_{j∈I} S_{j} and

denote by ρ_{i} the canonical surjection S ↠ S_{i}, i ∈ I. Then the following holds:

(i) If all transition maps ρ_{ij} are open, then the maps ρ_{i} are open, too. (ii) If S_{i} = S_{0i} ∪. {∗_{i}} is the one-point compactification of a discrete set S_{0i} and

ρ_{ij}^{−1}({∗_{j}}) = {∗_{i}} for all i ≥ j, then the maps ρ_{ij} and ρ_{i} are open.

Proof: Assertion (ii) follows from (i), since the maps ρ_{ij} are obviously open. In order to prove (i), it is sufficient to show that the set ρ_{i}(V ∩ S) is open in S_{i},

where V = ∏_{j∈I} V_{j} ⊆ ∏_{j∈I} S_{j}

with open and closed subsets V_{j} ⊆ S_{j} such that V_{j} = S_{j} for all j ∉ I_{0}, where I_{0} is a finite non-empty subset of I. For any i ∈ I we define a subseteq W_{i} ⊆ S_{i} as follows: Let k ∈ I such that k ≥ i and k ≥ j for all j ∈ I_{0} and set

W_{i} := ρ_{ki}( ⋂_{j∈I_{0}} ρ_{kj}^{−1}(V_{j})) ⊆ V_{i} ⊆ S_{i}.

Since I is directed, it follows that W_{i} is independent from the choice of k. Assuming that ρ_{ij} is open for all i ≥ j, W_{i} is an open and closed subset of S_{i}. We claim that W_{i} = ρ_{i}(V ∩ S). It is easy to see that ρ_{ij} maps W_{i} surjectively onto W_{j} for all i ≥ j. Let W = lim_{⟵}_{j∈I} W_{j} and denote by ρ̃_{i} the canonical surjection

W ↠ W_{i}. Since W ⊆ V ∩ S, we have

W_{i} = ρ̃_{i}(W) ⊆ ρ_{i}(V ∩ S).

Conversely, let x = (x_{j})_{I} ∈ V ∩ S and let k ∈ I such that k ≥ i and k ≥ j for all j ∈ I_{0}. Then ρ_{ki}(x_{k}) = x_{i} and ρ_{kj}(x_{k}) = x_{j} ∈ V_{j} for all j ∈ I_{0}, i.e. x_{k} ∈ ⋂_{j∈I_{0}} ρ_{kj}^{−1}(V_{j}). Therefore ρ_{i}(x) = x_{i} ∈ W_{i} which finishes the proof. □

Theorem 2.11 Let {(G_{i}, U_{i}), T_{i}, ϕ_{i,j}}_{I} be a projective system of corestricted bundles of pro-𝔠-groups, where T_{i} = T_{0i} ∪. {∗_{i}} is the one-point compactification of a discrete set T_{0i} and (G_{i})_{∗_{i}} = {∗_{G_{i}}} for all i ∈ I. Let T = lim_{⟵}_{i∈I} T_{i} and

(G, U) = lim_{⟵}_{i∈I} (G_{i}, U_{i})

as defined in section 2.2. Assume that the following holds.

(i) The morphisms ϕ_{ij} are fibrewise surjective and strict. (ii) For all i ≥ j the transition maps satisfy ϕ_{ij}^{−1}({∗_{j}}) = {∗_{i}}. Then

∗_{T} G = lim_{⟵}_{i∈I} ∗_{T_{i}}(G_{i}, U_{i}).

Proof: Using (2.9) and (2.10), we see that the canonical maps ϕ_{i}: G ↠ (G_{i}, U_{i}) are open. Now (1.10) (with G = 1) and its following remark give the desired result. □

In proposition (2.3) we have seen that every corestricted free product ∗_{T} (G, U) can be written as the completion of the unrestricted free product ∗_{T} (G, 1 ). In the case where T is the one-point compactification of a discrete set T_{0}, we will show that ∗_{T} (G, U) can also be obtained in a different way as projective limit involving unrestricted free products of quotient bundles with fibres G_{t}/U_{t}.

Assume that U_{t} is a normal subgroup of G_{t} for all t ∈ T. If S is a finite subset of T_{0}, then U^{S} denotes the compact subbundle ⋃ . _{t∈T \S} U_{t} of U. According to (2.1), we have a canonical (open) surjection of bundles

(G, U) ↠ (G/U^{S}, U/U^{S}) = (G/U^{S}, 1 ),

where as sets

G/U^{S} = ⋃ ._{t∈S} G_{t} ∪. ⋃ ._{t∈T \S} G_{t}/U_{t}.

