Free quotients of Demuškin groups with operators

by Kay Wingberg at Heidelberg

This paper concerns the problem of the existence of Galois extensions of a local number field k whose Galois groups are free pro-p groups.

If k is a 𝔭-adic field, then the Galois group G(k(p)|k) of the maximal pextension of k is a free pro-p group if k does not contain the group of p-th roots of unity, and otherwise G(k(p)|k) is a Demuškin group, i.e. a pro-p Poincaré group of dimension 2. These groups are defined as follows: a pro-p group G is called a Demuškin group if its cohomology has the following properties:

dim_{𝔽_{p}} H^{1}(G, ℤ/pℤ) < ∞,

dim_{𝔽_{p}} H^{2}(G, ℤ/pℤ) = 1, and the cup-product

H^{1}(G, ℤ/pℤ) × H^{1}(G, ℤ/pℤ) ⟶^{∪} H^{2}(G, ℤ/pℤ) is non-degenerate.

In the following we exclude the exceptional case that G ≅ ℤ/2ℤ. Then the dualizing module I of G is isomorphic to ℚ_{p}/ℤ_{p} as an abelian group and we have a canonical action of G on I.

Demuškin groups occur not only as Galois groups of the maximal p-extension of 𝔭-adic number fields (if these fields contain the group of p-th roots of unity) but also as the p-completion of the fundamental group of a compact oriented Riemann surface. In the first case the action of G on I is non-trivial whereas in the second case G acts trivially on I. We will only consider Demuškin groups acting nontrivially on its dualizing module and we are interested in free pro-p quotients of these groups. Possible ranks of such free quotients were first calculated in [7], [6] and [2].

In many cases of interest a finite group Δ of order prime to p acts on a Demuškin group. As an example consider the local field k = ℚ_{p}(ζ_{p}), where p is an odd prime number. Then G(k|ℚ_{p}) ≅ ℤ/(p − 1)ℤ acts on the Demuškin group G(k(p)|k). Of particular interest is the case where Δ is generated by an involution, e.g. G(k|ℚ_{p}(ζ_{p} + ζ_{p}^{−1})) ≅ ℤ/2ℤ acts on G(k(p)|k); see [8] where Demuškin groups with involution were considered.

In this paper we are interested in free pro-p quotients F of a Demuškin group G which are invariant under a given action of Δ on G and where the maximal abelian factor group F^{ab} = F/[F, F] of F has a prescribed action of Δ.

In particular, we will show the following: if p is odd and Δ ≅ ℤ/2ℤ acts on a p-Demuškin group G of rank n+2, then there exists a Δ-invariant free quotient F of G such that rank_{ℤ_{p}}(F^{ab})^{+} = 1 and rank_{ℤ_{p}}(F^{ab})^{−} = n/2 (here the (±)-eigenspaces of a ℤ_{p}[Δ]-module M are denoted by M^{±}). This situation occurs as the following example shows: Let p be an odd regular prime number and consider the CMfield k = ℚ(ζ_{p}) with maximal totally real subfield k^{+} = ℚ(ζ_{p} + ζ_{p}^{−1}). Then the Galois group G(k|k^{+}) ≅ ℤ/2ℤ acts on the Galois group G(k_{S_{p}}|k) of the maximal p-extension k_{S_{p}} of k which is unramified outside p. Let k_{𝔭} be the completion of k with respect to the unique prime 𝔭 of k above p and let k_{𝔭}(p) its maximal p-extension. Since we assume that k has no unramified p-extension, we have a surjection

G(k_{𝔭}(p)|k_{𝔭}) ↠ G(k_{S_{p}}|k)

of the Demuškin group G(k_{𝔭}(p)|k_{𝔭}) of rank p+1 onto the free pro-p group G(k_{S_{p}}|k) of rank (p + 1)/2 (see [5] X.6 example 2) and

rank_{ℤ_{p}}(G(k_{S_{p}}|k)^{ab})^{+} = 1, rank_{ℤ_{p}}(G(k_{S_{p}}|k)^{ab})^{−} = (p − 1)/2 ,

since (G(k_{S_{p}}|k)^{ab})^{+} ≅ G((k^{+})_{S_{p}}|k^{+})^{ab} ≅ ℤ_{p}.

It would be of interest under which conditions there exist large free quotients of G(k_{S_{p}}(p)|k) for an arbitrary CM-field k. If we assume that no prime 𝔭 above p splits in the extension k|k^{+}, such a quotient should be defined by free quotients of the local groups G(k_{𝔭}(p)|k_{𝔭}), 𝔭|p, with an acting of G(k_{𝔭}|k_{p}^{+}) ≅ G(k|k^{+}) as above.

1 Pro-p Operator Groups

Let p be a prime number. For a pro-p group G we denote its Frattini subgroup by G^{∗} = G^{p}[G, G] and its descending p-central series by

G^{(1)} = G and G^{(i+1)} = (G^{(i)})^{p}[G^{(i)}, G], i ≥ 1.

For the cohomology groups of G with coeffients in ℤ/pℤ we often set H^{i}(G) = H^{i}(G, ℤ/pℤ). If A is an abelian group, then A^{∨} denotes its Pontryagin dual.

Let

1 ⟶ H ⟶ H ⟶ Δ ⟶ 1

be an exact sequence of profinite groups where H is a pro-p group and the (supernatural) order of Δ is prime to p. Let G be a pro-p-H operator group (see [5] IV.3 ex.3).

