FIELD THEORY AND THE COHOMOLOGY OF SOME GALOIS GROUPS

ALEJANDRO ADEM, WENFENG GAO, DIKRAN B. KARAGUEUZIAN, AND JÁN MINÁČ

Abstract. We prove that two arithmetically significant extensions of a field F coincide if and only if the Witt ring W F is a group ring ℤ/n[G]. Furthermore, working modulo squares with Galois groups which are 2-groups, we establish a theorem analogous to Hilbert’s theorem 90 and show that an identity linking the cohomological dimension of the Galois group of the quadratic closure of F, the length of a filtration on a certain module over a Galois group, and the dimension over 𝔽_{2} of the square class group of the field holds for a number of interesting families of fields. Finally, we discuss the cohomology of a particular Galois group in a topological context.

1. Introduction

In recent years substantial breakthroughs have been made in the study of the cohomology of absolute Galois groups, culminating in Voevodsky’s proof of the Milnor conjecture [31]. These results severely restrict possible absolute Galois groups of fields and it seems virtually certain that they will have purely field-theoretic consequences. Such results are not easily derived, however, and in fact only a few theorems of this type have appeared (for some examples the reader may consult [18, 24, 30]). The goal of this paper is to obtain some results in field theory as consequences of Merkurjev’s theorem [23]. The techniques we use are somewhat varied; for example, we study the square class group of a field as a module for a Galois group and relate its socle series to the E_{∞}^{1,1}-term of a spectral sequence. We are then able to obtain information on this term of the spectral sequence using techniques from the theory of binary quadratic forms. In the study of particular examples our techniques become more specialized, for example, at one point we make essential use of the main theorem of local class field theory.

Throughout the paper, F will denote a field whose characteristic is not 2. We write F^{(2)} for the field obtained by adjoining all the square roots of elements of F. Now let F^{{3}} = (F^{(2)})^{(2)} and Gal(F^{(2)}/F) = G_{F}^{[2]}. We introduce the following definition: the V-group of F is the Galois group G_{F}^{{3}} = Gal(F^{{3}}/F). In [24] Mináč and Spira defined an extension of F whose Galois group is closely related to the Witt ring of F. Let F^{(3)} be the extension of F^{(2)} obtained by adjoining the square roots of elements α ∈ F^{(2)} such that F^{(2)}(\sqrt{α}) is Galois over F. F^{(3)} is a Galois extension of F and we recall that the W-group of F is the Galois group G_{F} = Gal(F^{(3)}/F). Of course F^{(3)} ⊂ F^{{3}}, so that G_{F} is a quotient of G_{F}^{{3}}. Furthermore, it follows immediately from the definitions that G_{F} = G_{F}^{{3}} if and only if F^{{3}} = F^{(3)}.

The first author was partially supported by an NSF grant, the MPIM-Bonn, and the CRM-Barcelona.

The third author was partially supported by an NSF postdoctoral fellowship, CRM-Barcelona, and MPIM-Bonn.

The fourth author was partially supported by the NSERC and the special Dean of Science fund at UWO. .

In this paper we use tools from the cohomology of groups to study these Galois groups and the field-theoretic information they may contain. Our first result is a characterization of C-fields (i.e. fields F such that the Witt ring W F is isomorphic to a group ring of the form ℤ/nℤ[G]):

Theorem. (3.1) The W-group of F is the same as the V-group of F if and only if F is a C-field.

The proof makes use of a theorem due to Merkurjev as well as a description of the E_{∞}^{1,1} term of the mod 2 Lyndon-Hochschild-Serre spectral sequence associated to the Frattini extension for G_{F}. We introduce a length invariant associated to F as follows. Let J := Ḟ^{(2)}/(Ḟ^{(2)})^{2}, regarded as a G_{F}^{[2]}-module. The socle series of J,

0 = J_{0} ⊂ J_{1} ⊂ J_{2} ⊂ ⋯ ⊂ J

has finite length l (i.e. J_{l} = J) if Ḟ /Ḟ^{2} is finite. This length is an invariant of the field, which we will write as l(F).

One of the key results used in our proof is

Proposition. (3.10) In the mod 2 Lyndon–Hochschild–Serre spectral sequence for the group extension 1 → Φ → G_{F} → G_{F}^{[2]} → 1, we have E_{∞}^{1,1} ≈ J_{2}/J_{1}.

Next we obtain a result reminiscent of Hilbert’s Theorem 90, describing the elements in the kernel of the differential d_{2}. While the proof of this theorem depends on Proposition 3.10 and hence indirectly on Merkurjev’s theorem, we provide a combinatorial, group-theoretic proof of “half” of the theorem in an appendix (§7).

Let {[a_{i}] | i ∈ I} be a basis for Ḟ /Ḟ^{2}, which we identify with H^{1}(G_{F}^{[2]}, 𝔽_{2}). Let {σ_{i} | i ∈ I} be a minimal set of generators of G_{F}^{[2]} such that σ_{i}(\sqrt{a_{j}})/\sqrt{a_{j}} = (−1)^{δ_{i,j}}, where i, j ∈ I and δ_{i,j} is the Kronecker delta function. One may view the a_{i} as a dual basis to the σ_{i}.

Theorem. (4.1) If {[j_{i_{k}}] | 1 ≤ k ≤ l} is a finite set of elements in J_{1}, then

∑_{k=0}^{l}[a_{i_{k}}]d_{2}^{0,1} ([j_{i_{k}}]) = 0

if and only if there exists an element [j] ∈ J_{2} such that 1. [σ_{i_{k}}(j)/j] = [j_{i_{k}}] for each k ∈ {1, …, l} and

2. [σ_{u}(j)] = [j] for each u ∉ {i_{1}, …, i_{l}}.

One should observe that this result is a practical criterion for the existence of elements [j] from J_{2} with prescribed images under the action of the Galois group G_{F}^{[2]}. We illustrate this in a detailed way in a special example when the W –group is the universal W –group on two generators (see example 4.2). Moreover this theorem fully describes the G_{F}^{[2]}–module J_{2}.

Let G_{q} := Gal(F_{q}/F), where F_{q} is a quadratic closure of F. Our next result provides a formula for the cohomological dimension of G_{q} for certain non-formally real fields in terms of the length function l(F):

Theorem. (5.2, 5.3, 5.15) Let F be a field of which is not formally real, and assume that |Ḟ /Ḟ^{2}| = 2^{n}. Then if F is a C-field, a local field, or a field such that G_{q} is free, we have l(F) + cd(G_{q}) = n + 1.

Finally in our last section we make an explicit analysis of the mod 2 cohomology of the Vgroup arising when G_{q} is the free pro-2-group on 2 generators (denoted V (2), a group of order 2^{7}). We obtain its Poincaré series as well as some rather interesting topological information.

Proposition. (6.4) The Poincaré series for H^{∗}(V (2), 𝔽_{2}) is

\frac{1 − x + x^{2}}{(1 − x)^{3}(1 − x^{2})^{2}} ,

which expands to 1+2x+6x^{2} +11x^{3} +22x^{4} +36x^{5} +60x^{6} +90x^{7} +135x^{8} +190x^{9} +266x^{10} +· · ·.

Our methods combine techniques from field theory with methods from the cohomology of groups. For background information on Galois cohomology, group cohomology, quadratic form theory, and local class field theory, we refer the reader to [28], [3, 7], [21, 22], and [5, 14] respectively. We have endeavored to provide specific references throughout the paper as often as possible.

This paper is organized as follows: in §2 we describe preliminary definitions and basic facts about the Galois groups considered here; in §3 we prove the characterization of C-fields and introduce the length of F; in §4 we prove our analogue of Hilbert’s Theorem 90; in §5 we discuss our results relating l(F) with cd(G_{q}) for examples such as local fields; and finally in §6 we provide a fairly complete cohomological analysis of the group V (2) and of a 5-dimensional crystallographic group associated to it.

A few comments about notation may be helpful. We shall have several uses for the cyclic group of order two, and thus as many notations for it. Generally, we denote this group by ℤ/2ℤ, but when we regard it as the group of square roots of unity in a field, we write it μ_{2}, and when we consider the field of two elements, used as coefficients for cohomology, we write 𝔽_{2}. Generally, we do not specify the coefficients for our cohomology theories, and in such cases 𝔽_{2}-coefficients are to be understood. When we use other coefficient rings, they will be specified.

2. Preliminaries

Let F denote a field of characteristic different from two, and denote by F^{(2)} the field obtained by adjoining all the square roots of elements of F. Now let F^{{3}} = (F^{(2)})^{(2)}. From Kummer theory it is easy to see that F^{(2)} is a Galois extension of F, and if B is a basis of Ḟ /Ḟ^{2}, then the Galois group of F^{(2)} over F is ∏_{B} ℤ/2ℤ. We denote this Galois group by G_{F}^{[2]}. A slightly less obvious fact is Lemma 2.1. F^{{3}} is a Galois extension of F.

Proof. Let σ: F^{{3}} → F̄ be an embedding of F^{{3}} into an algebraic closure F̄ of F containing F^{\{3\}}. It is enough to show that for each γ ∈ Ḟ^{(2)}, σ(\sqrt{γ}) ∈ F^{\{3\}}. However, σ(\sqrt{γ})^{2} = σ(γ) and therefore σ(\sqrt{γ}) = ±σ(\sqrt{γ}). But σ(γ) ∈ Ḟ^{(2)}, since F^{(2)}/F is Galois, so the lemma follows.