It follows that we obtain an isomorphism of bundles

(G, U) ⟶^{∼} lim_{⟵}_{S} (G/U^{S}, 1 ),

where S runs trough the finite subsets of T_{0}. Thus we get the following proposition, see also [4] Satz (2.2):

Proposition 2.12 Let T = T_{0} ∪. {∗} be the one-point compactification of a discrete set T_{0} and let (G, U) be a corestricted bundle of pro-𝔠-groups over T. Let U_{t} be normal in G_{t} for all t ∈ T. Then

∗_{t∈T}(G_{t}, U_{t}) = lim_{⟵}_{S} ( ∗_{t∈S} G_{t} ∗ ∗_{t∈T \S}(G_{t}/U_{t}, 1 )).

Proof: If S_{1} ⊆ S_{2} are finite subsets of T_{0}, then the morphism of bundles

ϕ_{S_{2},S_{1}} : (G/U^{S_{2}}, 1) ↠ (G/U^{S_{1}}, 1 )

is fibrewise surjective and ϕ_{S}: (G, U) ↠ (G/U^{S}, 1 ) is open. Using (1.10) (with G = 1), we obtain the desired result. □

3 Corestricted free products of families

Let G be a pro-𝔠-group and let (G_{t})_{t∈T} and (U_{t})_{t∈T} be families of closed subgroups of G indexed by the points of a profinite space T, where U_{t} is a closed subgroup of G_{t} for every t ∈ T. Assume that (U_{t})_{t∈T} is a continuous family, i.e.

U = {(g, t) ∈ G × T | g ∈ U_{t}}

is a compact bundle over T, see (1.7)(i). Let

G = {(g, t) ∈ G × T | g ∈ G_{t}}

and let pr_{G} be the restriction to G of the projection G × T ↠ T. Let G be equipped with the following topology: a set V ⊆ G is open if and only if

(i) V ∩ U is open in U, (ii) V ∩ G_{t} is an open subset of G_{t} for all t ∈ T.

(We identify the fiber G_{t} = (G_{t}, t) with G_{t}.) Observe that this topology on G is finer than the topology which is induced by the topology of the constant (compact) bundle (G × T, pr, T). We define the maps m, ι, e by restricting the corresponding maps from the constant bundle to G.

Lemma 3.1 With the notation and assumptions as above let x ∈ G and let V be an open neighborhood of x. Then there exists an open neighborhood V_{0} of x of the following form:

(i) if x ∈ U_{t}, then V_{0} = (xN × S) ∩ G, where N is an open normal subgroup of G and S open in T, and V_{0} ∩ U ⊆ V ∩ U,

(ii) if x ∈ G_{t}\U_{t}, then V_{0} = xN_{t}, where N_{t} is an open normal subgroup of G_{t}, and V_{0} ⊆ V.

Proof: If x ∈ U_{t}, then x ∈ W = V ∩ U. Since W is open in the compact bundle U (which equipped with the induced topology of the compact constant bundle G × T), there exists an open normal subgroup N of G and an open subset S of T such that (xN × S) ∩ U ⊆ W. Then V_{0} = (xN × S) ∩ G is an open neighborhood of x ∈ G and V_{0} ∩ U ⊆ V ∩ U.

Assume that x ∈ G_{t}\U_{t} and let N_{t} be an open normal subgroup of G_{t} such that xN_{t} ⊆ V_{t} = V ∩ G_{t}. We may assume that N_{t} is small enough such that xN_{t} ∩ U_{t} = ∅. It follows that the set V_{0} = xN_{t} ⊆ G_{t} ⊆ G is open in G. Indeed, for t′ ≠ t the set V_{0} ∩ G_{t′} = ∅ is open in G_{t′}, the set V_{0} ∩ G_{t} = xN_{t} is open in G_{t} and V_{0} ∩ U = ∅ is open in U. □

Lemma 3.2 Let G be a pro-𝔠-group, U_{0} ⊆ U are closed subgroups of G and g an element of G. Let V be an open subset of G containing gU_{0} such that

gU_{0} = V ∩ gU.

Then there exists an open subgroup H of G such that gU_{0} ⊆ gH ⊆ V and U_{0} = H ∩ U.