Proposition 1.1 With the notation as above the following is true:

(i) Let A ⊆ G/G^{∗} be a 𝔽_{p}[[H]]-submodule which is free as a 𝔽_{p}[[H]]-module. Then there exists a pro-p-H operator subgroup U of G such that the canonical map

U/U^{∗} ⟶ G/G^{∗}

is injective with image equal to A.

(ii) Let B ⊆ G^{ab} be a ℤ_{p}[[H]]-submodule which is free as a ℤ_{p}[[H]]-module. Then there exists a pro-p-H operator subgroup V of G such that the canonical map

V^{ab} ⟶ G^{ab}

is injective with image equal to B.

If H = Δ is finite of order prime to p and G is a finitely generated pro-p group with an action by Δ, then case (i) is a result of N. Boston, see [1], prop. (2.3).

Proof: We only prove case (i), since the proof of case (ii) is completely analogously. Let

U = {U ⊆ G | U is an H-invariant subgroup, UG^{∗}/G^{∗} = A}.

Then U is not empty since the full pre-image of A under the canonical H-invariant map G ↠ G/G^{∗} is an element of U. Furthermore, if {U_{λ}, λ ∈ I} is a totally ordered subset of U, then V = ⋂ U_{λ} is an H-invariant subgroup of G and

V G^{∗}/G^{∗} = \underleftarrow{lim} U_{λ}G^{∗}/G^{∗} = A,

and so V ∈ U. By Zorn’s lemma there exists a minimal element U_{0} ∈ U. The H-invariant map

U_{0}/U_{0}^{∗} ↠ U_{0}G^{∗}/G^{∗} = A

has an H-invariant splitting s. Indeed, by assumption, A is projective as a 𝔽_{p}[[H]]- module, and so A is a projective 𝔽_{p}[[H]]-module, see [5] V.2 ex. 7). Let W be the full pre-image of s(A) ⊆ U_{0}/U_{0}^{∗} under the canonical map U_{0} ↠ U_{0}/U_{0}^{∗}. Then W is H-invariant, W/U_{0}^{∗} = s(A) and we have the commutative diagram

[figure: U_{0}^{∗}; U_{0}; U_{0}/U_{0}^{∗}; U_{0}G^{∗}/G^{∗}; = A ⊆ G/G^{∗}; U_{0}^{∗}; W; W/U_{0}^{∗}; ∼]

Since W G^{∗}/G^{∗} = A, it follows that W ∈ U, and so W = U_{0} because of the minimality of U_{0}. We obtain that U_{0}/U_{0}^{∗} = W/U_{0}^{∗} ⟶^{∼} U_{0}G^{∗}/G^{∗} = A which finishes the proof of the proposition. □

If k is a field and Δ a finite group of order prime to the characteristic of k, then by Maschke’s theorem the category of k[Δ]-modules is semi-simple. If Δ is abelian und k is splitting field for Δ, then every simple k[Δ]-module has k-dimension equal to 1; one has a decomposition into eigenspaces

M = ∏_{χ∈Δ^{∨}} M^{χ} ,

where M^{χ} = {x ∈ M | x^{σ} = x^{χ(σ)} for all σ ∈ Δ} is the isotypic component of a k[Δ]-module M with respect to the character χ of Δ

Corollary 1.2 Let p be an odd prime number and let Δ be a finite abelian group of exponent p − 1 with character group Δ^{∨}. Let G be a pro-p-Δ operator group and let

G/G^{∗} = ∏_{χ∈Δ^{∨}}(G/G^{∗})^{χ},

be the decomposition of G/G^{∗} in χ-eigenspaces. Then there exist subsets M_{χ} of G such that

(i) ⋃_{χ∈Δ^{∨}} M_{χ} is a minimal set of generators of G,

(ii) \overline{M}_{χ} = \{x mod G^{∗} | x ∈ M_{χ}\} is a basis of (G/G^{∗})^{χ} for all χ ∈ Δ^{∨},

(iii) x^{σ} = x^{χ(σ)} for x ∈ M_{χ} and σ ∈ Δ.

Proof: This follows directly from proposition (1.1)(i) with H = 1 and A a 1-dimensional subspace of an eigenspace (G/G^{∗})^{χ}. □

Let Δ be a finite group of order prime to p and G a pro-p-Δ operator group which is finitely generated as a pro-p group. Let

1 ⟶ R ⟶ E ⟶^{φ} G ⟶ 1

be an exact sequence of pro-p groups such that the surjection φ induces an isomorphism E/E^{∗} ⟶^{∼} G/G^{∗}. A lemma, which we will need later, is the following.

Lemma 1.3 With the notation and assumptions as above there exists a continuous action of Δ on E extending the action on G, i.e. the surjection E ↠ G is Δ-invariant and R is a Δ-operator group.

