Triple Massey Products over Global Fields

Ján Mináč and Nguyễn Duy Tân^{1}

Received: March 18, 2014 Revised: October 8, 2015

Communicated by Ulf Rehmann

Abstract. Let K be a global field which contains a primitive p-th root of unity, where p is a prime number. M. J. Hopkins and K. G. Wickelgren showed that for p = 2, any triple Massey product over K with respect to _{p}, contains 0 whenever it is defined. We show that this is true for all primes p.

2010 Mathematics Subject Classification: 12G05, 55S30.

Keywords and Phrases: Massey products, Galois cohomology, local fields, global fields.

1. Introduction

Massey products were introduced by W. S. Massey in [M]. (We review the definition in Section 2.) Massey products were first used in topology where usual cohomology cup products would not detect some linking properties of knots but Massey products would. (See for example [Mo, page 98] or [GM, pages 154-158].) Further interest in Massey products arises as an obstruction to the "formality" of manifolds over real numbers. In the case of compact Kähler manifolds, formality formalizes the property that their homotopy type is a formal consequence of their real cohomology ring. (See [DGMS].) We treat Massey products also as obstructions to solving certain Galois embedding problems. Throughout this paper, we let p be a prime number. Let K be a field which we assume contains a fixed primitive p-th root of unity ζ_{p}. Let G_{K} be the absolute Galois group of K. Let C^{•} = C^{•}(G_{K}, _{p}) denote the differential graded algebra of _{p}-inhomogeneous cochains in continuous group cohomology of G_{K} (see e.g., [NSW, Chapter I, §2]). For any a ∈ K^{×} = K \ {0}, let χ_{a} denote the corresponding character via the Kummer map K^{×} → H^{1}(G_{K}, _{p}), i.e., χ_{a} is defined by σ(\sqrt[p]{a}) = ζ_{p}^{χ_{a}(σ)} \sqrt[p]{a}, for all σ ∈ G_{K}. In the work of M. J. Hopkins and K. G. Wickelgren [HW], the following fundamental result was proved. (By a global field we mean a finite extension of , or a function field in one variable over a finite field.)

1The first named author is partially supported by the Natural Sciences and Engineering Research Council of Canada (NSERC) grant R0370A01. The second named author is partially supported by the National Foundation for Science and Technology Development of Vietnam (NAFOSTED) grant 101.04-2014.34

Theorem 1.1 ([HW, Theorem 1.2]). Let the notation be as above. Assume that p = 2 and K is a global field of characteristic ≠ 2. Let a, b, c ∈ K^{×} . The triple Massey product ⟨χ_{a}, χ_{b}, χ_{c}⟩ contains 0 whenever it is defined.

In [MT1] we extend the result of Hopkins-Wickelgren to an arbitrary field K of characteristic ≠ 2, still assuming that p = 2.

Theorem 1.2 ([MT1, Theorem 1.2]). Let the notation be as above. Assume that p = 2 and K is an arbitrary field of characteristic ≠ 2. Let a, b, c ∈ K^{×} . The triple Massey product ⟨χ_{a}, χ_{b}, χ_{c}⟩ contains 0 whenever it is defined.

In this paper we extend the result of Hopkins-Wickelgren in Theorem 1.1 in another direction. We still consider a global field K but we let the prime p be arbitrary.

Theorem 1.3. Let the notation be as above. Assume that K is a global field containing a primitive p-th root of unity and a, b, c ∈ K^{×} . Then the triple Massey product ⟨χ_{a}, χ_{b}, χ_{c}⟩ contains 0 whenever it is defined.

Let us denote by _{4}( _{p}) the group of all upper-triangular unipotent 4-by-4-matrices with entries in _{p}. For a finite group G, by a G-Galois extension L/K, we mean a Galois extension with Galois group isomorphic to G. It is a classical problem to describe extensions M/K which can be embedded into a G-Galois extension L/K with a prescribed Galois group G. From Theorem 1.3 and its local version we can deduce the following contribution to this problem when G = _{4}( _{p}).

Corollary 1.4. Let K be a local or global field containing a primitive p-th root of unity. Let a, b, c ∈ K^{×} and assume that the classes [a], [b], [c] in the _{p}-vector space K^{×} /(K^{×} )^{p} are linearly independent. Assume further that χ_{a} ∪ χ_{b} = χ_{b} ∪ χ_{c} = 0 in H^{2}(G_{K}, _{p}). Then the Galois extension K(\sqrt[p]{a}, \sqrt[p]{b}, \sqrt[p]{c})/K can be embedded in a 4( _{p})-Galois extension L/K.