Proof: Since U_{0} = ⋂_{i∈I} H_{i}, where H_{i} is an open subgroup of G for every i ∈ I, we have G\V ⊆ ⋃_{i∈I}(G\gH_{i}). Since G\V is closed in G, hence compact, and the sets G\gH_{i} are open (H_{i} is open and closed), we get G\V ⊆ ⋃_{i=1}^{n}(G\gH_{i}), and so

gU_{0} ⊆ ⋂_{i=1}^{n} gH_{i} ⊆ V.

Thus H = ⋂_{i=1}^{n} H_{i} has the desired properties. □

Theorem 3.3 With the notation as above, the space G = (G, U) is a corestricted bundle of pro-𝔠-groups over T with respect to U, and we have continuous inclusions

\xymatrix{U \ar@{^{(}->}[r] & (G, U) \ar@{^{(}->}[r]^{ϕ} & G × T.}

Proof: Since the injective map (G, U) ↪^{ϕ} G × T is continuous and G × T is a totally disconnected Hausdorff space, the same is true for the space (G, U).

Obviously, the map pr_{G}: G → T is continuous: if S ⊆ T is open, then pr_{G}^{−1}(S)∩ G_{t} is empty or equal to G_{t} for all t ∈ T and pr_{G}^{−1}(S) ∩ U = pr_{U}^{−1}(S) is open in U. In particular, it follows that a fiber G_{t} is closed in (G, U) and that the topology on G_{t} induced by the topology of G is the pro-𝔠 topology of G_{t}.

Now we prove that the multiplication m: G ×_{T} G → G is continuous. Let (a, b) ∈ G ×_{T} G and let t_{0} ∈ T such that a, b ∈ G_{t_{0}}. Let V = ⋃ . _{t∈T} V_{t} ⊆ G be an open neighborhood of m(a, b) = ab. We consider the following two cases.

1. Let a ∉ U_{t_{0}} or b ∉ U_{t_{0}}. Assume that a ∉ U_{t_{0}} and let N_{t_{0}} be an open normal subgroup of G_{t_{0}} such that abN_{t_{0}} ⊆ V_{t_{0}}. We may assume that N_{t_{0}} is small enough such that aN_{t_{0}} ∩ U_{t_{0}} = ∅. It follows that the set V_{a} := aN_{t_{0}} ⊆ G_{t_{0}} ⊆ G is an open neighborhood of a in G, see lemma (3.1).

Let V_{b} := ⋃ . _{t≠t_{0}} G_{t} ∪. (bN_{t_{0}}). Then V_{b} is open in (G, U), since V_{b} ∩ U = ⋃ . _{t≠t_{0}} U_{t} ∪. (bN_{t_{0}} ∩ U_{t_{0}}) is open in U and V_{b} ∩ G_{t} is open in G_{t} for all t ∈ T. Furthermore, m(V_{a}, V_{b}) = abN_{t_{0}} ⊆ V_{t_{0}} ⊆ V.

2. Let a, b ∈ U_{t_{0}}. Using lemma (3.1), we replace V by (abN ×S)∩V, where N is an open normal subgroup of G and S an open subset of T, i.e. we may assume that

W = V ∩ U = (abN × S) ∩ U = ⋃ ._{t∈S}(abN ∩ U_{t}).

Thus V_{t} ∩ U_{t} = abN ∩ U_{t} for every t ∈ S. Let

W_{a} = ⋃ ._{t∈S}(aN ∩ U_{t}) and W_{b} = ⋃ ._{t∈S}(bN ∩ U_{t}).

Then W_{a} and W_{b} are open neighborhoods of a and b in U, respectively, and m_{U}(W_{a}, W_{b}) ⊆ W. Let

S_{a} := {t ∈ S|aN ∩ U_{t} ≠ ∅} and S_{b} := {t ∈ S|bN ∩ U_{t} ≠ ∅}.

For every t ∈ S_{a} resp. t ∈ S_{b} there are elements n_{t} ∈ N resp. m_{t} ∈ N such that an_{t} ∈ U_{t} and bm_{t} ∈ U_{t} and

aN ∩ U_{t} = an_{t}(N ∩ U_{t}) resp. bN ∩ U_{t} = am_{t}(N ∩ U_{t}).