Proof: We consider the natural homomorphism

Aut_{R}(E) ⟶ Aut(G)

where Aut_{R}(E) ⊆ Aut(E) denotes the group of automorphisms θ of E such that θ(R) ⊆ R. Recall that the kernel of the homomorphism Aut(G) → Aut(G/G^{∗}) is a pro-p group, cf. [4] 5.5. Therefore the commutative diagram

\xymatrix{ & Δ \ar[d] \\ Aut_{R}(E) \ar[r] \ar[d] & Aut(G) \ar[d] \\ Aut(E/E^{∗}) \ar[r]^{∼} & Aut(G/G^{∗})}

shows that the image of the prime-to-p group Δ in Aut(G) is contained in the image of Aut_{R}(E) → Aut(G). Since the kernel of Aut(E) ↠ Aut(E/E^{∗}) is a pro-p group, and Aut(E/E^{∗}) ⟶^{∼} Aut(G/G^{∗}) is an isomorphism, it follows that the kernel of Aut_{R}(E) → Aut(G) is a pro-p group. Using again that Δ is a prime-to-p group, we get a commutative diagram

\xymatrix{ & Δ \ar[d] \ar@{.>}[dl] \\ Aut_{R}(E) \ar[r] & Aut(G).}

This proves the lemma. □

Let p be a prime number and let

\xymatrix{1 \ar[r] & G \ar[r] & G \ar[r] & Δ \ar[l]^{s} \ar[r] & 1 ,}

be a split exact sequence of profinite groups where G is a pro-p group and Δ is a finite group of order prime to p. Thus G is the semi-direct product of Δ by G and G is a pro-p-Δ operator group where the action of Δ on G is defined via the splitting s. Conversely, given a pro-p-Δ operator group G, we get a semi-direct product G = G ⋊ Δ where the action of Δ on G is the given one.

Let G(p) be the maximal pro-p quotient of G and let G_{Δ} be the maximal quotient of G with trivial Δ-action. Observe that G_{Δ} is well-defined.

Proposition 1.4 With the notation and assumptions as above there is a canonical isomorphism

G_{Δ} ⟶^{∼} G(p) .

Furthermore, if Δ_{0} is a subgroup of Δ such that H^{2}(G, ℤ/pℤ)^{Δ_{0}} = 0, then G_{Δ_{0}} is a free pro-p group.

Proof: Consider the exact commutative diagram

[figure: 1; N ∩ G; N; Δ; s; 1; 1; G; G; Δ; s; 1; G̃; ∼; G(p); ,]

where N is the kernel of the canonical surjection G ↠ G(p) and G̃ denotes the quotient G/N ∩ G. Since Δ acts on N ∩ G via s, we obtain an induced action on G̃. This action is trivial because

g^{s(σ)−1} = [s(σ), g] ∈ N ∩ G for g ∈ G and σ ∈ Δ ,

and so we get a surjection

φ : G_{Δ} ↠ G̃ .

Consider the exact commutative diagram

[figure: 0; H^{1}(G̃); H^{1}(G_{Δ}); ∼; inf; H^{1}(ker φ)^{G_{Δ}}; H^{2}(G̃); H^{2}(G_{Δ}); inf; H^{1}(G)^{Δ}; H^{2}(G)^{Δ}; H^{1}(G(p)); ∼; _{inf _{1}}; ∼; res; H^{1}(G); ∼; res; H^{2}(G(p)); ∼; res; _{inf _{2}}; H^{2}(G); ∼; res; ,]

where the map inf_{1} is bijective and inf_{2} is injective because Hom(N, ℤ/pℤ) = 0. Therefore H^{1}(ker φ)^{G_{Δ}} = 0, and so by the Frattini argument, see [5] (1.7.4), ker φ = 1, i.e. G_{Δ} ≅ G̃ ≅ G(p). Furthermore, it follows that

H^{2}(G_{Δ}) ⟶^{inf} H^{2}(G)^{Δ}

is injective. Therefore, if H^{2}(G)^{Δ} = 0, then H^{2}(G_{Δ}) = 0, and so G_{Δ} is a free pro-p group.

For a subgroup Δ_{0} of Δ let G_{0} be the semi-direct product G ⋊ Δ_{0}. Replacing in the proof above Δ by Δ_{0} and G by G_{0}, we obtain the assertion for every subgroup. □

2 Demuškin Groups with Operators

In this section we assume that

Δ is a finite group of order prime to p and
G is a p-Demuškin group of rank n + 2, n ≥ 0, with dualizing module I

and an action by Δ.

Let G be the semi-direct product of Δ by G, i.e. the sequence

1 ⟶ G ⟶ G ⟶ Δ ⟶ 1

is split-exact.

The dualizing module I of G is defined as

I = \underrightarrow{lim}_{m} \underrightarrow{lim}_{U} H^{2}(U, ℤ/p^{m}ℤ)^{∨},

where the second limit is taken over the maps cor ^{∨}, the dual to the corestriction, and U runs through the open normal subgroups of G; the first limit is taken with respect to the multiplication by p.

Let

χ : G ⟶ Aut(I) ≅ ℤ_{p}^{×}

be the character given by the action of G on I. We denote the canonical quotient G/ ker(χ) by Γ, i.e.

χ_{0} : Γ ↪ Aut(I) .

In the following we assume that

G acts non-trivially on I

(thus Γ ≅ ℤ_{p}), and we define the (finite) invariant q of G by

q = #(I^{G}).

Then we have a Δ-invariant isomorphism

H^{2}(G, ℤ/qℤ) ≅ Hom(I^{G}, ℤ/qℤ) (≅ ℤ/qℤ as an abelian group)

and a Δ-invariant non-degenerate pairing

H^{1}(G, ℤ/qℤ) × H^{1}(G, ℤ/qℤ) ⟶^{∪} H^{2}(G, ℤ/qℤ) .

From the exact sequence 0 → ℤ/qℤ →^{q} ℤ/q^{2}ℤ → ℤ/qℤ → 0, we get the Bockstein homomorphism

B : H^{1}(G, ℤ/qℤ) ⟶ H^{2}(G, ℤ/qℤ)

which is surjective and Δ-invariant.