In fact for each _{4}( _{p})-extension L/K, there exist a, b, c ∈ K^{×} ∩ L^{p} such that the classes [a], [b], [c] in the _{p}-vector space K^{×} /(K^{×} )^{p} are linearly independent, and that χ_{a} ∪ χ_{b} = χ_{b} ∪ χ_{c} = 0 in H^{2}(G_{K}, _{p}). Thus we see that this hypothesis is both necessary and sufficient for embedding abelian extensions of degree p^{3} and exponent p into a _{4}( _{p})-extension. (See Section 4 for more detail.) In the case when p = 2, Corollary 1.4 was also proved in [GLMS, Section 4] for all fields K of characteristic not 2. (See also [MT1, Section 6].) Let us now recall briefly how Theorem 1.1 is established in [HW]. Let p = 2 and K be a field of characteristic not 2. In [HW], the authors construct for each a, b, c ∈ K^{×} , a K-variety X_{a,b,c} which has a K-rational point if and only if the triple Massey product ⟨χ_{a}, χ_{b}, χ_{c}⟩ is defined and contains 0 (see [HW, Theorem 1.1]). The authors then establish a local version of Theorem 1.1 by using the non-degeneracy property of the cup products and the indeterminacy of Massey products. Now assume that K is a global field and consider a, b, c ∈ K^{×} such that ⟨χ_{a}, χ_{b}, χ_{c}⟩ is defined. By applying a result of D. B. Leep and A. R. Wadsworth in [LW], the authors show that the splitting variety X_{a.b,c} satisfies the Hasse local-global principle (see [HW, Theorem 3.4]), and then the result follows from the local case. In our paper we also use the local-global principle but our method is different from the method used in the paper [HW]. Let p be any prime, and let K be a field containing a primitive p-th root of unity. Let a, b, c ∈ K^{×} such that the triple Massey product ⟨χ_{a}, χ_{b}, χ_{c}⟩ is defined. Now instead of constructing a splitting variety for ⟨χ_{a}, χ_{b}, χ_{c}⟩, we use the technique of Galois embedding problems to detect the vanishing property of triple Massey products. Namely, ⟨χ_{a}, χ_{b}, χ_{c}⟩ vanishes if certain kinds of embedding problems are solvable. This is true because of a result of W. G. Dwyer. We then use Hoechsmann’s lemma to translate the problem of solvability of embedding problems to the problem of showing the vanishing of some degree 2 cohomology classes. Then we establish a local-global principle for the vanishing of the cohomology classes (see Lemma 6.2). Theorem 1.3 then follows from its local version. This being said, our proof also provides another proof for Theorem 1.1 in the case p = 2.

Acknowledgments: We would like to thank Stefan Gille, Thong Nguyen Quang Do and Kirsten Wickelgren for their interest and correspondence. We are grateful to the an anonymous referee for his/her very careful reading of our paper and for providing us with insightful comments and valuable suggestions which we used to improve our exposition considerably. For example, Proposition 4.7 and Lemma 6.1 were formulated based on his/her report.

Addendum (October 2015): Since submitting of this paper there have been some new significant developments in this subject motivated and influenced by this paper and [MT1]. In [EM1] Efrat and Matzri proved a result which implies the main result Theorem 1.3 of this paper. In [Ma] Matzri extended our main result Theorem 1.3 to an arbitrary field K. Efrat and Matzri [EM2] and in parallel [MT3] gave a direct proofs of Matzri’s result, using only tools from Galois cohomology. In [MT4] the explicit constructions of _{4}( _{p})-Galois extensions over all fields which admit such extensions are provided. In [MT5] the authors also considered the vanishing property of higher Massey products over rigid fields.

2. Review of Massey products

In this section, we review some basic facts about Massey products, see [MT1] and references therein for more detail.

Let A be a unital commutative ring. Recall that a differential graded algebra (DGA) over A is a graded associative A-algebra

C^{•} = ⊕_{k≥0} C^{k} = C^{0} ⊕ C^{1} ⊕ C^{2} ⊕ ⋯

with product ∪ and equipped with a differential ∂: C^{•} → C^{•+1} such that (1) ∂ is a derivation, i.e.,

∂(a ∪ b) = ∂a ∪ b + (−1)^{k}a ∪ ∂b (a ∈ C^{k});

(2) ∂^{2} = 0. Then as usual the cohomology H^{•} of C^{•} is ker ∂/im∂. We shall assume that a_{1}, …, a_{n} are elements in H^{1}.

