On the Embedding Problem with Non-abelian Kernel

by Kay Wingberg at Heidelberg

Version: October 6, 2010

An embedding problem for a profinite group G is a diagram

\xymatrix{ &  &  & G \ar@{->>}[d]^{φ} \\ 1 \ar[r] & N \ar[r] & E \ar[r] & Ḡ \ar[r] & 1}

with an exact sequence of profinite groups and a surjective homomorphism φ. A solution of this problem is a homomorphism ψ: G → E such that the diagram commutes. A solution is called proper if ψ is surjective. In the following we denote an embedding problem for G only by its exact line.

In this note we consider pro-p groups and problems with not necessarily abelian kernel N and prove a theorem which generalizes a result of Lur’e for the Galois group of the maximal p-extension of a 𝔭-adic field [1]. We will follow the proof given in [3].

Let G be a pro-p group of finite cohomological dimension cd_{p} G = n. By d(G) we denote the generator rank dim_{𝔽_{p}}H^{1}(G, ℤ/pℤ) of G. We recall the notion of the dualizing module of G, [2] (3.4.4):

D(G) = \underrightarrow{lim}_{ν∈ℕ} \underrightarrow{lim}_{U} H^{n}(U, ℤ/p^{ν}ℤ)^{∨},

where^{∨} denotes the Pontryagin-dual, the second direct limit is taken over all open subgroups U of G and the transition maps are the duals of the corestriction maps. D(G) is a discrete G-module in a natural way. We have a natural isomorphism

H^{n}(G, A)^{∨} ⟶^{∼} Hom_{G}(A, D(G))

for every finite p-primary G-module A.

Theorem 1 Let G be a pro-p group of cohomological dimension cd_{p} G = 2 with dualizing module D = D(G). Let H be a normal subgroup of G and let E be a pro-p group with d(E) = d(G/H). Assume that one of the following assumptions is fulfilled:

(i) D ≅ ℚ_{p}/ℤ_{p} as abelian group. (ii) D = D^{H}.

Then the embedding problem

(1)  1 ⟶ N ⟶ E ⟶ G/H ⟶ 1

has a proper solution if and only if the abelian problem

(2),  1 ⟶ N/N^{p^{s}}[N, N] ⟶ E/N^{p^{s}}[N, N] ⟶ G/H ⟶ 1

has a proper solution, where p^{s} is the order of D^{H}, s ≤ ∞.

Here we put N^{p^{∞}} = 1. We will need the following

Lemma 2 Let G be a pro-p group of cohomological dimension cd_{p} G = 2 with dualizing module D. Then the embedding problem

1 ⟶ A ⟶ E ⟶ G/H ⟶ 1

with finite p-primary abelian kernel A is solvable if and only if the problem

1 ⟶ A/B ⟶ E/B ⟶ G/H ⟶ 1

is solvable, where B is a normal subgroup of E, contained in A, such that

Hom_{G}(A, D) = Hom_{G}(A/B, D).

Proof: Let π: A ↠ A/B be the natural surjection. Then by assumption

H^{0}(G, Hom(A/B, D)) _{∼} ⟶^{π_{∗}} H^{0}(G, Hom(A, D)).

It follows that π_{∗}: H^{2}(G, A) →^{∼} H^{2}(G, A/B) is an isomorphism. Using [2](3.5.9), the commutative diagram

\xymatrix{H^{2}(G, A) \ar[r]^{π_{∗}}_{∼} & H^{2}(G, A/B) \\ H^{2}(G/H, A) \ar[r]^{π_{∗}} \ar[u]^{inf} & H^{2}(G/H, A/B) \ar[u]_{inf}}

gives the result. □

Proof of the theorem: Passing to the projective limit by using standard arguments, we may assume that N is a finite p-group.

If D = D^{H}, then s = ∞. Otherwise, using the lemma and since

Hom_{G}(N^{ab}, D) = Hom_{G}(N^{ab}, D^{H}) = Hom_{G}(N/N^{p^{s}}[N, N], D),

the embedding problem (2) is solvable if and only if

(3)  1 ⟶ N^{ab} ⟶ E/[N, N] ⟶ G/H ⟶ 1

is solvable.

In order to show that the solvability of (3) implies the solvability of (1), we use induction on the order of N. Since d(E) = d(G/H), every solution will be proper.