Let P be a pro-p group. In this section we denote by P^{i}, i ≥ 1, the descending q-central series, i.e.

P^{1} = P and P^{i+1} = (P^{i})^{q}[P^{i}, P] for i ≥ 1 .

Let

1 ⟶ F ⟶ F ⟶ Δ ⟶ 1

be an exact sequence of profinite groups where F is a finitely generated pro-p group. Obviously, G^{i} and F^{i} are normal open subgroups of G and F respectively.

Proposition 2.1 With the notation as above let q > 2 and m ≥ 2. Assume that there exists a surjection

\xymatrix{φ_{m+1} : G \ar@{->>}[r] & F/F^{m+1} .}

Then there exists a surjection

\xymatrix{φ : G \ar@{->>}[r] & F}

inducing the surjection \xymatrix{φ_{m} : G \ar@{->>}[r]^{φ_{m+1}} & F/F^{m+1} \ar@{->>}[r]^{can} & F/F^{m} .}

Proof: Assume that we have already found a surjection

\xymatrix{φ_{i+1} : G \ar@{->>}[r] & F/F^{i+1}}

for i ≥ m which induces φ_{m}, and let \xymatrix{φ_{i} : G \ar@{->>}[r]^{φ_{i+1}} & F/F^{i+1} \ar@{->>}[r]^{can} & F/F^{i} .}

Let γ, x_{0}, …, x_{n} be a minimal system of generators of G such that x_{k} ∈ ker(χ) for k ≥ 0 and χ(γ) = 1 − q.

Claim: The group F^{i+1}/F^{i+2} is generated by elements of the form

w^{q}[w, γ̄] mod F^{i+2} , [w, x̄_{k}] mod F^{i+2}, k ≥ 0, w ∈ F^{i} ,

where γ̄, x̄_{k} ∈ F are lifts of the images of γ, x_{k} in F/F^{2} under the surjection G ↠ F/F^{2}.

This shown in [3] prop. 5(i) (observe, that we have a surjection G/G^{i+1} ↠ F/F^{i+1}, and so the group F/F^{i+1} is generated by the elements γ̄, x̄_{k} mod F^{i+1}).

Consider the diagram with exact line

(∗)  [figure: G; φ_{i}; 1; F^{i}/F^{i+2}; F/F^{i+2}; F/F^{i}; 1; .]

Since i ≥ m ≥ 2, we have

[F^{i}, F^{i}] ⊆ F^{2i} ⊆ F^{i+2} ,

and so the group F^{i}/F^{i+2} is abelian; we consider F^{i}/F^{i+2} as a G-module via φ_{i}. The canonical exact sequence

0 ⟶ F^{i+1}/F^{i+2} ⟶ F^{i}/F^{i+2} ⟶ F^{i}/F^{i+1} ⟶ 0

induces a Δ-invariant exact sequence

0 ⟶ Hom_{G}(F^{i}/F^{i+1}, I) ⟶ Hom_{G}(F^{i}/F^{i+2}, I) ⟶ Hom_{G}(F^{i+1}/F^{i+2}, I) .

Let f ∈ Hom_{G}(F^{i}/F^{i+2}, I). Then

f([w, x̄_{k}] mod F^{i+2}) = f(w mod F^{i+2})^{x_{k}−1} = 0 for k ≥ 0,
f(w^{q}[w, γ̄] mod F^{i+2}) = f(w mod F^{i+2})q + f(w mod F^{i+2})^{γ−1}
= f(w mod F^{i+2})(q − q) = 0 .

Using the claim, we see that f vanishes on F^{i+1}/F^{i+2}, and so

Hom_{G}(F^{i}/F^{i+1}, I) ⟶^{∼} Hom_{G}(F^{i}/F^{i+2}, I) .

By duality, cf. [5] (3.7.6), (3.7.1), (3.4.6), we get

H^{2}(G, F^{i}/F^{i+2}) ⟶^{∼} H^{2}(G, F^{i}/F^{i+1}) ,

and so

H^{2}(G, F^{i}/F^{i+2})^{Δ} ⟶^{∼} H^{2}(G, F^{i}/F^{i+1})^{Δ} .

Since the order of Δ is prime to p, the Hochschild-Serre spectral sequence

H^{i}(Δ, H^{j}(G, −)) ⇒ H^{i+j}(G, −)

degenerates, i.e. H^{j}(G, −)^{Δ} ≅ H^{j}(G, −). Therefore we obtain the isomorphism

H^{2}(G, F^{i}/F^{i+2}) ⟶^{∼} H^{2}(G, F^{i}/F^{i+1}) .

Now we prove that the embedding problem (∗) is solvable. For this we have to show that the 2-class

[β_{i}] ∈ H^{2}(F/F^{i}, F^{i}/F^{i+2})

is mapped to zero under the inflation map inf = φ_{i}^{∗},

H^{2}(F/F^{i}, F^{i}/F^{i+2}) ⟶^{inf} H^{2}(G, F^{i}/F^{i+2}) ,

where β_{i} is the 2-cocycle corresponding to the group extension in (∗), see [5] (9.4.2). From the commutative exact diagram

[figure: G; φ_{i}; φ_{i+1}; 1; F^{i}/F^{i+2}; can; F/F^{i+2}; F/F^{i}; 1; β_{i}; 1; F^{i}/F^{i+1}; F/F^{i+1}; F/F^{i}; 1; α_{i}]

we get a commutative diagram

[figure: H^{2}(F/F^{i}, F^{i}/F^{i+2}); can_{∗}; φ_{i}^{∗}; H^{2}(G, F^{i}/F^{i+2}); can_{∗}; ∼; H^{2}(F/F^{i}, F^{i}/F^{i+1}); φ_{i}^{∗}; H^{2}(G, F^{i}/F^{i+1}); .]