Definition 2.1. A collection M = (a_{ij}), 1 ≤ i < j ≤ n + 1, (i, j) ≠ (1, n +

1) of elements of C^{1} is called a defining system for the n-fold Massey product ⟨a_{1}, …, a_{n}⟩ if the following conditions are fulfilled:

(1) a_{i,i+1} represents a_{i}.

(2) ∂a_{ij} = ∑ _{l=i+1}^{j−1} a_{il} ∪ a_{lj} for i + 1 < j.

Then ∑ _{k=2}^{n} a_{1k} ∪ a_{k,n+1} is a 2-cocycle. (See for example [Fe, page 233].) Its cohomology class in H^{2} is called the value of the product relative to the defining system M, and is denoted by ⟨a_{1}, …, a_{n}⟩_{M} . The product ⟨a_{1}, …, a_{n}⟩ itself is the subset of H^{2} consisting of all elements which can be written in the form ⟨a_{1}, …, a_{n}⟩_{M} for some defining system M. The n-fold Massey product ⟨a_{1}, …, a_{n}⟩ is said to be defined if it has a defining system, i.e., the set ⟨a_{1}, …, a_{n}⟩ is non-empty. For n ≥ 2 we say that C^{•} has the vanishing n-fold Massey product property if every defined Massey product ⟨a_{1}, …, a_{n}⟩, where a_{1}, …, a_{n} ∈ C^{1}, necessarily contains 0. When n = 3 we will speak about triple Massey products and the vanishing triple Massey product property.

Now let G be a profinite group and let A be a finite commutative ring considered as a trivial discrete G-module. Let C^{•} = C^{•}(G, A) be the DGA of inhomogeneous continuous cochains of G with coefficients in A [NSW, Ch. I, §2]. We write H^{i}(G, A) for the corresponding cohomology groups.

Definition 2.2. We say that G has the vanishing n-fold Massey product property (with respect to A) if the DGA C^{•}(G, A) has the vanishing n-fold Massey product property.

3. Unipotent matrices

Let _{n+1} ( _{p}) be the group of all upper-triangular unipotent (n + 1) × (n + 1)- matrices with entries in _{p}. Let Z_{n+1} ( _{p}) be the subgroup of all such matrices with all off-diagonal entries being 0 except possibly at position (1, n + 1). We may identify the quotient _{n+1} ( _{p})/Z_{n+1} ( _{p}) with the group _{ n̄+1} ( _{p}) of all upper-triangular unipotent (n + 1) × (n + 1)-matrices with entries over _{p} with the (1, n + 1)-entry omitted.

For a representation ρ: G → _{n+1} ( _{p}) and 1 ≤ i < j ≤ n + 1, let ρ_{ij}: G → p be the composition of ρ with the projection from _{n+1} ( _{p}) to its (i, j)- coordinate. We use similar notation for representations ρ̄: G → _{ n̄+1} ( _{p}). Note that ρ_{i,i+1} (resp., ρ̄_{i,i+1} ) is a group homomorphism. Now we assume n = 3. We consider the following exact sequence of finite groups

1 −→ A −→ _{4}( _{p}) −−−−−−→^{(a_{12},a_{23},a_{34})} _{p}^{3} −→ 1,

here a_{ij}: _{4}( _{p}) → _{p} is the map sending a matrix to its (i, j)-coefficient. Explicitly,

A = \{ [ \begin{matrix}1 & 0 & a & b \\ 0 & 1 & 0 & c \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix}] : a, b, c ∈ _{p}\} .

We consider the action of _{4}( _{p}) on A by conjugation: g · a = gag^{−1} , ∀ g ∈ 4( _{p}), a ∈ A. Since A is abelian, this action induces an action of _{p}^{3} on A, i.e., we get a homomorphism ψ: _{p}^{3} → Aut(A). Let A′ = Hom(A, _{p}) be the dual _{p}^{3}-module of the _{p}^{3}-module A. Here the action of _{p}^{3} on A′is given by

(gϕ)(a) = ϕ(g^{−1} · a),

where ϕ ∈ Hom(A, _{p}), g ∈ _{p}^{3} and a ∈ A. (Here we write the group _{p}^{3} multiplicatively.) From this action, we get a homomorphism ψ′: _{p}^{3} → Aut(A′). The following lemma is a special case of a more general result on matrix representations of dual representations. For the convenience of the reader, we include a short proof.