Let t ∈ S_{a} ∩ S_{b}. Using lemma (3.2) and

an_{t}bm_{t}(N ∩ U_{t}) = abN ∩ U_{t} = V_{t} ∩ U_{t} = an_{t}bm_{t}U_{t} ∩ V_{t},

there exists an open subgroup H_{t} of G_{t} such that

an_{t}bm_{t}(N ∩ U_{t}) ⊆ an_{t}bm_{t}H_{t} ⊆ V_{t}

and N ∩ U_{t} = H_{t} ∩ U_{t}. Let

\begin{aligned}V_{a} & = & ⋃ ._{t∈S_{a}∩S_{b}} an_{t}(H_{t})^{bm_{t}} ∪. ⋃ ._{t∈S_{a}\S_{b}} aN, \\ V_{b} & = & ⋃ ._{t∈S_{a}∩S_{b}} bm_{t}H_{t} ∪. ⋃ ._{t∈S_{b}\S_{a}} bN, \end{aligned}

where (H_{t})^{bm_{t}} = (bm_{t})H_{t}(bm_{t})^{−1}. Then m(V_{a}, V_{b}) ⊆ V. We show that the sets V_{a} and V_{b} are open in (G, U). Obviously we have condition (ii). Furthermore,

V_{a} ∩ U = ⋃ ._{t∈S_{a}}(aN ∩ U_{t}) = W_{a}, and V_{b} ∩ U = ⋃ ._{t∈S_{b}}(bN ∩ U_{t}) = W_{b},

since

an_{t}(H_{t})^{bm_{t}} ∩ U_{t} = an_{t}(H_{t} ∩ U_{t})^{bm_{t}} = an_{t}(N ∩ U_{t})^{bm_{t}} = an_{t}(N ∩ U_{t}) = aN ∩ U_{t}.

In order to prove that the inversion map ι: G → G is continuous, let a^{−1} ∈ G and V ⊆ G an open neighborhood of a^{−1}. Then the set V^{−1} = {x ∈ G|x^{−1} ∈ V } contains a, ι(V^{−1}) = V and V^{−1} is open in G, since V^{−1} ∩ U = ι_{U}^{−1}(V ∩ U) is open in U as ι_{U} is continuous, and V^{−1} ∩ G_{t} = (V ∩ G_{t})^{−1} is an open subset of G_{t} for all t ∈ T.

Finally, since the unit e is equal to the composition T ⟶^{e_{U}} U ↪ G, the map e is continuous. □

The continuous map ϕ induces a homomorphism

ϕ_{∗} : ∗_{T}(G, U) ⟶ G.

Definition 3.4 Let (G_{t})_{t∈T} and (U_{t})_{t∈T} be families of closed subgroups of a pro𝔠-group G indexed by the points of a profinite space T. Assume that (U_{t})_{t∈T} is a continuous family and U_{t} is a closed subgroup of G_{t} for every t ∈ T. Let (G, U) be the corestricted bundle which is associated to these families.

We say that the pro-𝔠-group G is the corestricted free pro-𝔠-product of the family (G_{t})_{t∈T} with respect to the continuous family (U_{t})_{t∈T} of closed subgroups of G if ϕ_{∗} is an isomorphism. We write

G = ∗_{t∈T}(G_{t}, U_{t}).

Remarks: We consider the situation given in definition (3.4).

1. If (G, U) is a corestricted bundle of pro-𝔠-groups over the one-point compactification T is of a discrete set T_{0}. Then it can be considered as a corestricted bundle associated to the families (G_{t})_{t∈T} and (U_{t})_{t∈T} of closed subgroups of the pro-𝔠-group G = ∗_{T} (G, U), see corollary (1.5); indeed, since the bundle U is compact, the family (U_{t})_{t∈T} is continuous, see (1.7)(ii).

2. For all t ∈ T, let U′_{t} ⊆ U_{t} be a closed subgroup such that (U′_{t})_{t∈T} is a continuous family. If U ′ denotes the compact bundle associated to the family (U′_{t})_{t∈T}, then we have a canonical surjection ∗_{T} (G, U ′) −↠ ∗_{T} (G, U) and an isomorphism lim_{⟵}_{N}(∗_{T} (G, U ′))/N ⟶^{∼} ∗_{T} (G, U), see proposition (2.3).

3. If t_{0} ∈ T, then the canonical map ω_{G_{t_{0}}} : G_{t_{0}} → ∗_{t∈T}(G_{t}, U_{t}) is an injec-

tive group homomorphism. This follows from the fact that the composition G_{t_{0}} →∗_{t∈T} (G_{t}, U_{t}) → G is injective.