Since there exists the solution φ_{i+1} for the embedding problem α_{i}, we have φ_{i}^{∗}([α_{i}]) = 0, and so

can_{∗} ∘ φ_{i}^{∗}([β_{i}]) = φ_{i}^{∗} ∘ can_{∗}([β_{i}]) = φ_{i}^{∗}([α_{i}]) = 0 .

From the injectivity of the map can_{∗} on the right-hand side of the diagram above it follows that φ_{i}^{∗}([β_{i}]) = 0, and so there exists a solution

φ_{i+2} : G ⟶ F/F^{i+2}

of the embedding problem corresponding to β_{i}. This homomorphism is necessarily surjective and induces φ_{m}, because φ_{i} has these properties, cf. [5] (3.9.1).

Using a compactness argument, we get in the limit a surjection φ : G ↠ F inducing φ_{m}. This finishes the proof of the proposition. □

In the following let p be an odd prime number and let Δ = ⟨σ⟩ ≅ ℤ/2ℤ be cyclic of order 2. We denote, as usual, the (±)-eigenspaces of a ℤ_{p}[Δ]-module M by M^{±}.

Proposition 2.2 Let p be an odd prime number and let G be a p-Demuškin group of rank n + 2, n ≥ 0, with dualizing module I and invariant q = #(I^{G}) < ∞. Assume that Δ ≅ ℤ/2ℤ acts on G. Then the following holds:

(i) If H^{2}(G, ℤ/pℤ) = H^{2}(G, ℤ/pℤ)^{−}, then G_{Δ} is a free pro-p group of rank n/2 + 1.

(ii) If H^{2}(G, ℤ/pℤ) = H^{2}(G, ℤ/pℤ)^{+}, then G_{Δ} is a p-Demuškin group of rank m + 2, 0 ≤ m ≤ n, with invariant q and dualizing module I.

Proof: We start with the following remark. Since Aut(I) ≅ ℤ_{p}^{×} is abelian, the surjection G ↠ Γ factors through G_{Δ}. With the notation of the proof of proposition (1.4), it follows that N ∩ G has infinite index in G and therefore cd_{p}(N) = cd_{p}(N ∩ G) ≤ 1, cf. [5] III.7 ex.3. Using the Hochschild-Serre spectral sequence and the fact that Hom(N, ℤ/pℤ) = 0, we see that inf_{2} is an isomorphism, and so the commutative diagram in the proof of (1.4) shows the surjectivity of the map H^{2}(G_{Δ}) ↪ H^{2}(G)^{Δ}, hence

H^{2}(G_{Δ}) ≅ H^{2}(G)^{Δ}.

(i) By proposition (1.4) and H^{2}(G_{Δ}) = 0, G_{Δ} is a free pro-p group. Since the non-degenerate pairing

H^{1}(G) × H^{1}(G) ⟶^{∪} H^{2}(G) ≅ ℤ/pℤ

is Δ-invariant, it follows from H^{2}(G) = H^{2}(G)^{−} that

dim_{𝔽_{p}} H^{1}(G)^{±} = n/2 + 1 .

Therefore

dim_{𝔽_{p}} H^{1}(G_{Δ}) = dim_{𝔽_{p}} H^{1}(G)^{Δ} = n/2 + 1 .

(ii) If H^{2}(G) = H^{2}(G)^{+}, then H^{2}(G_{Δ}) ≅ H^{2}(G), and we obtain a non-degenerate pairing

H^{1}(G_{Δ}) × H^{1}(G_{Δ}) ⟶^{∪} H^{2}(G_{Δ}) ≅ ℤ/pℤ

showing that G_{Δ} is a p-Demuškin group. Finally, since G_{Δ} is non-trivial and its rank has to be even, it follows that dim_{𝔽_{p}} H^{1}(G_{Δ}) ≥ 2, and since ker(G ↠ G_{Δ}) acts trivially on I, we have #(I^{G_{Δ}}) = #(I^{G}) = q and I is also the dualizing module of G_{Δ}. □

3 Free Quotients of Demuškin Groups

As before, let G be a p-Demuškin group of rank n + 2 with dualizing module I and assume that 2 < q < ∞. We are interested in quotients of G which are free pro-p groups. First we calculate the possible ranks of such quotients.

Proposition 3.1 Let G be a Demuškin group of rank n + 2 with finite invariant q > 2 and let F be a free quotient of G. Then

(i) H^{1}(F, ℤ/qℤ) lies in the kernel of the Bockstein homomorphism and

(ii) H^{1}(F, ℤ/qℤ) is a totally isotropic free ℤ/qℤ-submodule of H^{1}(G, ℤ/qℤ) with

respect to the pairing given by the cup-product. In particular,

rank F ≤ \frac{n}{2} + 1 .