Using this we introduce

Definition 2.2. The V-group of F is the Galois group G_{F}^{{3}} = Gal(F^{{3}}/F).

In [24] Mináč and Spira defined the “Witt closure” of F, an extension of F whose Galois group is closely related to the Witt ring of F. Let F^{(3)} be the extension of F^{(2)} obtained by adjoining the square roots of elements α ∈ F^{(2)} such that F^{(2)}(\sqrt{α}) is Galois over F. F^{(3)} is a Galois extension of F and we recall

Definition 2.3. The W-group of F is the Galois group G_{F} = Gal(F^{(3)}/F).

Of course F^{(3)} ⊂ F^{{3}}, so that G_{F} is a quotient of G_{F}^{{3}}. We will frequently make use of this fact without explicit mention. Furthermore, it follows immediately from the definitions that G_{F} = G_{F}^{{3}} if and only if F^{{3}} = F^{(3)}.

Now we introduce the Galois group of the quadratic closure and one of its subgroups:

Definition 2.4. Let G_{q} := Gal(F_{q}/F), where F_{q} is a quadratic closure of F, and set G^{(2)} := Gal(F_{q}/F^{(2)}).

Note that G_{q}/G^{(2)} ≅ G_{F}^{[2]} and that the V-group is determined by G_{q}, namely if Φ(G^{(2)}) denotes the Frattini subgroup of G^{(2)}, then from our definition it follows that

G_{F}^{{3}} = G_{q}/Φ(G^{(2)}).

From the above we obtain that there is an extension of elementary abelian groups

1 → G_{F^{(2)}}^{[2]} → G_{F}^{{3}} → G_{F}^{[2]} → 1,

where the action on the kernel can be highly non-trivial. We will often abbreviate G_{F^{(2)}}^{[2]} by simply using the symbol A. From the point of view of group theory, the V-group of the field F is simply the extension of G_{F}^{[2]} obtained by taking the quotient of G_{q} by the Frattini subgroup of G^{(2)}.

We will be especially interested in the case when G_{q} is a free pro-2-group on n generators. From the analysis in [24] we know that in this case the W-group maps onto any W-group arising from a field with n-dimensional group of square classes, hence it is referred to as the universal W-group W (n). Similarly the V-group arising from this situation will enjoy an analogous universal property and we denote it by V (n).

In this case the group G^{(2)} can be identified with the Frattini subgroup of the free pro-2-group G_{q}, hence it is a free pro-2-group of rank 2^{n}(n − 1) + 1, where G_{F}^{[2]} ≅ (ℤ/2ℤ)^{n}. More explicitly we have G^{(2)} = G_{q}^{2}[G_{q}, G_{q}] (in fact, [G_{q}, G_{q}] ⊂ G_{q}^{2}, so G^{(2)} = G_{q}^{2}). Our next step is to “abelianize” this extension, namely we factor out the Frattini subgroup of G^{(2)} This yields the quotient V (n) as an extension

1 → A → V (n) → G_{F}^{[2]} → 1.

Moreover we can identify the G_{F}^{[2]}-module A explicitly using the methods in [20]; indeed A ≅ Ω^{2}(𝔽_{2}), the second dimension shift of the trivial module (for the definition of the “Heller translate” Ω^{2} see [7, p. 8, v. I]). Alternatively, via a detailed study of higher commutators, one can determine the first two stages in a minimal resolution of the dual of A, which also establishes the isomorphism A ≅ Ω^{2}(𝔽_{2}). The group V (n) corresponds to the unique non-trivial element in H^{2}(G_{F}^{[2]}, Ω^{2}(𝔽_{2})); this element in fact restricts non-trivially on every cyclic subgroup, whence the extension above is totally non-split.

Now for the W-group we have a extension of elementary abelian groups

1 → Φ(G_{F}) → G_{F} → G_{F}^{[2]} → 1,

where Φ denotes the Frattini Subgroup of G_{F}. Moreover, elaborating on what we have mentioned this can be expressed as an extension

1 → U → G_{F}^{{3}} → G_{F} → 1

where U = Gal(F^{{3}}/F^{(3)}).

Definition 2.5. A field whose Witt ring is isomorphic to some group ring of the form ℤ/nℤ[G] is said to be a C-field (here n is permitted to take the value 0).

In [32, 1.9] a number of equivalent conditions are given for a field to be a C-field. We remark that the p-adic field ℚ_{p}, where p is an odd prime, is a C-field. Furthermore, ℝ((t_{1})) ⋯ ((t_{n})), ℂ((t_{1})) ⋯ ((t_{n})), and direct limits of such fields, are also C-fields. Further examples can be found in [22, p. 46].

Now suppose that K ⊂ L is a Galois extension of fields, and that there is a subgroup N of K̇/K̇^{2} such that L = K[\sqrt{N}]. Again from Kummer theory we know that there is a “perfect pairing”

N × Gal(L/K) → μ_{2}.

Suppose further that F ⊂ K is another field, and that the extensions F ⊂ K and F ⊂ L are also Galois. Then there is a conjugation action of Gal(K/F) on Gal(L/K), and an action of Gal(K/F) on N. These actions are compatible in the sense that the following lemma holds, see for example [18, p. 101] or [34].

Lemma 2.6. If σ ∈ Gal(K/F), n ∈ N, γ ∈ Gal(L/K), and ⟨·, ·⟩ is the Kummer pairing, then

⟨σn, γ⟩ = ⟨n, γ^{σ^{−1}}⟩,

i.e. the action of Gal(K/F) on Gal(L/K) and N is compatible with the Kummer pairing.

3. The V-group of a C-field

In this section we will prove one of our main theorems, namely

Theorem 3.1. If F is a field of characteristic different from two, then the W-group of F is the same as the V-group of F if and only if F is a C-field.

We briefly describe the plan of the proof. To show that whenever F is a C-field we have G_{F} = G_{F}^{{3}}, we use a classification of C-fields [32, 1.1] and work case-by-case to show that F^{{3}} = F^{(3)}. To prove the converse, we note that by [32, 1.9], if F is not a C-field, then there exists a binary quadratic form of a certain type over F, which implies the existence of a certain permanent cycle in E_{∞}^{1,1}, the mod 2 Lyndon-Hochschild-Serre spectral sequence for the group extension 1 → Φ → G_{F} → G_{F}^{[2]} → 1. An identification of E_{∞}^{1,1} using Merkurjev’s theorem [23] then shows that G_{F}^{[2]} must act nontrivially on G_{F^{(2)}}^{[2]}, and so it follows that G_{F} ≠ G_{F}^{{3}}.

As mentioned above our proof depends on Ware’s classification of C-fields, specifically the following result (recall that s(F) denotes the level of the field F):

Proposition 3.2. Let F be a C-field. If F is formally real, then F is superpythagorean, while if F is not formally real, s(F) = 1 or 2.

Remark 3.3. A field F is pythagorean if F^{2} + F^{2} = F^{2}. A formally real field F is superpythagorean if it is pythagorean and given any subgroup S of index 2 in Ḟ such that −1 ∉ F, S is an ordering of F. For further details, see [22, p. 44].

It is clear from the proposition above that we can prove one implication in Theorem 3.1 by studying three cases: the superpythagorean case, where s(F) = ∞, the case s(F) = 1, and the case s(F) = 2. Indeed, it is a consequence of the following three lemmas.

Lemma 3.4. If F is a superpythagorean field then G_{F} = G_{F}^{{3}}.

Lemma 3.5. If F is a C-field with s(F) = 1 then G_{F} = G_{F}^{{3}}.

Lemma 3.6. If F is a C-field with s(F) = 2 then G_{F} = G_{F}^{{3}}.

Remark 3.7. The proofs of Lemmas 3.4, 3.5, and 3.6 borrow heavily from the study of the W-groups of the relevant types of fields in [25]. In that paper, the possible W-groups of a C-field were determined, and their structure turns out to depend on the same sort of structural properties of Ḟ /Ḟ^{2} as are used in the proofs of the following lemmas. To clarify things, we provide a list of the possible W-groups of a C-field.

Proof of Lemma 3.5. First we discuss the possible W-groups of C-fields F with s(F) = 1. If |Ḟ /Ḟ^{2}| = 1, F is quadratically closed, so F^{{3}} = F^{(2)} = F and G_{F} = 1. If |Ḟ /Ḟ^{2}| = 2, then G_{F} = ℤ/4ℤ or ℤ/2ℤ; by [25, 3.12.3] if G_{F} = ℤ/2ℤ, then F is euclidean, and in particular −1 is not a square in F, so this case is impossible for a C-field of level 1. Thus we see that if F is a C-field of level 1 and |Ḟ /Ḟ^{2}| = 1 or 2, then G_{F} = 1 or ℤ/4, accordingly.

Now let us suppose that Ḟ /Ḟ^{2} = ⊕_{I}ℤ/2ℤ, where |I| ≥ 2. Then from [25, 3.13] we have that the W-group of F is ∏_{I} ℤ/4ℤ.

So we must prove that when G_{F} = ∏_{I} ℤ/4ℤ, F^{{3}} = F^{(3)}. As we noted in the first paragraph of this proof, if I = ∅, this is trivial, so (although it is not logically necessary) we can eliminate the need to keep track of trivial cases by assuming that |I| ≥ 1.