Lemma 3.1. Assume that {e_{1}, e_{2}, e_{3}} is a basis for the _{p}-vector space A. Let g be any element _{p}^{3}. Suppose that ψ(g) is given by matrix X with respect to e_{1}, e_{2}, e_{3}. Then the matrix of ψ′(g) with respect to the dual basis is (X^{−1} )^{T}.

Proof. We write X^{−1} = (x_{ij}). Let {e′_{1}, e′_{2}, e′_{3}} be the dual basis of the basis {e_{1}, e_{2}, e_{3}}. Then

(ψ′(g)(e′_{i}))(e_{j}) = e′_{i}(ψ(g^{−1} )(e_{j})) = e′_{i}(∑_{k} x_{kj}e_{k}) = x_{ij} = (∑_{k} x_{ik}e′_{k})(e_{j}).

Hence ψ′(g)(e′_{i}) = ∑ _{k}x_{ik}e′_{k}, and the lemma follows. □

Lemma 3.2. There exists an _{p}-basis of A′ such that with respect to this basis the map ψ′: _{p}^{3} → Aut(A′) becomes a map _{p}^{3} → GL_{3}( _{p}) which sends (x, y, z) ∈ _{p}^{3}

to [ \begin{matrix}1 & 0 & −x \\ 0 & 1 & z \\ 0 & 0 & 1 \end{matrix} ] .

1472 Ján Mináč and Nguyễn Duy Tân Proof. We first describe the action of _{p}^{3} on A, i.e., we describe the map ψ: _{p}^{3} → Aut(A), as follows. Let e_{1} = I + E_{24}, e_{2} = I + E_{13}, e_{3} = I + E_{14}. We have ψ(x, y, z)(e_{1}) =

= [ \begin{matrix}1 & x & 0 & 0 \\ 0 & 1 & y & 0 \\ 0 & 0 & 1 & z \\ 0 & 0 & 0 & 1 \end{matrix}] [ \begin{matrix}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix}] [ \begin{matrix}1 & x & 0 & 0 \\ 0 & 1 & y & 0 \\ 0 & 0 & 1 & z \\ 0 & 0 & 0 & 1 \end{matrix}]^{−1}= [ \begin{matrix}1 & 0 & 0 & x \\ 0 & 1 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix}] = e_{1} + xe_{3};

ψ(x, y, z)(e_{2}) =

= [ \begin{matrix}1 & x & 0 & 0 \\ 0 & 1 & y & 0 \\ 0 & 0 & 1 & z \\ 0 & 0 & 0 & 1 \end{matrix}] [ \begin{matrix}1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix}] [ \begin{matrix}1 & x & 0 & 0 \\ 0 & 1 & y & 0 \\ 0 & 0 & 1 & z \\ 0 & 0 & 0 & 1 \end{matrix}]^{−1}= [ \begin{matrix}1 & 0 & 1 & −z \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix} ] = e_{2} − ze_{3};

ψ(x, y, z)(e_{3}) =

= [ \begin{matrix}1 & x & 0 & 0 \\ 0 & 1 & y & 0 \\ 0 & 0 & 1 & z \\ 0 & 0 & 0 & 1 \end{matrix}] [ \begin{matrix}1 & 0 & 0 & 1 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix}] [ \begin{matrix}1 & x & 0 & 0 \\ 0 & 1 & y & 0 \\ 0 & 0 & 1 & z \\ 0 & 0 & 0 & 1 \end{matrix}]^{−1}= [ \begin{matrix}1 & 0 & 0 & 1 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix}] = e_{3}.

Thus with respect to the _{p}-basis {e_{1}, e_{2}, e_{3}} of A, the element (x, y, z) ∈ _{p}^{3} is sent to the matrix [ \begin{matrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ x & −z & 1 \end{matrix}] ∈ GL_{3}( _{p}). Now we consider the _{p}^{3}-module A′. By Lemma 3.1, the structure map ψ′: _{p}^{3} → Aut(A′) describing the action of _{p}^{3} on A′with respect to the dual basis of (e_{1}, e_{2}, e_{3}), is given by:

(x, y, z) ↦ ( [ \begin{matrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ x & −z & 1 \end{matrix}]^{−1} ) ^{T} = [ \begin{matrix}1 & 0 & x \\ 0 & 1 & −z \\ 0 & 0 & 1 \end{matrix} ]^{−1} = [ \begin{matrix}1 & 0 & −x \\ 0 & 1 & z \\ 0 & 0 & 1 \end{matrix} ] .