Proof: Since F is free, H^{1}(F, ℤ/qℤ) is a free ℤ/qℤ-module. The commutative diagram

\xymatrix{H^{1}(G, ℤ/qℤ)×H^{1}(G, ℤ/qℤ) \ar[r]^{∪} & H^{2}(G, ℤ/qℤ) \\ H^{1}(F, ℤ/qℤ)×H^{1}(F, ℤ/qℤ) \ar[r]^{∪} \ar@{^{(}->}[u]^{(inf ,inf )} & H^{2}(F, ℤ/qℤ) \ar[u]_{inf} & = 0}

shows that H^{1}(F, ℤ/qℤ) is a totally isotropic ℤ/qℤ-submodule of H^{1}(G, ℤ/qℤ), and so dim_{𝔽_{p}} H^{1}(F, ℤ/pℤ) = rank_{ℤ/qℤ}H^{1}(F, ℤ/qℤ) ≤ n/2 + 1. From the commutative diagram

\xymatrix{H^{1}(G, ℤ/qℤ) \ar[r]^{B} & H^{2}(G, ℤ/qℤ) \\ H^{1}(F, ℤ/qℤ) \ar[r]^{B} \ar@{^{(}->}[u]^{inf} & H^{2}(F, ℤ/qℤ) \ar[u]_{inf} & = 0}

follows that H^{1}(F, ℤ/qℤ) ⊆ ker(B). □

Recall that Γ is the canonical quotient G/ ker(χ) of G, where χ : G ⟶ Aut(I) is the character given by the action of G on I, i.e. Γ ↪ Aut(I).

Lemma 3.2 The submodules H^{1}(Γ, ℤ/qℤ) and ker B of H^{1}(G, ℤ/qℤ) are orthogonal to each other, more precisely

H^{1}(Γ, ℤ/qℤ) = (ker B)^{⊥} .

Proof: Consider the commutative diagram of non-degenerate pairings

[figure: H^{1}(G,_{q}I); ×; H^{1}(G,; B; ℤ/qℤ); ∪; H^{2}(G,_{q}I); ⟶^{∼} ℤ/qℤ; H^{0}(G,_{q}I); δ; ×; H^{2}(G,; ℤ/qℤ); ∪; H^{2}(G,_{q}I); ⟶_{∼} ℤ/qℤ]

which is induced by the exact sequences

0 ⟶ ℤ/qℤ ⟶^{q} ℤ/q^{2}ℤ ⟶ ℤ/qℤ ⟶ 0 and 0 ⟶_{q}I ⟶_{q^{2}}I ⟶^{q}_{q}I ⟶ 0 .

The image of H^{0}(G,_{q}I) = H^{0}(Γ,_{q}I) under δ is contained in H^{1}(Γ,_{q}I), and from (_{q^{2}}I)^{Γ} = _{q}I it follows that im δ = H^{1}(Γ,_{q}I). Since the pairings above are non-degenerated, H^{1}(Γ, ℤ/qℤ) = H^{1}(Γ,_{q}I) is orthogonal to ker B. Since rank_{ℤ/qℤ} ker B = rank_{ℤ/qℤ}H^{1}(G, Z/qℤ) − 1, and so rank_{ℤ/qℤ}(ker B)^{⊥} = 1, we prove the lemma. □

Proposition 3.3 Let G be a p-Demuškin group of rank n+2 with finite invariant q > 2 and let F be a free factor of G of rank n/2+1. Then the canonical surjection G ↠ Γ factors through F, i.e. there is a commutative diagram

[figure: G; Γ; F; .]

Proof: Suppose the contrary. Then there exists an open subgroup G′ of G which has a surjection

(G′)^{ab} ↠ (F′)^{ab} × Γ′ ,

where F′ is the image of G′ in F under the projection G ↠ F and Γ′ is the image of G′ under the projection G ↠ Γ. Let q′ = #(I^{G′}) = #(I^{Γ′}) and let

B′ : H^{1}(G′, Z/q′ℤ) ⟶ H^{2}(G′, Z/q′ℤ)

be the corresponding Bockstein map. Since F′ is free, it follows that H^{1}(F′, ℤ/q′ℤ) is a totally isotropic submodule of H^{1}(G′, ℤ/q′ℤ) and contained in ker B′ by proposition (3.1). From lemma (3.2) we know that H^{1}(Γ′, ℤ/q′ℤ) is orthogonal to ker B′, and so also to H^{1}(F′, ℤ/q′ℤ). Thus H^{1}(F′, ℤ/q′ℤ) ⊕ H^{1}(Γ′, ℤ/q′ℤ) is totally isotropic. But H^{1}(F′, ℤ/q′ℤ) is a maximal totally isotropic ℤ/q′ℤsubmodule of H^{1}(G′, ℤ/q′ℤ) of rank d · n/2 + 1, where d = (G : G′). This contradiction proves the proposition. □

For the existence of free quotients of Demuškin groups we have the following Theorem 3.4 Let G be a p-Demuškin group of rank n + 2 with finite invariant q > 2 and let Δ be a finite abelian group of exponent p − 1 acting on G. Let V be a ℤ/qℤ-submodule of H^{1}(G, ℤ/qℤ) such that

(i) V is ℤ/qℤ-free and Δ-invariant, (ii) V is totally isotropic with respect to the pairing given by the cup-product, (iii) V lies in the kernel of the Bockstein map B : H^{1}(G, ℤ/qℤ) → H^{2}(G, ℤ/qℤ).

Then there exists a Δ-invariant surjection

\xymatrix{G \ar@{->>}[r] & F}

onto a free quotient F of G such that H^{1}(F, ℤ/qℤ) = V.