If the dualizing module D is isomorphic to ℚ_{p}/ℤ_{p} as abelian group, then we get for the kernel of the homomorphism

φ : G ⟶ Aut(D) ≅ ℤ_{p}^{×},

which is given by the action of G on D, the inclusion G/ ker φ ⊆ ℤ_{p}^{×}. We may assume that G/ ker φ ⊆ 1 + pℤ_{p} ≅ ℤ_{p} (for p ≠ 2 this is always the case) and that s ≥ 2, if p ≠ 2, and s ≥ 3, if p = 2, i.e. G/ ker φ ⊆ 1 + p^{2}ℤ_{p}. otherwise we consider the embedding problem

1 ⟶ N ⟶ E′ ⟶ G/H′ ⟶ 1,

where E′ = E × ℤ/pℤ and H′ is defined by D^{H′} = _{p^{2}}D if p ≠ 2 (resp. E′ = E × (ℤ/2ℤ)^{ε}, where ε ≤ 2 such that D^{H′} = _{8}D); hence G/H′ = G/H × ℤ/pℤ (resp. G/H′ = G/H × (ℤ/2ℤ)^{ε}); observe that d(E′) = d(G/H′).

Now we assume that (3) is solvable. Let Ñ = N^{p}[N, E]. If N ⊇ N′ ⊇ Ñ, then N′ is normal in E and we get a commutative and exact diagram

[figure: 1; N^{ab}; E/[N, N]; G/H; 1; 1; N′/[N, N]; E/[N, N]; E/N′; 1; N/N′.]

Since the solution of (3) is proper, we get a normal subgroup H′ of G contained in H such that E/N′ ≅ G/H′. We consider two cases:

1. Assume that there exists a subgroup N ⊇ N′ ⊇ Ñ, (N : N′) = p such that D^{H} = D^{H′}. Since N/N′ is cyclic, we obtain

Hom_{G}((N′)^{ab}, D) = Hom_{G}((N′)^{ab}, D^{H′})
= Hom_{G}((N′)^{ab}, D^{H})
= Hom_{G}(N′/[N′, N], D^{H}) = Hom_{G}(N′/[N, N], D).

Using the lemma, the solvability of (3) implies the solvability of

1 ⟶(N′)^{ab} ⟶ E/[N′, N′] ⟶ G/H′ ⟶ 1

By the induction hypothesis the result follows. This finishes already the proof of the theorem in the case that D^{H} = D.

2. The assumption of 1. is not fulfilled, i.e. D^{H} ≠ D^{H′} for all subgroups N ⊇ N′ ⊇ Ñ. Hence we assume that D ≅ ℚ_{p}/Z_{p} and we have seen above that φ(H) ≅ ℤ_{p}, φ : G → Aut(D). It follows that (N : Ñ) = p. As above let H̃ be obtained via a solution of (3).

Let N =< β, Ñ >, and f ∈ Hom_{G}(Ñ^{ab}, DH̃). It follows that

Ñ =< β^{p}, Ñ^{p}, [N, E] > .

If ᾱ denotes the class α[Ñ, Ñ], α ∈ Ñ, then f(β̄^{p})^{β} = f(β̄^{p}), hence f(β̄^{p}) ∈ (DH̃)^{<β>} = D^{H}. Furthermore, f(Ñ^{p}) ∈ (DH̃)^{p} = D^{H}. Since

[N, E]^{p} ⊆ [N^{p}, E][[N, E], N] ⊆ [N^{p}[N, E], E] = [Ñ, E],

we obtain

f([N, E]/[Ñ, Ñ])^{p} ⊆ f(Ñ^{ab})^{E−1} ⊆ (DH̃)^{G−1} ⊆ (DH̃)^{p^{2}} = (D^{H})^{p},

(since s ≥ 2, if p ≠ 2, and s ≥ 3, if p = 2), hence f([N, E]/[Ñ, Ñ]) ⊆ D^{H}. It follows that

Hom_{G}(Ñ^{ab}, D) = Hom_{G}(Ñ/[Ñ, N], D^{H}) = Hom_{G}(Ñ/[N, N], D).

As in case 1. the result follows. □

References

[1] Lur’e, B. B.B. Problem of immersion of local fields with non-abelian kernel. J. Soviet Math. 6 (1976), 298-306

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

[3] Wingberg, K. Eine Bemerkung zum Einbettungsproblem mit nichtabelschem Kern. J. reine u. angew. Math. 331 (1982), 146-150

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

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