□

4. Embedding problems

A weak embedding problem E for a profinite group G is a diagram

E := \xymatrix{ & G \ar[d]^{α} \\ U \ar[r]^{f} & Ū}

which consists of profinite groups U and Ū and homomorphisms α: G → Ū, f: U → Ū with f being surjective. (All homomorphisms of profinite groups considered in this paper are assumed to be continuous.) A weak solution of E is a homomorphism β: G → U such that fβ = α. We call E a finite weak embedding problem if the group U is finite. The kernel of E is defined to be M := ker(f). Let ϕ_{1}: G_{1} → G be a homomorphism of profinite groups. Then ϕ_{1} induces the following weak embedding problem

E_{1} := \xymatrix{ & G_{1} \ar[d]^{α∘ϕ_{1}} \\ U \ar[r]^{f} & Ū.}

If β is a weak solution of E then β ∘ ϕ_{1} is a weak solution of E_{1}. The following result is due to W. Dwyer. We will use this result to reformulate the vanishing Massey product property in terms of weak embedding problems.

Theorem 4.1 ([Dwy, Theorem 2.4]). Let α_{1}, …, α_{n} be elements of H^{1}(G, _{p}). There is a one-one correspondence M ↔ ρ̄_{M} between defining systems M for ⟨α_{1}, …, α_{n}⟩ and group homomorphisms ρ̄_{M} : G → _{ n̄+1} ( _{p}) with (ρ̄_{M} )_{i,i+1} = −α_{i}, for 1 ≤ i ≤ n. Moreover ⟨α_{1}, …, α_{n}⟩_{M} = 0 in H^{2}(G, _{p}) if and only if the dotted homomorphism exists in the following commutative diagram

[figure: G; ρ̄_{M}; 0; _{p}; _{n+1} ( _{p}); _{n̄+1} ( _{p}); 1.]

Explicitly, the one-one correspondence in Theorem 4.1is given by: For a defining system M = (a_{ij}) for ⟨α_{1}, …, α_{n}⟩, ρ̄_{M} : G → _{ n̄+1} ( _{p}) is defined by letting (ρ̄_{M} )_{ij} = −a_{ij} (see [Dwy, Proof of Theorem 2.4]).

Lemma 4.2. Let G be a profinite group, and n ≥ 3 an integer. Then the following statements are equivalent:

(1) G has the vanishing n-fold Massey product property with respect to _{p}.

(2) For every homomorphism ρ̄: G → _{ n̄+1} ( _{p}), the finite weak embedding problem

[figure: G; (ρ̄_{12},...,ρ̄_{n,n+1} ); 0; A; _{n+1} ( _{p}); _{p}^{n}; 1,]

has a weak solution, i.e., (ρ̄_{12}, ρ̄_{23}, …, ρ̄_{n,n+1} ) can be lifted to a homomorphism ρ: G → _{n+1} ( _{p}).

Proof. This follows from Theorem 4.1. □

Corollary 4.3. Let G be a profinite group. Let χ_{1}, χ_{2}, χ_{3} ∈ H^{1}(G, _{p}) be _{p}- linearly independent. Assume that G has the vanishing triple Massey product and that χ_{1} ∪ χ_{2} = χ_{2} ∪ χ_{3} = 0 ∈ H^{2}(G, _{p}). Then there is a continuous surjective homomorphism ρ: G → _{4}( _{p}) such that ρ_{12} = χ_{1}, ρ_{23} = χ_{2} and ρ_{34} = χ_{3}.

Proof. Since χ_{1} ∪ χ_{2} = χ_{2} ∪ χ_{3} = 0 ∈ H^{2}(G, _{p}), there exist a_{12}, a_{23} ∈ C^{1}(G, _{p}) such that ∂a_{12} = χ_{1} ∪ χ_{2} and ∂a_{23} = χ_{2} ∪ χ_{3}. This implies that the triple Massey product ⟨χ_{1}, χ_{2}, χ_{3}⟩ is defined. By Theorem 4.1, we have a homomorphism ρ̄: G → _{ ¯ 4} ( _{p}) such that ρ̄_{12} = χ_{1}, ρ̄_{23} = χ_{2} and ρ̄_{34} = χ_{3}. By Lemma 4.2, there exists a homomorphism ρ: G → _{4}( _{p}) such that

ρ_{12} = ρ̄_{12} = χ_{1}, ρ_{23} = ρ̄_{23} = χ_{2}, ρ_{34} = ρ̄_{34} = χ_{3}.