Proof: Let

1 ⟶ R ⟶ F_{n+2} ⟶ G ⟶ 1

be a minimal presentation of G, where F_{n+2} is a free pro-p group of rank n + 2. Using lemma (1.3), we extend the action of Δ to F_{n+2}. Let γ, x_{0}, …, x_{n} be a basis of F_{n+2} such that

(i) each element of the basis of F_{n+2} generates a Δ-invariant subgroup isomorphic to ℤ_{p} on which Δ acts by some character ψ : Δ → μ_{p−1},

(ii) R, as a normal subgroup of F_{n+2}, is generated by the element

w = (x_{0})^{q}[x_{0}, γ][x_{1}, x_{2}][x_{3}, x_{4}] · · · · · [x_{n−1}, x_{n}] · f, where f ∈ (F_{n+2})^{3},

(iii) V^{∨} = Hom(V, ℤ/qℤ) has a basis {v_{i} mod (F_{n+2})^{2}, 1 ≤ i ≤ r = rank_{ℤ/qℤ}V }

such that

{v_{1}, …, v_{r}} is a subset of {γ, x_{1}, …, x_{n}}

and, if v_{i} = x_{j(i)}, then x_{j(i)+1} ∉ {v_{1}, …, v_{r}} or x_{j(i)−1} ∉ {v_{1}, …, v_{r}} according to whether j(i) is odd or even.

Such a basis exists: by corollary (1.2), we find a basis of F_{n+2} with the property (i). Let ψ_{0} : Δ ⟶ μ_{p−1} be the character such that H^{2}(G, ℤ/qℤ) = H^{2}(G, ℤ/qℤ)^{ψ_{0}}. The Δ-invariance of the cup-product gives us the perfect pairing

H^{1}(G, ℤ/qℤ)^{ψ} × H^{1}(G, ℤ/qℤ)^{ψ^{−1}ψ_{0}} ⟶^{∪} H^{2}(G, ℤ/qℤ)

for every character ψ ∈ Δ^{∨} and the Bockstein homomorphism restricts to a surjection

H^{1}(G, ℤ/qℤ)^{ψ_{0}} ↠ H^{2}(G, ℤ/qℤ).

Applying the usual procedure in order to get a basis with property (ii), see [5] (3.9.16), on the eigenspaces H^{1}(G, ℤ/qℤ)^{ψ}, ψ ∈ Δ^{∨}, we find a basis satisfying (i) and (ii). Using the assumptions on V, we can also satisfy (iii).

Let N be the normal subgroup of F_{n+2} generated by the set

{γ, x_{k}, 0 ≤ k ≤ n} ∖ {v_{1}, …, v_{r}},

then F := F_{n+2}/N is a free pro-p group of rank r, N is Δ-invariant and we have

R ⊆ N(F_{n+2})^{3}

by the properties (ii) and (iii) of the basis γ, x_{0}, …, x_{n}. Thus the Δ-invariant surjection

\xymatrix{F_{n+2} \ar@{->>}[r] & F/F^{3} = F_{n+2}/N(F_{n+2})^{3}}

factors through a Δ-invariant surjection \xymatrix{G \ar@{->>}[r] & F/F^{3} .} Applying proposition (2.1), we get a Δ-invariant surjection from G onto a free pro-p group F which induces a surjection G ↠ F/F^{2} ≅ F_{n+2}/N(F_{n+2})^{2}.

By construction, we have F/F^{2} ≅ V^{∨}, and so H^{1}(F, ℤ/qℤ) = V. This finishes the proof of the theorem. □

Now we consider free quotients of a Demuškin group G which are invariant under a given Δ-action of G, where Δ is a group of order 2.

Corollary 3.5 Let p be an odd prime number and let G be a p-Demuškin group of rank n + 2, n ≥ 0, with finite invariant q. Let Δ ≅ ℤ/2ℤ acting on G such that H^{2}(G, ℤ/qℤ) = H^{2}(G, ℤ/qℤ)^{−}. Let

u^{+}, u^{−} ≥ 0 be integers such that u^{+} + u^{−} = n/2 .

Then there exists a Δ-invariant surjection

\xymatrix{φ : G \ar@{->>}[r] & F}

such that

(i) F is a free pro-p group of rank n/2 + 1, (ii) rank_{ℤ_{p}}(F^{ab})^{+} = u^{+} + 1 and rank_{ℤ_{p}}(F^{ab})^{−} = u^{−}.

Proof: Since H^{2}(G, ℤ/qℤ) = H^{2}(G, ℤ/qℤ)^{−}, the submodules H^{1}(G, ℤ/qℤ)^{±} are maximal totally isotropic with respect to the cup-product pairing, and so

rank_{ℤ/qℤ}H^{1}(G, ℤ/qℤ)^{±} = n/2 + 1.

Let

V = V^{+} ⊕ V^{−},

where V^{+} is a free ℤ/qℤ-submodule of H^{1}(G, ℤ/qℤ)^{+} ⊆ ker B of rank 1 + u^{+} containing H^{1}(Γ, ℤ/qℤ), and V^{−} is defined as follows. By lemma (3.2)

H^{1}(Γ, ℤ/qℤ) ⊆ (ker B^{−})^{⊥},

and since

rank_{ℤ/qℤ}H^{1}(G, ℤ/qℤ)^{+} − rank_{ℤ/qℤ}V^{+} = n/2 + 1 − (1 + u^{+}) = u^{−} ,

there exists a free ℤ/qℤ-submodule V^{−} of (ker B)^{−} of rank u^{−} which is orthogonal to V^{+}. It follows that V is maximal totally isotropic and contained in ker B.