To show that F^{\{3\}} = F^{(3)} we will show show for each γ ∈ Ḟ^{(2)} that F^{(2)}(\sqrt{γ})/F is Galois, so that F^{\{3\}} ⊂ F^{(3)}, and hence F^{\{3\}} = F^{(3)}. Our method of demonstrating that F^{(2)}(\sqrt{γ})/F is Galois will be to prove something a bit stronger, namely that for each γ ∈ Ḟ^{(2)} there is an a ∈ Ḟ such that [γ] = [\sqrt{a}] in Ḟ^{(2)}/(Ḟ^{(2)})^{2}. From this it follows that F^{(2)}(\sqrt{γ}) = F^{(2)}(\sqrt[4]{a}), which is a Galois extension of F since \sqrt{−1} ∈ F^{(2)}.

So, let us take γ ∈ Ḟ^{(2)} and seek an element a ∈ Ḟ such that [γ] = [\sqrt{a}] in Ḟ^{(2)}/(Ḟ^{(2)})^{2}. Note first that γ ∈ F (\sqrt{a_{1}}, …, \sqrt{a_{n}}) for some elements a_{1}, …, a_{n} of F, and that we may take [a_{1}], …, [a_{n}] to be linearly independent in Ḟ /Ḟ^{2}. Writing K := F (\sqrt{a_{1}}, …, \sqrt{a_{n}}) and using the proof of [25, 3.13], it follows that we may complete our partial basis of Ḟ /Ḟ^{2} to a basis B = \{[a_{1}], …, [a_{n}]\}∪\{[a_{j}] | j ∈ Ω−\underline{n}\} of Ḟ /Ḟ^{2} with the property that B̃ = \{[\sqrt{a_{1}}], …, [\sqrt{a_{n}}]\}∪ \{[a_{j}] | j ∈ Ω − \underline{n}\} is a basis of K̇/K̇^{2}. (Here Ω is an ordinal number, used as an indexing set, and \underline{n} := \{1, …, n\}.) Then in K̇/K̇^{2} we have

γ = ∏_{i=1}^{n}[\sqrt{a_{i}}]^{ϵ_{i}} · ∏_{j∈Ω−\underline{n}}[a_{j}]^{ϵ_{j}},

where ϵ_{i} = 0 or 1 and all but finitely many terms in the product are 1, so that

[γ] = [\sqrt{∏_{i=1}^{n}[a_{i}]^{ϵ_{i}} · ∏_{j∈Ω−\underline{n}}[a_{j}]^{2ϵ_{j}}}]

in K̇/K̇^{2}, and hence also in Ḟ^{(2)}/(Ḟ^{(2)})^{2} since K ⊂ F^{(2)}.

This completes the proof that F^{{3}} = F^{(3)} for C-fields of level 1.

The proofs of Lemmas 3.4 and 3.6 are quite similar to the proof above and so we give a somewhat abbreviated presentation.

Proof of Lemma 3.4. We follow the same plan: we show that for each γ ∈ Ḟ^{(2)} there is an a ∈ Ḟ such that [γ] = [\sqrt{a}] in Ḟ^{(2)}/(Ḟ^{(2)})^{2}. It follows that for any γ ∈ Ḟ^{(2)} that F^{(2)}(\sqrt{γ})/F is Galois, so that F^{{3}} = F^{(3)}. The key point is again that there is a basis of Ḟ /Ḟ^{2} of the form \{[−1]\} ∪ \{[a_{i}] | i ∈ I\} such that the set \{[\sqrt{a_{i}}] | i ∈ I\} is a basis of Ḟ^{(2)}/(Ḟ^{(2)})^{2}.

Remark 3.8. The failure to list \sqrt{−1} as a basis element of Ḟ^{(2)}/(Ḟ^{(2)})^{2} is not a mistake. If L is a field of characteristic not equal to 2, then \sqrt{2} ∈ L^{(2)} and \sqrt{−1} ∈ L^{(2)}, so that \sqrt[4]{−1} = \frac{\sqrt{2}}{2}(1 + \sqrt{−1}) ∈ L^{(2)}, i.e. \sqrt{−1} is a square in L^{(2)}.

The proof that F^{{3}} = F^{(3)} is then completed my mimicking the details at the end of the proof of Lemma 3.5.

The proof of Lemma 3.6 will involve an appeal to the following fact, whose proof can be found in [32].

Lemma 3.9. Let K/F be a Galois extension and a an element of K̇. Then the extension K(\sqrt{a})/F is Galois if and only if σ(a) · a is a square in K̇ for every σ ∈ Gal(K/F).

Proof of Lemma 3.6. As in the proof of Lemma 3.5, it is enough to show that for each γ ∈ Ḟ^{(2)} that F^{(2)}(\sqrt{γ})/F is Galois. Since in the case at hand, F is a C-field of level 2, by [25, p. 527], we have a basis for Ḟ^{(2)}/(Ḟ^{(2)})^{2} of the form B̃ = \{[y], [\sqrt{a_{i}}] | i ∈ Ω\}, where y ∈ F (\sqrt{−1}) and such that B = {[−1], [a_{i}] | i ∈ Ω} is a basis of Ḟ /Ḟ^{2}.

We claim that in order to show that for every γ ∈ Ḟ^{(2)}, F^{(2)}(\sqrt{γ})/F is Galois, it suffices to prove this fact for γ in B̃.

Proof of claim. Suppose γ = γ_{1} ⋯ γ_{s}, where γ_{i} ∈ B̃. To show that F^{(2)}(\sqrt{γ})/F is Galois, by Lemma 3.9 it is enough to show that σ(γ) · γ is a square in F^{(2)} for each σ ∈ G_{F}^{[2]}. But if this is true for each γ_{i} ∈ B̃, the calculation

σ(γ) · γ = σ(γ_{1} ⋯ γ_{s}) · γ_{1} ⋯ γ_{s} = σ(γ_{1}) · γ_{1} ⋯ σ(γ_{s}) · γ_{s} = x_{1}^{2} ⋯ x_{s}^{2}

shows that it is also true for γ.

So we must only show that for each γ ∈ B̃ that F^{(2)}(\sqrt{γ})/F is Galois. If γ = \sqrt{a_{i}}, this follows from the fact that \sqrt{−1} ∈ F^{(2)}, while if we take γ = y, we may use Lemma 3.9, and note that σ(y) · y is either y^{2}, which is obviously a square, or N_{F (\sqrt{−1})/F} (y), which is an element of F and therefore a square in F^{(2)}.

For the other half of the proof we will use group cohomology and a result due to Merkurjev. Our basic goal is the identification of the E_{∞}^{1,1}-term of the Lyndon-Hochschild-Serre spectral sequence for the group extension

1 → Φ → G_{F} → G_{F}^{[2]} → 1

in terms of information in the V-group of F, G_{F}^{{3}}. To describe this information, we first introduce some notation.

Recall that there is a group extension 1 → A → G_{F}^{{3}} → G_{F}^{[2]} → 1 where the Pontrjagin dual of A is isomorphic to Ḟ^{(2)}/(Ḟ^{(2)})^{2}. This elementary abelian 2-group is a G_{F}^{[2]}-module.

Let J := Ḟ^{(2)}/(Ḟ^{(2)})^{2}, regarded as a G_{F}^{[2]}-module. The socle series of J,

0 = J_{0} ⊂ J_{1} ⊂ J_{2} ⊂ ⋯ ⊂ J

has finite length l (i.e. J_{l} = J) if Ḟ /Ḟ^{2} is finite. This length is an invariant of the field, which we will write as l(F). In this socle series, J_{1} = J^{G_{F}^{[2]}}, J_{2}/J_{1} = (J/J_{1})^{G_{F}^{[2]}}, …, J_{i+1}/J_{i} = (J/J_{i})^{G_{F}^{[2]}}, etc. We will loosely refer to this as the length of the field F.

Now we can state our key result

Proposition 3.10. In the mod 2 Lyndon–Hochschild–Serre spectral sequence for the group extension 1 → Φ → G_{F} → G_{F}^{[2]} → 1, we have that E_{∞}^{1,1} ≈ J_{2}/J_{1}.

Note that the group extension is one for the W-group, but that the group J_{2}/J_{1} comes from the V-group. Thus, this fact is more subtle than it looks. In fact its proof involves the use of Merkurjev’s theorem in an essential way.

Consider the commutative diagram of extensions:

[figure: 1; G^{(2)}; G_{q}; G_{F}^{[2]}; 1; 1; Φ; G_{F}; G_{F}^{[2]}; 1.]

We shall denote the spectral sequences for the mod 2 cohomology of G_{F} and G_{q} by E and Ē respectively. There is an induced map from E_{2}^{1,1} to Ē_{2}^{1,1}, for which we will write γ.

Lemma 3.11. d̄_{2}^{1,1} is injective.

Proof. By Merkurjev’s theorem, the inflation map from H^{2}(G_{F}^{[2]}) to H^{2}(G_{q}) is surjective, so Ē_{∞}^{1,1} = 0, but this is just the kernel of d̄_{2}^{1,1}.

Lemma 3.12. ker(γ) = ker(d_{2}^{1,1}).

Proof. This follows by considering the following commutative diagram and applying the previous lemma.