Note that the Frattini subgroup of _{4}( _{p}) is A. Hence by the Frattini argument ρ: G → _{4}( _{p}) is surjective. □

Remark 4.4. Let ρ: G → _{4}( _{p}) be a surjective homomorphism. Let χ_{1} = ρ_{12}, χ_{2} = ρ_{23} and χ_{3} = ρ_{34}. Since (ρ_{12}, ρ_{23}, ρ_{34}): G → _{p} × _{p} × _{p} is surjective, we see that χ_{1}, χ_{2} and χ_{3} are _{p}-linearly independent. Furthermore since ρ is group homomorphism, we see that χ_{1} ∪ χ_{2} = χ_{2} ∪ χ_{3} = 0 ∈ H^{2}(G, _{p}).

Lemma 4.5 (Hoechsmann). Let E be a finite weak embedding problem for G with finite abelian kernel M. Let ϵ ∈ H^{2}(Ū, M) be the cohomology class corresponding to the embedding problem E. Then E has a weak solution if and only if α^{∗}(ϵ) = 0 ∈ H^{2}(G, M).

Proof. See [Ho, Statement 1.1, page 82]. (See also [NSW, Chapter 3, §5, Proposition 3.5.9].) □

Corollary 4.6. Let E(G) = (α: G → Ū, f: U → Ū) be a finite weak embedding problem for G with abelian kernel M. Let ϕ_{i}: G_{i} → G, i ∈ I, be a family of homomorphisms of profinite groups. Assume that the natural homomorphism

H^{2}(G, M) → ∏_{i} H^{2}(G_{i}, M),

is injective. Then the weak embedding problem E(G) has a weak solution if and only if for every i ∈ I the induced weak embedding problem E(G_{i}) has a weak solution.

Proof. We consider the following sequence

[figure: H^{2}(Ū, M); α^{∗}; H^{2}(G, M); ∏ _{i∈I} H^{2}(G_{i}, M).]

The statement follows from Lemma 4.5. □

Proposition 4.7. Suppose that G_{i}, i ∈ I, are closed subgroups of a profinite group G, and that for every map α: G → _{p}^{3} the map

Res: H^{2}(G, A) −→ ∏_{i∈I} H^{2}(G_{i}, A)

is injective, where the action is via ψ ∘ α: G → Aut(A). If each G_{i} has the triple vanishing Massey product property, then G also has the triple vanishing Massey product property.

Proof. We shall prove the condition (2) in Lemma 4.2. Let ρ̄: G → _{ ¯ 4} ( _{p}) be any homomorphism. We consider the weak embedding problem

(E) [figure: G; (ρ̄_{12},ρ̄_{23},ρ̄_{34}); 0; A; _{4}( _{p}); ( _{p})^{3}; 1.]

By assumption for every i ∈ I the induced weak embedding problem (E_{i})

(E_{i}) [figure: G_{i}; (ρ̄_{12},ρ̄_{23},ρ̄_{34}); 0; A; _{4}( _{p}); ( _{p})^{3}; 1,]

has a weak solution. By Corollary 4.6, (E) has a weak solution also. □

5. The vanishing of a certain cohomology group

Let G be a profinite group, and let M be a discrete G-module. We define

H_{∗}^{1}(G, M) = ker(H^{1}(G, M) → ∏_{C} H^{1}(C, M)),

where the product is over all closed cyclic subgroups (in the profinite sense) of G. (The definition of H_{∗}^{1}(G, M) is due to Tate (see [Se, §2]). This definition also appeared in [DZ, §2], in which the authors used the notation H_{loc}^{1} instead of using H_{∗}^{1}.) The following lemma is a special case of [DZ, Lemma 3.3]. It is a simple lemma and therefore we also omit a proof.

Lemma 5.1. Let V be a vector space of finite dimension over a field k. Let φ_{1}, φ_{2} be elements in the dual k-vector space V^{∗} := Hom(V, k). If ker φ_{1} ⊆ ker φ_{2} then there exists λ ∈ k such that φ_{2} = λφ_{1}.