3.1 Abelianization of corestricted free products

Let ∏ ∐_{t∈T_{0}}(A_{t}, B_{t}) = {(a_{t})_{t∈T_{0}} ∈ ∏_{t∈T_{0}}A_{t} | a_{t} ∈ B_{t} for almost all t ∈ T_{0}}

be the restricted product over a discrete set T_{0} of abelian locally compact groups A_{t} with respect to closed subgroups B_{t}. The topology is given by the subgroups V such that

(i) V ∩ A_{t} is open in A_{t} for all t ∈ T_{0}, (ii) V ⊇ B_{t} for almost all t ∈ T_{0}. Then we call

∏ ∐_{t∈T_{0}}^{c}(A_{t}, B_{t}) := lim_{⟵}_{V} ( ∏ ∐_{t∈T_{0}}(A_{t}, B_{t}))/V,

the compactification of ∏ ∐_{t∈T_{0}}(A_{t}, B_{t}), where V runs through all open subgroups of finite index in ∏ ∐_{t∈T_{0}}(A_{t}, B_{t}). The canonical map ∏ ∐_{t∈T_{0}}(A_{t}, B_{t}) → ∏ ∐_{t∈T_{0}}^{c}(A_{t}, B_{t}) has dense image.

We define the discretization of ∏ ∐_{t∈T_{0}}(A_{t}, B_{t}) by

∏ ∐_{t∈T_{0}}^{d}(A_{t}, B_{t}) := \underrightarrow{lim}_{W} W

where W runs through the finite subgroups of ∏ ∐_{t∈T_{0}}(A_{t}, B_{t}). If the subgroups B_{t} of A_{t}, t ∈ T_{0}, are open and compact, then ∏ ∐_{t∈T_{0}}(A_{t}, B_{t}) is locally compact. Using the equality (∏ ∐_{t∈T_{0}}(A_{t}, B_{t}))^{∨} = ∏ ∐_{t∈T_{0}}(A_{t}^{∨}, (A_{t}/B_{t})^{∨}), we obtain

∏ ∐_{t∈T_{0}}^{d}(A_{t}, B_{t}) = ((( ∏ ∐_{t∈T_{0}}(A_{t}, B_{t}))^{∨})^{c})^{∨} and ∏ ∐_{t∈T_{0}}^{c}(A_{t}, B_{t}) = ((( ∏ ∐_{t∈T_{0}}(A_{t}, B_{t}))^{∨})^{d})^{∨},

where ^{∨} denotes the Pontryagin-dual.

Proposition 3.5 Let T the one-point compactification of the discrete set T_{0}.

(i) Let A_{t}, t ∈ T_{0}, be discrete abelian torsion groups such that their exponents have a common finite bound. Then we have the equality of abstract groups

∏ ∐_{t∈T_{0}}^{d}(A_{t}, B_{t}) = ∏ ∐_{t∈T_{0}}(A_{t}, B_{t}).

However, ∏ ∐_{t∈T_{0}}^{d}(A_{t}, B_{t}) is endowed with the discrete topology in contrast to ∏ ∐_{t∈T_{0}}(A_{t}, B_{t}).

(ii) Let each A_{t}, t ∈ T_{0}, be a profinite abelian group. Then the canonical map ∏ ∐_{t∈T_{0}}(A_{t}, B_{t}) ↪ ∏ ∐_{t∈T_{0}}^{c}(A_{t}, B_{t}) is injective. Setting A_{∗} = B_{∗} = {0}, then (B_{t})_{t∈T} is a continuous familiy of closed subgroups of ∏ ∐_{t∈T_{0}}^{c}(A_{t}, B_{t}) and

∗_{t∈T}(A_{t}, B_{t}) ≅ ∏ ∐_{t∈T_{0}}^{c}(A_{t}, B_{t})

where ∗_{t∈T} (A_{t}, B_{t}) is the corresponding corestricted free pro-abelian product.

(iii) Let ∗_{t∈T} (G_{t}, U_{t}) be the corestricted free pro-𝔠-product of the family {G_{t}}_{t∈T} with respect to the continuous family {U_{t}}_{t∈T} where G_{∗} = U_{∗} = {∗}. Then

( ∗_{t∈T}(G_{t}, U_{t}))^{ab} = ∏ ∐_{t∈T}^{c}(G_{t}^{ab}, Ũ_{t}) = lim_{⟵}_{N} ( ∏ ∐_{t∈T}(G_{t}^{ab}, Ū_{t}))/N,

where G_{t}^{ab} = G_{t}/[G_{t}, G_{t}] and Ū_{t} = U_{t}[G_{t}, G_{t}]/[G_{t}, G_{t}] and N runs through the subgroups of ∏ ∐_{t∈T} (G_{t}^{ab}, Ū_{t}) of finite index such that N ∩ G_{t}^{ab} is open in

G_{t}^{ab} for all t ∈ T and N ⊇ Ū_{t} for almost all t ∈ T.