[figure: Ē_{2}^{1,1}; d̄_{2}^{1,1}; H^{3}(G_{F}^{[2]}); E_{2}^{1,1}; d_{2}^{1,1}; γ; H^{3}(G_{F}^{[2]})]

Lemma 3.13. ker(γ) = J_{2}/J_{1}

Proof. We have Ē_{2}^{1,1} = H^{1}(G_{F}^{[2]}, H^{1}(G^{(2)})) and E_{2}^{1,1} = H^{1}(G_{F}^{[2]}, H^{1}(Φ)). Since J_{1} ≈ H^{1}(Φ) and J ≈ H^{1}(G^{(2)}), the result follows from considering the long exact sequence in cohomology associated to the short exact sequence of coefficients J_{1} ↪ J ↠ J/J_{1}, since γ is just the map H^{1}(G_{F}^{[2]}, J_{1}) → H^{1}(G_{F}^{[2]}, J).

Now the proof of Proposition 3.10 is merely a matter of stringing the lemmas together.

Proof of Proposition 3.10. Note that E_{∞}^{1,1} = E_{3}^{1,1} = ker(d_{2}^{1,1}), as there are no further differentials in this part of the spectral sequence. By Lemma 3.12, ker(d_{2}^{1,1}) = ker(γ), which is J_{2}/J_{1} by Lemma 3.13.

We can now show that if F is not a C-field, then the V-group of F is not the W-group of F. The proof of this fact is based on an examination of the spectral sequence for the cohomology of the W-group (3.15) and an identification of part of the E_{∞}-term of that spectral sequence with information contained in the V-group (Lemma 3.14).

Lemma 3.14. The following conditions are equivalent:

1. J = J_{1}

2. F^{{3}} = F^{(3)}

3. J_{2} = J_{1}.

Proof. 1 ⇔ 2. We need the fact that an extension F^{(2)}(\sqrt{γ})/F is Galois if and only if [γ] ∈ J_{1}. This follows from Lemma 3.9.

We have shown that F^{(3)} is obtained from F^{(2)} by adjoining the square roots of elements in J_{1}, while F^{{3}} is obtained by adjoining the square roots of elements in J. Therefore we have F^{{3}} = F^{(3)} if and only if J = J_{1}.

1 ⇔ 3. To show that J_{2} = J_{1} implies J = J_{1} we note that for any element j ∈ J, the orbit of j under the action of G_{F}^{[2]} is finite. This is because we may choose a representative of j, which is an element of Ḟ^{(2)}, hence algebraic over F, so that any image of this element under the action of the Galois group satisfies the same minimal polynomial. Therefore, if j̄ ∈ J/J_{1}, the G_{F}^{[2]} submodule generated by j̄ is finite and has a fixed point by the usual counting argument. Thus if J/J_{1} ≠ 0, J_{2} ≠ J_{1}, which is the contrapositive of the desired statement. The converse is trivial.

Now we turn to the study of the spectral sequence for the group extension

1 → Φ → G_{F} → G_{F}^{[2]} → 1.

We will show that if F is not a C-field, then there are classes in the spectral sequence that survive to E_{∞}^{1,1}. More precisely we will prove:

Proposition 3.15. Suppose F is not a C-field. Then in the spectral sequence for the cohomology of G_{F}, E_{∞}^{1,1} ≠ 0.

In the proof of this proposition we will need the following lemma which follows from [21, theorem 2.7, p. 58] and [26, p. 255].

Lemma 3.16. Let L be a field and a, b elements of L̇. Writing [a] and [b] for the classes of a and b in L̇/L̇^{2}, or equivalently in H^{2}(Gal(L_{q}/L); 𝔽_{2}), we have [a] ∪ [b] = 0 if and only if ax^{2} + by^{2} − z^{2} = 0 has a nontrivial solution (x, y, z) ∈ L^{3}.

Remark 3.17. More precisely, we need the fact that this lemma remains true if the Galois group of the quadratic closure L_{q} is replaced by the W-group G_{L}. But this is true by the construction of the W-group and Merkurjev’s theorem. See [2, 3.14].

Proof of Proposition 3.15. Since F is not a C-field, by [32, 1.9] there exists an anisotropic binary form B over F such that the set D of values of B, regarded as a subgroup of Ḟ /Ḟ^{2}, has at least three elements. (As Ḟ /Ḟ^{2} is an elementary abelian 2-group, so are any of its subgroups, so “D has at least three elements” immediately implies “D has at least four elements”. The fact that D is a subgroup follows from the identity (1 − ax^{2})(1 − ay^{2}) = (1 + axy)^{2} − a(x + y)^{2}.) We may assume that B = x^{2} − ay^{2} for some a ∈ Ḟ by an appropriate transformation. Because B is anisotropic, we know that [a] ≠ [1] in Ḟ /Ḟ^{2}.

Since |D| ≥ 4, there exist elements a_{2}, a_{3} in D which are linearly independent over 𝔽_{2}. The statement “a_{2} is a value of B in Ḟ /Ḟ^{2}” means that there exist x, y, and z in F such that a_{2}x^{2} + ay^{2} − z^{2} = 0. By Lemma 3.16 we see that [a][a_{2}] = 0 ∈ H^{2}(G_{F}), and similarly that [a][a_{3}] = 0.

This information allows us to construct directly a nonzero element in the spectral sequence for H^{∗}(G_{F}). Because the relations [a][a_{2}] = 0 and [a][a_{3}] = 0 exist in H^{∗}(G_{F}) but not in H^{∗}(G_{F}^{[2]}), there exist z_{2}, z_{3} in H^{1}(Φ) such that d_{2}^{0,1}(z_{2}) = [a][a_{2}] and d_{2}^{0,1}(z_{3}) = [a][a_{3}]. Then, setting λ = [a_{2}] ⊗ z_{3} + [a_{3}] ⊗ z_{2} ∈ E_{2}^{1,1} = H^{1}(G_{F}^{[2]}) ⊗ H^{1}(Φ), we can compute

d_{2}^{1,1}(λ) = [a_{2}] · d_{2}^{0,1}(z_{3}) + [a_{3}] · d_{2}^{0,1}(z_{2}) = [a_{2}][a][a_{3}] + [a_{3}][a][a_{2}] = 0.

Thus λ survives to E_{3}^{1,1} and hence to E_{∞}^{1,1} by its position in the spectral sequence.

The results above immediately imply the converse implication in the statement of 3.1 and so the proof is complete.

4. Surjectivity in Merkurjev’s Theorem and an Analogue of Hilbert’s Theorem 90

We have seen how methods from group cohomology can be used to characterize certain fields. In this section we make an explicit analysis of the kernel of the cohomological differential d_{2} used previously and interpret the result in field-theoretic terms. From this we obtain a result which is reminiscent of Hilbert’s Theorem 90.

Let {[a_{i}] | i ∈ I} be a basis for Ḟ /Ḟ^{2}, which we identify with H^{1}(G_{F}^{[2]}). Let {σ_{i} | i ∈ I} be a minimal set of generators of G_{F}^{[2]} such that σ_{i}(\sqrt{a_{j}})/\sqrt{a_{j}} = (−1)^{δ_{i,j}}, where i, j ∈ I and δ_{i,j} is the Kronecker delta function. One may view the a_{i} as a dual basis to the σ_{i}.

We are now able to state the result:

Theorem 4.1. Let {[j_{i_{k}}] | 1 ≤ k ≤ l} be a finite set of elements of J_{1}. Then

∑_{k=0}^{l}[a_{i_{k}}]d_{2}^{0,1} ([j_{i_{k}}]) = 0 (1)

if and only if there exists an element [j] ∈ J_{2} such that 1. [σ_{i_{k}}(j)/j] = [j_{i_{k}}] for each k ∈ {1, …, l} and

2. [σ_{u}(j)] = [j] for each u ∉ {i_{1}, …, i_{l}}.

It is worth noting that one implication of the theorem (the “if” part) can be proved directly, without the use of Merkurjev’s theorem or spectral sequences. We indicate how this can be done in an appendix (§7).

Proof of Theorem 4.1. Recall that we have the exact sequence

0 → J_{2}/J_{1} → H^{1}(G_{F}^{[2]}, J_{1}) → H^{1}(G_{F}^{[2]}, J)

arising from the long exact sequence in cohomology associated to the short exact sequence of coefficients J_{1} ↪ J ↠ J/J_{1}. Note that since the action of G_{F}^{[2]} on the coefficient group J_{1} is trivial, we may identify H^{1}(G_{F}^{[2]}, J_{1}) with Hom(G_{F}^{[2]}, J_{1}). In other words J_{2}/J_{1} ≅ Ker(Hom(G_{F}^{[2]}, J_{1}) → H^{1}(G_{F}^{[2]}, J)). We shall make this isomorphism explicit. Let [j] ∈ J_{2}/J_{1}. Then we can associate to this [j] the function f_{[j]} : G_{F}^{[2]} → J_{1} which is given by the formula:

f_{[j]}(σ) = [\frac{σ(j)}{j}] ∈ J_{1} .

It follows from the definition of socle series that [σ(j)/j] ∈ J_{1}. The connecting homomorphism J_{2}/J_{1} → H^{1}(G_{F}^{[2]}, J_{1}) is given by [j] ↦ f_{[j]}. In other words, we have a one-to-one correspondence J_{2}/J_{1} → Ker(Hom(G_{F}^{[2]}, J_{1}) → H^{1}(G_{F}^{[2]}, J)) given by [j] ↦ f_{[j]}. Using the basis {a_{i}} introduced earlier each function f_{[j]} can be written as

f_{[j]} = [a_{i_{1}}] ⊗ [j_{i_{1}}] + ⋯ + [a_{i_{k}}] ⊗ [j_{i_{k}}] ∈ H^{1}(G_{F}^{[2]}) ⊗ J_{1},

simply because Hom(G_{F}^{[2]}, J_{1}) = Hom(G_{F}^{[2]}, ℤ/2ℤ) ⊗ J_{1}. Observe that our function f_{[j]} has the values:

f_{[j]}(σ_{i_{1}}) = [j_{i_{1}}], …, f_{[j]}(σ_{i_{k}}) = [j_{i_{k}}]

and

f_{[j]}(σ_{l}) = [1] , for each l ≠ i_{1}, …, i_{k} .