Lemma 5.2. Let

G = \{ [ \begin{matrix}1 & 0 & a \\ 0 & 1 & b \\ 0 & 0 & 1 \end{matrix}] : a, b ∈ _{p}\} ,

and let _{p}^{3} act on G by matrix multiplication. Then H_{∗}^{1}(G, ( _{p})^{3}) = 0.

Proof. Let (Z_{σ}) be a cocycle representing an element in H_{∗}^{1}(G, ( _{p})^{3}). Then for each σ ∈ G, there exists W_{σ} ∈ ( _{p})^{3} such that

Z_{σ} = (σ − 1)W_{σ}.

Writing Z_{σ} = [ \begin{matrix}x_{σ} \\ y_{σ} \\ z_{σ} \end{matrix} ], W_{σ} = [ \begin{matrix}u_{σ} \\ v_{σ} \\ t_{σ} \end{matrix} ] and σ = [ \begin{matrix}1 & 0 & a_{σ} \\ 0 & 1 & b_{σ} \\ 0 & 0 & 1 \end{matrix}] , we have

[ \begin{matrix}x_{σ} \\ y_{σ} \\ z_{σ} \end{matrix} ] = [ \begin{matrix}0 & 0 & a_{σ} \\ 0 & 0 & b_{σ} \\ 0 & 0 & 0 \end{matrix}] [ \begin{matrix}u_{σ} \\ v_{σ} \\ t_{σ} \end{matrix} ] = [ \begin{matrix}t_{σ}a_{σ} \\ t_{σ}b_{σ} \\ 0 \end{matrix}] .

Hence

(1)  x_{σ} = t_{σ}a_{σ}, y_{σ} = t_{σ}b_{σ}, z_{σ} = 0.

By the cocycle condition, σ ↦ x_{σ} and σ ↦ y_{σ} are homomorphisms. Also, σ ↦ a_{σ} and σ ↦ b_{σ} are homomorphisms. From (1), one has ker a_{σ} ⊆ ker x_{σ} and ker b_{σ} ⊆ ker y_{σ}. Hence by Lemma 5.1, there exist λ, μ ∈ _{p} such that

(2)  x_{σ} = λa_{σ}; y_{σ} = μb_{σ}.

We consider the matrix σ_{0} = [ \begin{matrix}1 & 0 & 1 \\ 0 & 1 & 1 \\ 0 & 0 & 1 \end{matrix}] , i.e., a_{σ_{0}} = b_{σ_{0}} = 1. Then (1) and (2)

imply that

x_{σ_{0}} = t_{σ_{0}} = λ, and y_{σ_{0}} = t_{σ_{0}} = μ.

Thus λ = μ. Hence for all σ ∈ G we have Z_{σ} = (σ − 1)W, with W = (0, 0, λ)^{t}. Therefore (Z_{σ}) is cohomologous to 0, as desired. □

6. The injectivity of a localization map

Let K be a global field containing a primitive p-th root of unity. For any G_{K}- module M with the structure map ρ: G_{K} → Aut(M), let K(M) be the smallest splitting field of M, explicitly K(M) is the fixed field of the separable closure K^{sep} under ker(ρ). For each prime v of K, let K_{v} denote the completion of K at v. We will fix an embedding ι_{v}: G_{K_{v}} ↪ G_{K} which is induced by choosing an embedding of K^{sep} in K_{v}^{sep}. Then for each i, ι_{v}’s induce a homomorphism

β^{1}(K, M): H^{i}(G_{K}, M) → ∏_{v} H^{i}(G_{K_{v}}, M).

This homomorphism does not depend on the choice of embeddings K^{sep} ↪ K_{v}^{sep}, and it is called the localization map.

Lemma 6.1. Let F be a finite Galois extension of K containing K(M). Then we can inject the group ker β^{1}(K, M) into the group H_{∗}^{1}(Gal(F/K), M).

(See [Se, Proposition 8] for a similar statement.)

Proof. By [Mi, Chapter I, Lemma 9.3] and/or [Ja, Lemma 1], we have the following diagram

\xymatrix{ & ker β^{1}(K, M) \ar@{^{(}->}[d] \\ H_{∗}^{1}(Gal(F/K), M) \ar[r]^{≃} & H_{∗}^{1}(G_{K}, M).}

The lemma then follows. □

Now let α: G_{K} → _{p}^{3} be any (continuous) homomorphism. We consider A as a G_{K}-module via

ψ ∘ α: G_{K} →^{α} _{p}^{3} →^{ψ} Aut(A).