Proof: Under the assumption of (i) every element of ∏ ∐_{t∈T_{0}}(A_{t}, B_{t}) generates a finite subgroup. Thus (i) is obvious.

In order to prove (ii), we fix t_{0} ∈ T_{0}. If V_{t_{0}} is an open subgroup of A_{t_{0}}, then Ṽ_{t_{0}} := V_{t_{0}} × ∏ ∐_{t≠t_{0}}(A_{t}, B_{t}) is an open subgroup of ∏ ∐_{t∈T_{0}}(A_{t}, B_{t}) of finite index. If V_{t_{0}} runs through a basis of neighborhoods of the unit of A_{t_{0}}, then the t_{0}-component of the intersection of the open subgroups Ṽ_{t_{0}} is {0}. Varying t_{0}, it follows that the intersection ⋂ V of all open subgroups V of finite index in ∏ ∐_{t∈T_{0}}(A_{t}, B_{t}) is zero. This proves the first assertion of (ii). The second assertion of (ii) follows immediately from the definitions. Finally, (iii) follows from (ii) and (2.5). In fact, noting that {s} ⊆ T is open and closed in T for all s ∈ T_{0}, by (1.4) (ii) there exists a continuous splitting of G_{s} → ∗_{t∈T}(G_{t}, U_{t}). □

4 Cohomology of corestricted free products

Now we consider the cohomology of a corestricted free pro-𝔠-product

G = ∗_{t∈T}(G_{t}, U_{t})

of a family (G_{t})_{t∈T} with respect to a continuous family (U_{t})_{t∈T} of closed subgroups of a pro-𝔠-group G̃ in the case where T = T_{0} ∪. {∗} is the one-point compactification of a discrete set T_{0}. By Ũ_{t} we denote the normal closure of U_{t} in G_{t}.

Lemma 4.1 With the above notation (Ũ_{t})_{t∈T} is a continuous family. Furthermore, if G = ⋃ . _{t∈T_{0}} G_{t} ∪. {∗}, U = ⋃ . _{t∈T_{0}} U_{t} ∪. {∗} and Ũ = ⋃ . _{t∈T_{0}} Ũ_{t} ∪. {∗}, then the (continuous) morphism of bundles id : (G, U) → (G, Ũ) induces an isomorphism of pro-𝔠-groups

∗_{t∈T}(G_{t}, U_{t}) ⟶^{∼} ∗_{t∈T}(G_{t}, Ũ_{t}).

Proof: Let V be an open neighborhood of the identity in G̃. Then V contains a normal subgroup N of G̃. Since {U_{t}}_{t∈T} is a continuous family, the set

T (N) = {t ∈ T | U_{t} ⊆ N} = {t ∈ T | Ũ_{t} ⊆ N}

is an open subset of T containing {∗}. Since T is the one-point compactification of the discrete set T_{0}, it follows that also the set

T (V) = {t ∈ T | Ũ_{t} ⊆ V }

containing T (N) is open in T. Hence, the family (Ũ_{t})_{t∈T} is also continuous and we have the continuous (not necessarily open) morphism of bundles id : (G, U) → (G, Ũ). In order to show that the corresponding corestricted free products coincide, it remains to show that any morphism (G, U) → H, where H is a finite 𝔠-group, is continuous with respect to the topology of (G, Ũ). However, this is a direct consequence of remark 2 following definition (3.4). □

Let A be a discrete ∗_{t∈T} (G_{t}, U_{t})-module. Let H_{nr}^{i}(G_{t}, A) be defined as the image of the inflation map H^{i}(G_{t}/Ũ_{t}, AŨ_{t})→H^{i}(G_{t}, A). It is easy to see that the map

res_{H} : H^{i}(G, A) → ∏_{t∈T} H^{i}(G_{t}, A)

has image in ∏ ∐_{T}^{d}(H^{1}(G_{t}, A), H_{nr}^{1}(G_{t}, A)), see [4] §4.