(Recall that each continuous homomorphism G_{F}^{[2]} → J_{1} has only finitely many values by definition of the Krull topology on G_{F}^{[2]}.)

Combining the above with the fact that J_{2}/J_{1} ≡ E_{∞}^{1,1} (3.10) and the definition of E_{∞}^{1,1} = ker d_{2}^{1,1} completes the proof.

Example 4.2. Let F be a field such that G_{F} ≅ W (2). In this case G_{q} is the free pro-2-group on two generators, and as we saw before, the V-group is a quotient expressible as an associated “abelianized extension”.

We can choose a basis {[a_{1}], [a_{2}]} of Ḟ /Ḟ^{2} and we can choose as our generating k-invariants of W (2) the following elements of H^{2}(G_{F}^{[2]}):

q_{1} = d_{2}^{0,1}([j_{1}]) = [a_{1}][a_{1}], q_{2} = d_{2}^{0,1}([j_{2}]) = [a_{2}][a_{2}], q_{3} = d_{2}^{0,1}([j_{3}]) = [a_{1}][a_{2}].

Thus J_{1} = ⟨[j_{1}], [j_{2}], [j_{3}]⟩ ⊂ J.

Set also λ_{1} = [a_{1}] ⊗ [j_{3}] + [a_{2}] ⊗ [j_{1}] ∈ E_{2}^{1,1}, λ_{2} = [a_{1}] ⊗ [j_{2}] + [a_{2}] ⊗ [j_{3}] ∈ E_{2}^{1,1}.

We see that ⟨λ_{1}, λ_{2}⟩ = ker d_{2}^{1,1} and therefore λ_{1}, λ_{2} form a basis of E_{∞}^{1,1}. We will identify λ_{1}, λ_{2} with elements [λ_{1}], [λ_{2}] ∈ J_{2}/J_{1} (here we are using 3.10).

We pick as usual generators σ_{1}, σ_{2} for G_{F}^{[2]}. Then we have σ_{1}([λ_{1}]) = [λ_{1}][j_{3}] and σ_{2}([λ_{1}]) = [λ_{1}][j_{1}] similarly the expression for λ_{2} as an element of E^{1,1} determines the action of G_{F}^{[2]} on [λ_{2}]: σ_{2}[λ_{2}] = [j_{3}][λ_{2}] and σ_{1}[λ_{2}] = [λ_{2}][j_{2}].

It is convenient to picture our G_{F}^{[2]}-module J_{2} as follows:

\xymatrix{ & [λ_{1}] \ar[dr]^{Σ_{1}} \ar[dl]_{Σ_{2}} &  & [λ_{2}] \ar[dr]^{Σ_{1}} \ar[dl]_{Σ_{2}} \\ [j_{1}] &  & [j_{3}] &  & [j_{2}]}

Here Σ_{2}[λ_{2}] := (σ_{2} − 1)[λ_{2}] := σ_{2}[λ_{2}]/[λ_{2}], etc.

As mentioned above, G_{q} is the free pro-2-group on two generators (see [28, 4.3, pp. 33,34]). Recall that the well-known formula, due to Schreier, on the number of any minimal set of generators of any subgroup of finite index of a free group has a pro-p-analogue [19, p. 49, 6.3]. Set G^{(2)} := Gal(F_{q}/F^{(2)}). Then G^{(2)} is an open subgroup of G_{q}, since here |Ḟ /Ḟ^{2}| = 4 < ∞. Then by the aforementioned formula we have

dim_{𝔽_{2}} H^{1}(G^{(2)}) = 4(dim H^{1}(G_{q}) − 1) + 1.

Hence we see that dim_{𝔽_{2}} H^{1}(G^{(2)}) = 5. From Pontrjagin duality we know that

J = Ḟ^{(2)}/(Ḟ^{(2)})^{2} ≅ H^{1}(G^{(2)}).

Thus dim_{𝔽_{2}} J = 5. Since dim_{𝔽_{2}} J_{1} = 3 and dim_{𝔽_{2}} J_{2}/J_{1} = 2 we see that l(J) = 2. This means J_{2} = J.

Therefore we have the following extension

1 → A → G_{F}^{{3}} → ℤ/2ℤ × ℤ/2ℤ → 1

Our group A ≅ (ℤ/2ℤ)^{5} = ⟨σ̂_{1}^{2}, σ̂_{2}^{2}, [σ̂_{1}, σ̂_{2}], [σ̂_{1}^{2}, σ̂_{2}], [σ̂_{2}^{2}, σ̂_{1}]⟩. J is just the dual of A. This means J ≅ H^{1}(A) ≅ Ω^{−2}(𝔽_{2}). Observe also that our basis {[j_{1}], [j_{2}], [j_{3}], [λ_{1}], [λ_{2}]} is dual to the basis of A given above with respect to the Kummer pairing. Thus we may write

[j_{1}] = (σ̂_{1}^{2})^{∗}, [j_{2}] = (σ̂_{2}^{2})^{∗}, [j_{3}] = ([σ_{1}, σ_{2}])^{∗} ,

[λ_{1}] = ([σ_{1}^{2}, σ_{2}])^{∗}, [λ_{2}] = ([σ_{2}^{2}, σ_{1}])^{∗} .

It is not hard to use the description above to make explicit the existence of a central extension 1 → ℤ/2ℤ × ℤ/2ℤ → V (2) → W (2) → 1.

5. Length and Cohomological Dimension

We turn now to the case where Ḟ /Ḟ^{2} is finite, so the order of this group is 2^{n} for some n. In this case J is a finite-dimensional G_{F}^{[2]}-module. Therefore its socle series J_{1} ⊂ J_{2} ⊂ ⋯ ⊂ J_{l} has length l for some l ∈ ℕ. This number is an interesting field-theoretic invariant and there are conjectural connections between l(F) and the cohomological dimension of G_{q}. More generally, for G_{F}^{[2]}-modules M, we will write l(M) for the length of the socle series of M, provided that this length is finite.

If G is a profinite group, its cohomological dimension cd(G) is defined as sup cd_{p}(G) where cd_{p}(G) is the smallest integer n such that the p-primary component of H^{q}(G, A) is zero for all discrete, torsion G-modules A and all integers q > n (see [28, 3.1]). Recall that if G is a pro-2-group then cd(G) = cd_{2}(G) ≤ n iff H^{n+1}(G, 𝔽_{2}) = 0 (see [28, p. 27]).

One of the basic connections we wish to consider appears in [17, 6.2.8]:

Question 5.1. For which non-formally real fields F with |Ḟ /Ḟ^{2}| = 2^{n} is it true that l(F) + cd(G_{q}) = n + 1?

We shall provide some examples of classes of fields for which equality holds in the equation above. In the remainder of this section we show that equality holds in the following cases: first, if F (in addition to satisfying the hypotheses of the question) is also a C-field, second, if G_{F} ≈ W (n), and third, if F is a local field.

Proposition 5.2. Let F be a C-field which is not formally real, and suppose in addition that |Ḟ /Ḟ^{2}| = 2^{n}. Then l(F) + cd(G_{q}) = n + 1.

Proof. We have shown that for C-fields, G_{F} = G_{F}^{{3}}, so that J = J_{1} and l(F) = 1. So we must show that cd(G_{q}) = n. From the isomorphism between Galois cohomology and the Witt ring given by the Milnor conjecture, it is enough to show that I^{n}F/I^{n+1}F ≠ 0 and that I^{n+1}F/I^{n+2}F = 0, where I^{n}F is the n-th power of the fundamental ideal in the Witt ring. Ware [32] has shown for fields satisfying our hypotheses that I^{n}F ≠ 0, while it follows from a result of Kneser [21, p. 317] and the basic theory of Pfister forms that I^{n+1}F = 0 in this case.

We shall show in what follows that the G_{F}^{[2]}-module J can be studied in a very concrete way. In the following examples we will use the identification of J_{2}/J_{1} as E_{∞}^{1,1} to calculate dim_{𝔽_{2}}(J_{2}/J_{1}) in some interesting special cases.

Example 5.3. Suppose that G_{F} = W (n). Then it follows from [2, 6.12] that dim_{𝔽_{2}}(E_{∞}^{1,1}) = n(n+1)(n−1)/3. In fact in this case the group G_{q} is a free pro-2-group, which means that it has cohomological dimension equal to one. In this case we have identified the module J with the G_{F}^{[2]} module Ω^{2}(𝔽_{2}), which is known to have a socle series of length n. Hence l(F) + cd(G_{q}) = n + 1, as asserted above.

Remark 5.4. To see that the l(Ω^{2}(𝔽_{2})) = n, one can use Remark 5.13 to see that it is enough to prove the surjectivity of N_{F^{(2)}/F}. This surjectivity follows by repeatedly applying surjectivity for quadratic extensions K which connect F to F^{(2)}. The surjectivity for quadratic extensions follows from the fact that each quaternion algebra defined over K will split over K. To see that this last assertion is true, note that G_{q}(F) is free, and hence that G_{q}(K) ⊂ G_{q}(F) is also free.