By theorem (3.4), we obtain a free Δ-invariant quotient F of G of rank n/2+1 such that

H^{1}(F, ℤ/qℤ) = V ≅ (ℤ/qℤ[Δ]^{+})^{u^{+}+1} ⊕ (ℤ/qℤ[Δ]^{−})^{u^{−}}.

Since F^{ab} is a free ℤ_{p}-module, we obtain assertion (ii). □

Remark: Explicitly, we get a submodule V with the properties as above in the following way: let

1 ⟶ R ⟶ F_{n+2} ⟶ G ⟶ 1

be a minimal presentation of G, where F_{n+2} is a free pro-p group of rank n + 2 with the extended action of Δ. Let γ, x_{0}, …, x_{n} be a basis of F_{n+2} such that R is generated by the element

w = (x_{0})^{q}[x_{0}, γ][x_{1}, x_{2}][x_{3}, x_{4}] · · · · · [x_{n−1}, x_{n}] · f ,

f ∈ (F_{n+2})^{3}, and

\begin{aligned}γ^{σ} & = & γ · a , & x_{i}^{σ} & = & x_{i} · a_{i} & for i = 2, 4, …, n , \\ x_{0}^{σ} & = & x_{0}^{−1} · b, & x_{i}^{σ} & = & x_{i}^{−1} · b_{i} & for i = 1, 3, 5, …, n − 1 . \end{aligned}

with a, b, a_{i}, b_{i} ∈ (F_{n+2})^{2}. Such a basis exists by the Δ-invariance of the cupproduct and the Bockstein homomorphism, cf. [8] lemma 3. If we put

\begin{aligned}γ′ & := & γ · a^{\frac{1}{2}} , & x′_{i} & := & x_{i} · ai^{\frac{1}{2}} & for i = 2, 4, …, n , \\ x′_{0} & := & b^{− \frac{1}{2}} · x_{0} , & x′_{i} & := & b_{i}^{− \frac{1}{2}} · x_{i} & for i = 1, 3, 5, …, n − 1 , \end{aligned}

then

(γ′)^{σ} = γ′ , (x′_{i})^{σ} = x′_{i} for i ≥ 2 even,
(x′_{0})^{σ} = (x′_{0})^{−1}, (x′_{i})^{σ} = (x′_{i})^{−1} for i ≥ 1 odd,

and

w = (x′_{0})^{q}[x′_{0}, γ′][x′_{1}, x′_{2}][x′_{3}, x′_{4}] · · · · · [x′_{n−1}, x′_{n}] · f′

where f′ ∈ (F_{n+2})^{3}. Let u = 2u^{+} − 1. If we denote x mod F^{2} by x̄, then the dual of

\begin{aligned}V^{∨} & := & ℤ/qℤ · γ̄ ⊕ ⨁_{i=1,3,...,u} ℤ/qℤ · x̄_{i+1} ⊕ ⨁_{i=u+3,...,n} ℤ/qℤ · x̄_{i−1} \\ & ≅ & (ℤ/qℤ[Δ]^{+})^{u^{+}+1} ⊕ (ℤ/qℤ[Δ]^{−})^{u^{−}} \end{aligned}

gives an example for a submodule with the properties (i)-(iii) in the proof of corollary (3.5). The free quotient of G is obtained in the following way: if

N = (x′_{0}, x′_{1}, x′_{3}, …, x′_{u}, x′_{u+3}, …, x′_{n}) ⊴ F_{n+2} , _{u^{+}-times} _{u^{−}-times}

then F = F_{n+2}/N is a free pro-p group of rank n/2 + 1, N is Δ-invariant, R ⊆ N(F_{n+2})^{3} and V^{∨} = F/F^{2}. Using proposition (2.1) we get the desired quotient of G.

With the notation and assumptions of corollary (3.5), we make for a Δinvariant free quotient F of G of rank n/2 + 1 the following

Definition 3.6 We call the tuple (u^{+}, u^{−}) the signature of F, if

F/F^{2} ≅ (ℤ/qℤ[Δ]^{+})^{u^{+}+1} ⊕ (ℤ/qℤ[Δ]^{−})^{u^{−}} .

One can show that in general the signature of a maximal free quotient F of G does not determine F. But if the signature is equal to (n/2, 0), then we have the following proposition.

Proposition 3.7 Let p be an odd prime number and let Δ be of order 2. Let G be a p-Demuškin group of rank n + 2 with finite invariant q on which Δ acts such that H^{2}(G, ℤ/pℤ)^{Δ} = 0. Let F be a free Δ-invariant quotient of G of rank n/2 + 1, i.e. the canonical surjection

\xymatrix{G \ar@{->>}[r] & F}

is Δ-invariant. If the induced action of Δ on F/F^{2} is trivial, i.e. F has signature (n/2, 0), then F is equal to the maximal quotient G_{Δ} of G with trivial Δ-action. In particular, a free quotient of G with the properties above is unique.

Proof: As in the remark after the proof of corollary (3.5), we find generators of F on which Δ acts trivially, and so F has a trivial Δ-action. Thus we have a surjection \xymatrix{φ : G_{Δ} \ar@{->>}[r] & F .} Since G_{Δ} is free of rank n/2+1 = dim_{𝔽_{p}} H^{1}(F, ℤ/pℤ) by proposition (2.2)(i), it follows that φ is an isomorphism. Thus F is the maximal quotient of G with trivial Δ-action. □

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

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