Theorem 4.2 With the notation and assumptions as above let A be a finite ∗_{t∈T} (G_{t}, U_{t})-module. Then there is an exact sequence

0 →A/A^{G} →∏ ∐_{T}^{d}(A/A^{G_{t}}, AŨ_{t}/A^{G_{t}}) → H^{1}(G, A) →∏ ∐_{T}^{d}(H^{1}(G_{t}, A), H_{nr}^{1}(G_{t}, A))→ 0

and an isomorphism

H^{2}(G, A) ⟶^{∼} ∏ ∐_{T}^{d}(H^{2}(G_{t}, A), H_{nr}^{2}(G_{t}, A)).

If the cohomological dimension of G_{t}/Ũ_{t} is equal or less than 1 for all t ∈ T, then

H^{i}(G, A) ⟶^{∼} ⨁_{T}H^{i}(G_{t}, A), i ≥ 3.

Proof: By lemma (4.1), we may assume without loss of generality that U_{t} = Ũ_{t}, i.e. U_{t} is a normal subgroup of G_{t} for any t ∈ T. If i ≤ 2, the assertion follows along the same lines as in [4] Satz (4.1); but one has to be careful with the topology and has to replace the restricted product by its discretization, see (3.5)(i).

If i ≥ 3, we use dimension shifting: Let A′ be defined by the exact sequence 0 → A → Coind_{G}A → A′ → 0, where Coind_{G}A denotes the coinduced G-module consisting of all continuous functions from G to A. By assumption we have H_{nr}^{i}(G_{t}, A) = 0 for all t ∈ T and i ≥ 2. Thus the assertion follows since we get compatible isomorphisms ϕ: H^{i−1}(G, A′) →^{∼} H^{i}(G, A) and ϕ_{t}: H^{i−1}(G_{t}, A′) →^{∼} H^{i}(G_{t}, A), see also [4] Satz (4.2). □

Now we consider in the following case which has an application in number theory. Let I be a directed set and let

G = lim_{⟵}_{λ∈I} G_{λ},

be a pro-𝔠-group given as projective limit of pro-𝔠-groups G_{λ}. Let T = lim_{⟵}_{λ∈I} T_{λ},

where T_{λ} = T_{0λ} ∪. {∗_{λ}} is the one-point compactification of a discrete set T_{0λ}. Let {G, U} = {(G_{λ}, U_{λ})}_{λ} be a projective system of corestricted bundles (G_{λ}, U_{λ}) over T_{λ} with transition maps ϕ_{μλ}, where G_{λ} = ⋃ . _{t_{λ}∈T_{0λ}} G_{t_{λ}} ∪. {∗_{λ}} and U_{λ} = ⋃ . _{t_{λ}∈T_{0λ}} U_{t_{λ}} ∪. {∗_{λ}} and U_{t_{λ}} ⊆ G_{t_{λ}} ⊆ G_{λ} are closed subgroups, i.e. the diagrams

\xymatrix{U_{t_{μ}} \ar@{^{(}->}[r] \ar[d] & G_{t_{μ}} \ar@{^{(}->}[r] \ar[d] & G_{μ} \ar[d] \\ U_{t_{λ}} \ar@{^{(}->}[r] & G_{t_{λ}} \ar@{^{(}->}[r] & G_{λ}}

commute for μ ≥ λ and ϕ_{μλ}(t_{μ}) = t_{λ}. Let U = lim_{⟵}_{λ} U_{λ} and G = lim_{⟵}_{λ} G_{λ} be the projective limits and let G_{t} = lim_{⟵}_{λ} G_{t_{λ}}, U_{t} = lim_{⟵}_{λ} U_{t_{λ}}. Assume that the following holds:

(i) the morphisms ϕ_{μλ} are fibrewise surjective and strict, (ii) for all μ ≥ λ the transition maps satisfy ϕ_{μλ}^{−1}({∗_{λ}}) = {∗_{μ}}. By theorem (2.11) we have

∗_{T}(G, U) = ∗_{t∈T}(G_{t}, U_{t}) ⟶^{∼} lim_{⟵}_{λ} ∗_{t_{λ}∈T_{λ}}(G_{t_{λ}}, U_{t_{λ}})

and we have a canonical homomorphism

φ : ∗_{t∈T}(G_{t}, U_{t}) ⟶ G.

Remark: In the number theoretical situation we have in mind, one can also argue with theorem (2.6) in order to establish the isomorphism above.

Proposition 4.3 With the notation and assumptions as above let p be a prime number and let 𝔠 be the class of finite p-groups. Then the following are equivalent.