Lemma 6.2. The localization map

H^{2}(G_{K}, A) → ∏_{v} H^{2}(G_{K_{v}}, A),

is injective.

Proof. First note that if we consider A′ = Hom(A, _{p}) as a G_{K}-module via

the composition map β = ψ′∘ α: G_{K} → _{p}^{3} →^{ψ′} Aut(A′), then A′ is the dual G_{K}-module of the G_{K}-module A. We shall choose an _{p}-basis of A′ as in Lemma 5.2. Clearly, after identifying A′with _{p}^{3}, and Aut(A′) with GL_{3}( _{p}), the action of G_{K} on A′via the image im(β) is the matrix multiplication. By Poitou-Tate duality ([NSW, Theorem 8.6.7]), it is enough to show that

(3)  ker(H^{1}(G_{K}, A′) → ∏_{v} H^{1}(G_{K_{v}}, A′)) = 0.

Let F = (K^{sep})^{ker β} be the smallest splitting field of A′. Then Gal(F/K) ≃ im(β) ⊆ imψ′ = G, where G is the group defined in Lemma 5.2. Here the equality imψ′= G follows from Lemma 3.2. If Gal(F/K) ≃ imβ = G, then by Lemma 5.2, H_{∗}^{1}(Gal(F/K), A′) = 0. If Gal(F/K) ≃ imβ ≠ G, then Gal(F/K) is of order dividing p because |G| = p^{2}. Thus Gal(F/K) is cyclic. In this case, it is clear that H_{∗}^{1}(Gal(F/K), A′) = 0. Thus in all cases we have H_{∗}^{1}(Gal(F/K), A′) = 0. Therefore Lemma 6.1 implies that (3) is true, as desired. □

7. Triple Massey products over local and global fields

Recall that a pro-p-group G is call a Demushkin group if its cohomology H^{i}(G, _{p}) has the following properties: (1) dim_{_{p}} H^{1}(G, _{p}) < ∞, (2) dim_{_{p}} H^{2}(G, _{p}) = 1 and (3) the cup product H^{1}(G, _{p}) × H^{1}(G, _{p}) → H^{2}(G, _{p}) is non-degenerate.

Theorem 7.1. Let K be a local field containing a primitive p-th root of unity. Let n be an integer greater than 2. Then every n-fold Massey product contains 0 whenever it is defined.

Proof. Let G = G_{K}(p) be the maximal pro-p quotient of the absolute Galois group of K. If either K ≃ or K ≃ and p ≠ 2, then G is trivial. Clearly then G has the vanishing n-fold Massey product property. If K ≃ and p = 2 then G ≃ /2 , which is a Demushkin group by [NSW, Proposition 3.9.10]. Now assume that K is not isomorphic to either or , then by [NSW, Proposition 7.5.9 and Theorem 7.5.11], G is also a Demushkin group. Hence, in the both main cases when G is non-trivial, G has the vanishingn-fold Massey product property by [MT1, Theorem 4.3]. □

Proof of Theorem 1.3.Theorem 1.3 follows from Proposition 4.7, Lemma 6.2 and Theorem 7.1. □

Proof of Corollary 1.4.Corollary 1.4 follows from Theorems 7.1-1.3 and Corollary 4.3. □

Remark 7.2. If F is a local field containing a primitive p-th root of unity, then the situation in Corollary 1.4 actually occurs precisely when F is a finite extension of the field _{p}. Indeed, let G = G_{F}(p) be the maximal pro-p quotient of the absolute Galois group of F. Then [NSW, Proposition 7.5.9 and Theorem 7.5.11] imply that G is a Demushkin group of rank ≥ 3 precisely when F is a finite extension of the field _{p}. The statement then follows from [MT2, Proposition 3.1 and Lemma 3.6].

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Ján Mináč Department of Mathematics Western University London, Ontario Canada N6A 5B7 minac@uwo.ca Nguyễn Duy Tân Department of Mathematics Western University London, Ontario Canada N6A 5B7 and Institute of Mathematics Vietnam Academy of Science and Technology 18 Hoang Quoc Viet 10307 Hanoi Vietnam dnguy25@uwo.ca