It seems natural to explore the situation for 2-dimensional groups. We recall

Definition 5.5. A Demuškin group (at the prime 2) is a pro-2-group G which is a twodimensional Poincaré duality group, i.e. H^{2}(G, 𝔽_{2}) = 𝔽_{2}, and the cup product

H^{1}(G, 𝔽_{2}) × H^{1}(G, 𝔽_{2}) → H^{2}(G, 𝔽_{2})

is a perfect pairing.

Example 5.6. Suppose that F is a local field and |Ḟ /Ḟ^{2}| = 2^{n}. Then G_{q} is a Demuškin group (see [28, 4.5], and in particular we have cd G_{q} = 2 and dim_{𝔽_{2}} H^{2}(G_{q}) = 1. It follows from this and [2, 3.12] that in the extension 1 → Φ → G_{F} → G_{F}^{[2]} → 1 we have dim_{𝔽_{2}} Φ = \binom{n+1}{2} − 1. We can now calculate dim_{𝔽_{2}} E_{2}^{1,1} = n·(\binom{n+1}{2}−1) and dim_{𝔽_{2}} E_{2}^{3,0} = n(n+1)(n+2)/6; since H^{3}(G_{q}) = 0, it follows that d_{2}^{1,1} is surjective and that dim_{𝔽_{2}}(E_{∞}^{1,1}) = n(n − 2)(n + 2)/3 = dim_{𝔽_{2}} J_{2}/J_{1}. To make this example more concrete, we note that this calculation of course applies to the special cases F = ℚ_{p} (p an odd prime), F = ℚ_{2}, and F = ℚ_{2}(\sqrt{2}). In the case F = ℚ_{p}, our calculations give dim_{𝔽_{2}} J_{2}/J_{1} = 0, i.e. J_{1} = J_{2}. Another way to phrase this fact is to say that ℚ_{p} is a C-field. In the case F = ℚ_{2}, we obtain dim_{𝔽_{2}} J_{2}/J_{1} = 5, while in the case F = ℚ_{2}(\sqrt{2}), dim_{𝔽_{2}} J_{2}/J_{1} = 16.

We have thus calculated |J_{1}| = |Φ^{∗}| (here Φ^{∗} is the Pontrjagin dual of Φ) as well as |J_{2}/J_{1}| for the case of a local field; in fact our calculations are valid for any field F such that G_{q} is a Demuškin group.

Let G be a Demuškin group and let H be an open subgroup of G. Let r_{G} = dim_{𝔽_{2}} H^{1}(G) and r_{H} = dim_{𝔽_{2}} H^{1}(H). Then one can verify (see [28, p. 44, ex. 6]) that

r_{H} − 2 = [G : H](r_{G} − 2) .

Conversely, this property characterises Demuškin groups (see [15]). We use this characterization to prove

Proposition 5.7. If G_{q} is a Demuškin group and |Ḟ /Ḟ^{2}| = 2^{n}, then

dim_{𝔽_{2}} J = 2^{n}(n − 2) + 2

Proof. Suppose that |Ḟ /Ḟ^{2}| = 2^{n}. Let H = G_{F}^{(2)} := Gal(F_{q}/F^{(2)}). Then H is an open subgroup of G_{q} and [G_{q} : H] = 2^{n}. Therefore from the above we obtain r_{H} = 2^{n}(n − 2) + 2. However from Kummer theory we know that r_{H} = dim_{𝔽_{2}} (Ḟ^{(2)}/(Ḟ^{(2)})^{2}) = dim_{𝔽_{2}} J.

To further our understanding of the G_{F}^{[2]}-module J we look again at some of the examples above:

Example 5.8. Let F = ℚ_{2}. Then Ḟ /Ḟ^{2} = ⟨[−1], [2], [5]⟩, so n = 3 and dim_{𝔽_{2}} J = 2^{3}(3 − 2) + 2 = 10. Notice that this is the same as dim_{𝔽_{2}} Φ + dim_{𝔽_{2}} J_{2}/J_{1} = 6 − 1 + 5. In particular l(ℚ_{2}) = 2, and so the desired equality holds in this case.

Example 5.9. Let F = ℚ_{2}(\sqrt{2}), so n = 4. Then dim_{𝔽_{2}} J = 16 · 2 + 2 = 34. We know already 5.6 that dim_{𝔽_{2}} Φ = \binom{5}{2} − 1 = 9 and that dim_{𝔽_{2}} J_{2}/J_{1} = 16. If the equality holds in this case, then we have l(F) = 5 − 2 = 3, so that J = J_{3} and dim_{𝔽_{2}} J_{3}/J_{2} = dim_{𝔽_{2}} J/J_{2} = 34 − 25 = 9.

Example 5.10. Suppose that F is a local field with |Ḟ /Ḟ^{2}| = 2^{n}. Then we can compute dim_{𝔽_{2}}(J/J_{2}):

dim_{𝔽_{2}}(J/J_{2}) = dim_{𝔽_{2}} J − dim_{𝔽_{2}} J_{2}/J_{1} − dim_{𝔽_{2}} J_{1}
= 2^{n}(n − 2) + 2 − \frac{n(n − 2)(n + 2)}{3} − \binom{n + 1}{2} + 1
= 2^{n}(n − 2) + 3 − \frac{n(n − 2)(n + 2)}{3} − \binom{n + 1}{2}.

We will now make a more in-depth analysis of the module J. We recall the following basic fact from local class field theory:

Lemma 5.11. Let F be a local field such that |Ḟ /Ḟ^{2}| = 2^{n}. If N_{F^{(2)}/F} : Ḟ^{(2)} → Ḟ is the usual norm map then its image lies in Ḟ^{2}. If F ⊂ K ⊂ F^{(2)} is a proper quadratic extension of F, then the image of N_{F^{(2)}/K} : Ḟ^{(2)} → K̇ does not lie in K̇^{2}.

Proof. From local class field theory (see [14]) we know that there is a natural isomorphism Gal(F^{(2)}/L) ≅ L̇/ Im N_{F^{(2)}/L} for any intermediate extension F ⊂ L ⊂ F^{(2)}. If L = F we obtain the first statement. For L = K we observe (see [21], page 202) that for any proper quadratic extension K of F, |K̇/K̇^{2}| ≥ |Ḟ /Ḟ^{2}|; however | Gal(F^{(2)}/K)| < | Gal(F^{(2)}/F)| = |G_{F}^{[2]}|. We conclude that Im N_{F^{(2)}/K} cannot be contained in K̇^{2}.

This relates to our length invariant via the following

Proposition 5.12. Let F be a field such that |Ḟ /Ḟ^{2}| = 2^{n}. Then, in the socle series J_{1} ⊂ J_{2} ⊂ ⋯ ⊂ J_{l(F)}, the submodule J_{n−1} is equal to the kernel of the homomorphism N : J → Ḟ /Ḟ^{2} induced by the norm map above.

Remark 5.13. The proposition above can be extended to a characterization of the complete socle series {J_{i}}; a proof by induction, using [16, p. 133], which we omit, shows that J_{i} = {[j] ∈ J | N_{F^{(2)}/L}(j) = [1] ∈ L̇/L̇^{2} for all L with [F^{(2)} : L] = 2^{i+1}}.

Proof of Proposition 5.12. Let F ⊂ L ⊂ F^{(2)} denote an extension of F such that [F^{(2)} : F] = 2^{k}. Consider Gal(F^{(2)}/L); then by restriction J will also be a module over this group. Denote the usual module-theoretic norm map by T_{L} : J → J. From the definition of the socle series for J it follows that J_{k−1} = ∩ ker T_{L}, where the intersection is taken over all extensions as above, of co-degree 2^{k}. Let L denote a proper quadratic extension of F. The norm map T_{L} is induced from the composition of the field-theoretic norm map N_{F^{(2)}/L}: Ḟ^{(2)} → L̇ with the maps c_{L}: L̇ → L̇/L̇^{2} and i_{L/F} : L̇/L̇^{2} → J (the latter induced by the inclusion L ⊂ F^{(2)}). Hence if [j] ∈ J, we see that [j] ∈ ker T_{L} if and only if c_{L}(N_{F^{(2)}/L}(j)) ∈ ker i_{L/F}. However from Kummer theory (see [5], page 21, theorem 3) we have that ker i_{L/F} = Ḟ L̇^{2}/L̇^{2}, and so T_{L}(j) = 0 if and only if N_{F^{(2)}/L}(j) ∈ Ḟ L̇^{2}. However, from [21], theorem 3.4, pp. 202, we know that μ ∈ L̇ belongs to Ḟ L̇^{2} if and only if N_{L/F} (μ) ∈ Ḟ^{2}. Therefore, from the transitivity property of the norm we see that T_{L}([j]) = 0 if and only if N_{F^{(2)}/F} (j) ∈ Ḟ^{2}.

From this (5.12) we obtain

Corollary 5.14. Let F denote a local field with |Ḟ /Ḟ^{2}| = 2^{n}; then l(F) = n − 1.

Proof. From our description of J_{n−1} and the triviality of the map induced by the norm, we infer that J = J_{n−1}, whence l(F) ≤ n − 1. Taking a quadratic extension F ⊂ K ⊂ F^{(2)} we see that in the socle series for J as a Gal(F^{(2)}/K)-module, J_{n−2} ≠ J. Hence l(F) = n − 1.

This can be restated as

Theorem 5.15. Let F denote a local field with |Ḟ /Ḟ^{2}| = 2^{n}. Then l(F) + cd(G_{q}) = n + 1.