(i) ∗_{t∈T}(G_{t}, U_{t}) _{∼} ⟶^{φ} G is an isomorphism.

(ii) The induced maps

φ_{∗} : H^{1}(G, ℤ/pℤ) →^{∼} \underrightarrow{lim}_{λ} ∏ ∐_{T_{λ}}^{d}(H^{1}(G_{t_{λ}}, ℤ/pℤ), H_{nr}^{1}(G_{t_{λ}}, ℤ/pℤ))

φ_{∗} : H^{2}(G, ℤ/pℤ) ↪ \underrightarrow{lim}_{λ} ∏ ∐_{T_{λ}}^{d}(H^{2}(G_{t_{λ}}, ℤ/pℤ), H_{nr}^{2}(G_{t_{λ}}, ℤ/pℤ))

are bijective resp. injective.

Proof: Using (4.2), we have

\begin{aligned}H^{i}(∗_{t∈T}(G_{t}, U_{t}), ℤ/pℤ) & = & \underrightarrow{lim}_{λ} H^{i}( ∗_{t_{λ}∈T_{λ}}(G_{t_{λ}}, U_{t_{λ}}), ℤ/pℤ) \\ & = & \underrightarrow{lim}_{λ} ∏ ∐_{T_{λ}}^{d}(H^{i}(G_{t_{λ}}, ℤ/pℤ), H_{nr}^{i}(G_{t_{λ}}, ℤ/pℤ)) \end{aligned}

for i = 1, 2. Now the usual argument, see [5] (1.6.15), gives the result. □

We have the following application in number theory. Let p be a prime number and let k be number field and T a set of primes of k. We use the following notation.

k(p)  is the maximal p-extension of k,
k^{T}  is the maximal p-extension of k which is completely decomposed

at every prime of T,

G(k(p)|k^{T}) is the Galois group of the extension k(p)|k^{T},
G_{𝔓}(k)  is the decomposition group of G(k(p)|k) with respect to 𝔓,
I_{𝔓}(k)  is and inertia group of G(k(p)|k) with respect to 𝔓,

where 𝔓 is an extension of a prime 𝔭 of k to k(p).

Theorem 4.4 Let p be an odd prime number and let k be a number field of CMtype containing the group μ_{p} of all p-th roots of unity, with maximal totally real subfield k^{+}, i.e. k = k^{+}(μ_{p}) is totally imaginary and [k : k^{+}] = 2. Let

T = {𝔭 | 𝔭 ∩ k^{+} is inert in k|k^{+}} ∪ {𝔭|p}.

Then the Galois group G(k(p)|k^{T}) is the corestricted free pro-p-product of the family (G_{𝔓}(k))_{𝔓∈T} with respect to the continuous family (I_{𝔓}(k))_{𝔓∈T}, i.e. the canonical map

∗_{𝔓∈T} (G_{𝔓}(k), I_{𝔓}(k)) ⟶^{∼} G(k(p)|k^{T})

is an isomorphism. Here T = lim_{⟵}_{K} T̄_{K} is the projective limit of the one-point

compactifications T̄_{K} of the discrete sets T_{K} of all prolongations of T to K and K runs through all finite Galois extensions inside k^{T} |k.

This follows from (4.3) and the results of [7]: (2.4), (2.5), (2.2).

References

[1] Gildenhuys, D., Ribes, L. A Kurosh subgroup theorem for free pro-𝔠-products of pro-𝔠-groups. Trans. Amer. Math. Soc. 186 (1973) 309–329

[2] Haran, D. On closed subgroups of free products of profinite groups. Proc. London Math. Soc. (3) 55 (1987) 266–298

[3] Melnikov, O. V. Subgroups and homology of free products of profinite groups (in Russian). Izv. Akad. Nauk SSSR 53 (1989). English translation in Math. USSR Izv. 34 (1990) 97–119

[4] Neukirch, J. Freie Produkte pro-endlicher Gruppen und ihre Kohomologie. Archiv der Math. 22 (1971) 337–357

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

[6] Pontryagin, L. S. Topological Groups. New York, London, Paris 1966

[7] Wingberg, K. Sets of Completely Decomposed Primes in Extensions of Number Fields. Preprint 2013

Mathematisches Institut e-mail: gaertner (at) mathi.uni-heidelberg.de der Universität Heidelberg wingberg (at) mathi.uni-heidelberg.de Im Neuenheimer Feld 288 69120 Heidelberg Germany