It seems rather complicated to verify the relationship above for other types of fields. We briefly outline a more cohomological approach.

Given any finitely generated 𝔽_{2}G_{F}^{[2]}-module M, there is a minimal power I^{t} of the augmentation ideal (which is nilpotent) such that I^{t}M = 0; t = λ(M) is called the Loewy length of M. Note that if M ≠ 0, then 1 ≤ λ(M) ≤ n + 1. Moreover M is a trivial G_{F}^{[2]}-module if and only if λ(M) = 1. The length of the socle series of a module is equal to the Loewy length of its dual. Consider the following result, which follows from a theorem due to G. Carlsson ([13]):

Proposition 5.16. Suppose that H and G are pro-2-groups, which are topologically finitely generated, G of finite (continuous) cohomological dimension k at the prime 2, and which have finite total mod 2 cohomology. Assume in addition that E is an elementary abelian 2-group of rank equal to n, and that

1 → H → G → E → 1

is an extension. Then it follows that

λ(H^{1}(H, 𝔽_{2}) + ⋯ + λ(H^{k}(H, 𝔽_{2})) ≥ n.

In particular if H and G are 2-dimensional Poincaré duality groups, H^{1}(H, 𝔽_{2}) is self-dual, λ(H^{1}(H, 𝔽_{2})) ≥ n − 1, and therefore l(H^{1}(H, 𝔽_{2})) ≥ n − 1.

As an immediate consequence of the above we obtain a different proof of the inequality l(F) ≥ n − 1 for fields F such that G_{q} is Demuškin. As indicated by this cohomological method, the filtration lengths of higher cohomology groups will probably play a role in any generalization of this question to groups of larger cohomological dimension.

6. The Cohomology of V (2)

In this section we study the cohomology of the universal V-group V (2) defined previously. We will first describe some basic facts about V (n) expressing it in terms of certain group extensions which will be useful in computing and interpreting cohomology. Our point of view will closely follow the analysis of W-groups made in [2], so we will be brief. Many of the calculations described below can be done partially or entirely using a computer algebra system such as Magma [8], so almost all detailed justifications are omitted.

We begin by taking the standard surjection from the free group on n-generators F_{n} onto the elementary abelian group E_{n} = (ℤ/2ℤ)^{n}, with kernel the free group on 2^{n}(n−1)+1 generators. Associated to this we have the free abelianized extension 1 → M → X(n) → E_{n} → 1 where M is a ℤE_{n} lattice of rank as above. It is elementary to verify (see [20] for details) that, as a ℤE_{n}– module, M is isomorphic to Ω^{2}(ℤ), the second dimension shift of the trivial module. Factoring out the submodule 2M, we recover an expression for V (n) as an extension of finite groups, 1 → M/2M → V (n) → E_{n} → 1. This has a simple interpretation: the module M/2M ≅ Ω^{2}(𝔽_{2}) has a unique non zero class in its second cohomology group H^{2}(E_{n}, Ω^{2}(𝔽_{2})) = 𝔽_{2}; the group V (n) realizes this extension class. Similarly, the group X(n) is a Bieberbach group (see [35]) corresponding to the canonical generator in H^{2}(E_{n}, Ω^{2}(ℤ)) ≅ ℤ/|E_{n}|; it is also expressed as an extension 1 → 2M → X(n) → V (n) → 1.

The comments above show that the group V (n) acts freely on a 2^{n}(n − 1) + 1-dimensional torus, with quotient the classifying space of X(n). In turn this space can be obtained as the orbit space of a free E_{n}-action on such a torus. The cohomology of these euclidean space forms is not easy to compute, and we shall see that even the case n = 2 poses some interesting technical problems.

We should also point out that from the definition of the V-groups we can also express it as an extension 1 → U_{r} → V (n) → W (n) → 1 where U_{r} is a subgroup isomorphic to (ℤ/2ℤ)^{r}, r = 2^{n}(n−1)−n−\binom{n}{2}+1. An important thing to note is that the rank of the largest elementary abelian subgroup in V (n) is precisely equal r + n + \binom{n}{2}, or in other words the rank of V (n) differs from the rank of W (n) exactly by the quantity r.

We will now concentrate on the case n = 2. Our first result is about the group X(2): we study its mod 2 cohomology via the spectral sequence for the extension 1 → ℤ^{5} → X(2) → ℤ/2ℤ × ℤ/2ℤ → 1. The ℤ/2ℤ × ℤ/2ℤ-module structure of the kernel is known, so we can obtain the ℤ/2ℤ × ℤ/2ℤ-module structure of the cohomology from the fact that H^{∗}(ℤ^{5}, 𝔽_{2}) is an exterior algebra. Let k = 𝔽_{2} denote the coefficient field, then this exterior algebra can be written as a sum of indecomposable k[ℤ/2ℤ × ℤ/2ℤ]-modules as follows:

Degree Decomposition
0  k
1  Ω^{−2}k
2  Ω^{1}k ⊕ Ω^{1}k ⊕ F
3  Ω^{−1}k ⊕ Ω^{−1}k ⊕ F
4  Ω^{2}k
5  k

From the table it is straightforward to determine the E_{2}-term of the spectral sequence and even the d_{2}-differential. It turns out that there are no further possible differentials, so the spectral sequence collapses at E_{3}. The next few paragraphs sketch a proof of this fact and record some immediate consequences.

The E_{2}-term is not hard to understand as a H^{∗}(ℤ/2ℤ × ℤ/2ℤ, 𝔽_{2})-module since all the modules appearing in the table are either free, or dimension shifts of the trivial module. Furthermore, the d_{2}^{∗,1}-differential realizes the cohomology isomorphism H^{∗}(ℤ/2ℤ × ℤ/2ℤ, Ω^{−2}k) ≈ H^{∗+2}(ℤ/2ℤ × ℤ/2ℤ, k), since d_{2}^{0,1} is an isomorphism by the definition of X(2) (specifically, the extension class). By duality it follows that the d_{2}^{∗,5}-differential realizes the cohomology isomorphism H^{∗}(ℤ/2ℤ × ℤ/2ℤ, k) ≈ H^{∗+2}(ℤ/2ℤ × ℤ/2ℤ, Ω^{2}k). Since we know that the classifying space for X(2), and hence its cohomology, is finite-dimensional, it follows that the d_{2}^{∗,3}-differential realizes the cohomology isomorphism H^{∗}(ℤ/2ℤ × ℤ/2ℤ, Ω^{−1}k ⊕ Ω^{−1}k) ≈ H^{∗+2}(ℤ/2ℤ × ℤ/2ℤ, Ω^{1}k ⊕ Ω^{1}k).

We now note that all this implies

Proposition 6.1. In the spectral sequence for the extension 1 → ℤ^{5} → X(2) → ℤ/2ℤ × ℤ/2ℤ → 1, E_{3}^{p,q} = 0 for p > 1, and E_{3} = E_{∞}.

A little more attention to detail gives also

Proposition 6.2. The Poincaré Series for the cohomology of X(2) is 1+2t+5t^{2}+5t^{3}+2t^{4}+t^{5}.

A method similar to that of the previous section could be used to study the cohomology of V (2). However, this group has order only 2^{7}, and its cohomology can thus be studied in great detail with a computer. In particular, information on this group is available at Carlson’s well-known web site [12].

As the details of the calculation are complicated, we present only an outline of the work necessary to determine by hand calculation the cohomology of V (2). The main reason for outlining this work is to point out the existence of a phenomenon which the careful reader will already have noted in the previous section on X(2).

Now let us sketch the calculation. Studying the cohomology of V (2) via the spectral sequence for the extension 1 → (ℤ/2ℤ)^{5} → V (2) → ℤ/2ℤ × ℤ/2ℤ → 1, we must write the symmetric algebra of Ω^{−2}k as a direct sum of indecomposable modules. If we write P for the direct sum of the three nontrivial permutation ℤ/2ℤ × ℤ/2ℤ-modules which are not free, and F for the free ℤ/2ℤ × ℤ/2ℤ-module of rank 1, the multiplicities of the various modules in the symmetric algebra can be given in terms of Poincaré series:

Module  Poincaré Series
k  (1 − t^{4})^{−2}(1 + t^{3})
Ω^{−2}k  (1 − t^{4})^{−2}t
Ω^{2}k  (1 − t^{4})^{−2}t^{2}
P  (1 − t^{4})^{−2}(1 − t^{2})(t^{2} + t^{3} + t^{4} + t^{5})
F  (1 − t^{4})^{−2}(1 − t)^{−2}[(1 − t)^{−1}t^{2} + 4t^{3} + 4t^{5} + (1 − t)^{−1}3t^{6}]

It turns out that, up to projective summands, S^{2j+1}Ω^{−2}k ≈ Ω^{−2}S^{2j}Ω^{−2}k, and that the differential d_{2}^{i,2j+1} is the natural isomorphism H^{i}(ℤ/2ℤ × ℤ/2ℤ, S^{2j+1}) ≈ H^{i+2}(ℤ/2ℤ × ℤ/2ℤ, S^{2j}). (To study this differential in an ad hoc manner one can use the restrictions to index 2 subgroups of V (2)—all of these are isomorphic to ((ℤ/2ℤ)^{2} × (ℤ/2ℤ)^{2}) ⋊ ℤ/4ℤ, and these groups are well understood since they are not far from being wreath products; in fact the cohomology of these groups is detected on abelian subgroups.) As in the previous section, the spectral sequence associated to the extension collapses at E_{3} and E_{3}^{p,q} = 0 if p > 1.

Remark 6.3. The interest of the method above lies in the odd coincidences that arise in the computations—that Ω^{−2}S^{2j} should be isomorphic to S^{2j+1}, and that the differential d_{2} should contain the associated cohomology isomorphism. A similar phenomenon for the group X(2) was observed in the previous section. It is natural to wonder if these facts can be obtained by some simple method that does not involve ad hoc computations.

In any case, whether from a computer analysis, or following the outline above, one can obtain a number of facts about the cohomology of V (2), which are catalogued below.

Proposition 6.4. The Poincaré series of H^{∗}(V (2)) is

\frac{1 − x + x^{2}}{(1 − x)^{3}(1 − x^{2})^{2}} ,

which expands to 1+2x+6x^{2} +11x^{3} +22x^{4} +36x^{5} +60x^{6} +90x^{7} +135x^{8} +190x^{9} +266x^{10} +· · ·.

Proposition 6.5. H^{∗}(V (2)) is detected on abelian subgroups.

Remark 6.6. Note that the proposition above does not claim detection on elementary abelian subgroups.

We end our analysis of the groups V (n) on a topological note. One of the key properties of finite 2-groups with central involutions is that they act freely on a product of s spheres, where s is the rank of the largest elementary abelian subgroup. This number is known to be minimal for this property and interesting results can be derived from this (see [4]). One can prove that the universal V-groups also satisfy this, whence they seem to be rather special. In fact one can use the extension 1 → U_{r} → V (n) → W (n) → 1 to prove:

Proposition 6.7. Let s = dim Ω^{2}(𝔽_{2}) denote the rank of the largest elementary abelian subgroup in V (n). Then there exist 2^{n}-dimensional real representations Z_{1}, …, Z_{s} of V (n) such that its diagonal action on the product of associated linear spheres X = 𝕊(Z_{1}) × ⋯ × 𝕊(Z_{s}) is free.

Proof. Let x_{1}, …, x_{r} denote a basis for the subgroup U_{r}. For each i one can construct a 1– dimensional real representation of the Frattini subgroup of V (n) which restricts non-trivially to < x_{i} >. This can then be induced to yield a 2^{n}-dimensional representation Z_{i} for V (n). Doing the analogous construction for W (n) and pulling back to V (n), we obtain the representaions Z_{r+1}, …, Z_{s}. It is elementary to verify that every involution in V (n) must acts freely on the product of associated spheres; indeed the unique maximal elementary abelian subgroup satisfies this by construction.

7. Appendix : An Elementary Proof of Half of Theorem 4.1

In this section we will provide an elementary proof of “half” of Theorem 4.1 (i.e. the proposition below), and in doing so develop a connection between J_{2}/J_{1} and certain triple commutators.

Proposition 7.1. If {[j_{i_{k}}] | 1 ≤ k ≤ l} is a finite set of elements in J_{1}, and there is an element [j] ∈ J such that

1. [σ_{i_{k}}(j)/j] = [j_{i_{k}}] for each k ∈ {1, …, l} and

2. [σ_{u}(j)] = [j] for each u ∉ {i_{1}, …, i_{l}}, then

∑_{k=0}^{l}[a_{i_{k}}]d_{2}^{0,1} ([j_{i_{k}}]) = 0 .

Remark 7.2. The observant reader will note that d_{2}^{0,1} is merely a notation for a homomorphism determined by our W-group, and that we make no real use of spectral sequences.

Since this is an alternate proof, for the sake of simplicity we will assume |Ḟ /Ḟ^{2}| = 2^{n}; this means that we can simplify the notation by assuming that {i_{1}, …, i_{l}} = {1, …, n}. We leave to the reader the minor modifications necessary for the case in which Ḟ /Ḟ^{2} is infinite. Recall from the notational conventions in the proof of Theorem 4.1 that to each [j] ∈ J_{2}/J_{1} we can associate an element λ_{j} ∈ H^{1}(G_{F}^{[2]}) ⊗ J_{1}, which we shall assume to have the form: λ_{j} = ∑_{k=1}^{n}[a_{k}]⊗[j_{k}]. With the notation of this paragraph, equations 1 and 2 of the proposition above become [σ_{k}(j)/j] = [j_{k}] ∈ J_{1} for each k = 1, 2, …, n.

Notation 7.3. F_{j} := F^{(3)}(\sqrt{j}), which is a Galois extension of F, and G_{j} := Gal(F_{j}/F).

We will identify the elements of J_{1} with “k-invariants of G_{F} ” via d_{2}^{0,1}.

Notation 7.4. In the situation above, if in the expression of [j] ∈ J_{1} as a sum of monomials, the coefficient of [a_{i}][a_{j}] is nonzero, we will say: “[a_{i}][a_{j}] enters the expression of [j] ∈ J_{1}”, or more concisely “[a_{i}][a_{j}] enters [j].”

We will now use the Kummer pairing J_{1} × Φ → μ_{2}, given by ⟨[j], γ⟩ = γ(\sqrt{j})/\sqrt{j}. From the discussion of this pairing in [24, pp. 42–48] we have:

Lemma 7.5. Let 1 ≤ i ≤ k ≤ n, then 1. ⟨[j], [σ_{i}, σ_{k}]⟩ = −1 iff [a_{i}][a_{k}] enters [j].

2. ⟨[j], σ_{i}^{2}⟩ = −1 iff [a_{i}][a_{i}] enters [j].

Notation 7.6. We denote by θ the nontrivial element of the kernel of quotient map G_{j} ↠ G_{F}.

Let σ̂_{k}, k = 1, …, n be extensions of the generators σ_{k} of G_{F} to G_{j}. Let {σ̂_{i}, σ̂_{k}, σ̂_{l}}, 1 ≤ i < k < l ≤ n, be a triple consisting of generators of G_{j}. A direct calculation proves:

Lemma 7.7. Using the terminology of 7.4, we have the following identities:

1. [σ̂_{i}, [σ̂_{k}, σ̂_{l}]] = θ iff [a_{k}][a_{l}] enters [j_{i}].

2. [σ̂_{i}, σ̂_{k}^{2}] = θ iff [a_{k}][a_{k}] enters [j_{i}].

3. [a_{i}][a_{i}] does not enter [j_{i}].

Another combinatorial lemma which we will need is:

Lemma 7.8. For σ, τ, γ elements of G_{j}, we have the following additional identities: 1. [σ, [τ, γ]][τ, [γ, σ]][γ, [σ, τ]] = 1.

2. [σ, τ]^{2} = 1 and [σ, τ] = [τ, σ]

3. [σ^{2}, τ] = [σ, [σ, τ]] = [τ, σ^{2}]

Observe that part 1. of this lemma is the Jacobi identity, which is valid in any metabelian group (see [6]).

Proof of Proposition 7.1. Suppose that [j] ∈ J is as in the statement of our Proposition. In order to show that ∑_{β=1}^{n}[a_{β}]d_{2}^{0,1}([j_{β}]) = 0 it is enough to observe that for each monomial [a_{i}][a_{k}][a_{l}] that can occur in the expression of the left hand side as an element of H^{3}(G_{F}^{[2]}), the corresponding coefficient is zero. This can be proved using a case by case analysis, in which division into cases depends on the multiplicities of the [a_{i}] which appear in the monomial. We limit ourselves to providing a complete analysis for the case where all the multiplicities are 1; the other cases are very similar and left to the reader.

Assume that [1 ≤ i < k < l ≤ n] then the term [a_{i}][a_{k}][a_{l}] ∈ H^{3}(G_{F}^{[2]}) can occur as the summand of the following terms of our sum ∑_{β=1}^{n}[a_{β}]d_{2}^{0,1}([j_{β}]):

[a_{i}]d_{1}^{0,1}[j_{i}] ,  [a_{k}]d_{2}^{0,1}[j_{k}]  and [a_{l}]d_{2}^{0,1}[j_{l}].

From Lemma 7.7.1 we see that when this term does occur, one of our triple commutators [σ̂_{i}, [σ̂_{k}, σ̂_{l}], [σ̂_{k}, [σ̂_{i}, σ̂_{l}]], or [σ̂_{l}, [σ̂_{i}, σ̂_{k}]] will be θ and not the identity. However from the identity 7.8.1:

[σ̂_{k}, [σ̂_{i}, σ̂_{l}]][σ̂_{i}, [σ̂_{k}, σ̂_{l}]][σ̂_{l}, [σ̂_{i}, σ̂_{k}]] = 1

and the relation θ^{2} = 1 we see that the term [a_{i}][a_{k}][a_{l}] occurs in our sum ∑_{β=1}^{n}[a_{β}]d_{2}^{0,1}[j_{β}] either zero or two times.

This completes the analysis of the case when all the multiplicities are 1; an analysis of the other cases proves that all terms [a_{i}][a_{k}][a_{l}] occur an even number of times in the sum and thus we have

∑_{β=1}^{n}[a_{β}]d_{2}^{0,1}([j_{β}]) = 0

as desired.

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Mathematics Department, University of Wisconsin, Madison, Wisconsin, 53706

  E-mail address: adem@math.wisc.edu

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Mathematics Department, University of Wisconsin, Madison, Wisconsin, 53706

  E-mail address: dikran@math.wisc.edu

Mathematics Department, University of Western Ontario, London, Ontario, Canada N6A 5B7

  E-mail address: minac@math.uwo.ca.edu