Near-final version. November 2002. Comments appreciated

                                                    http://www.math.uiuc.edu/K-theory/0589/

GALOIS COHOMOLOGY OF CERTAIN FIELD EXTENSIONS AND THE DIVISIBLE CASE OF MILNOR–KATO CONJECTURE

LEONID POSITSELSKI

Let F be a field and m ⩾ 2 be an integer not divisible by the characteristic of F. Consider the absolute Galois group G_{F} = Gal(\overline{F} /F), where \overline{F} denotes the (separable) algebraic closure of F. The famous Milnor–Kato conjecture [6, 5] claims that the natural homomorphism of graded rings

K_{n}^{M}(F) ⊗ ℤ/m ⟶ H^{n}(G_{F} , μ_{m}^{⊗n})

from the Milnor K-theory of the field F modulo m to the Galois cohomology of F with cyclotomic coefficients is an isomorphism (here, as usually, μ_{m} denotes the group of m-roots of unity in \overline{F}). One can see that it suffices to verify this conjecture in the case when m is a prime number.

Several years ago V. Voevodsky outlined his approach to the Milnor–Kato conjecture in the preprint [10]. The first step of his argument dealt with a prime number ℓ, a field F having no finite extensions of degree prime to ℓ, and an integer n ⩾ 1 such that the group K_{n+1}^{M}(F) is ℓ-divisible. Assuming that the conjecture holds in degree less or equal to n for m = ℓ and any field containing F, Voevodsky was proving that H^{n+1}(G_{F} , ℤ/ℓ) = 0. The main goal of this paper is to give a simplified elementary version of Voevodsky’s proof of this step. In particular, we do not need to consider fields transcendental over F and we make no use of motivic cohomology at all. Our main result is formulated as follows.

Theorem 1. Let ℓ be a prime number, n ⩾ 1 be an integer, and F be a field of characteristic not equal to ℓ, having no finite extensions of degree prime to ℓ. Suppose that the homomorphism K_{n}^{M}(L)/ℓK_{n}^{M}(L) ⟶ H^{n}(G_{L}, ℤ/ℓ) is an isomorphism for any finite extension L of the field F. Furthermore, assume that K_{n+1}^{M}(F) = ℓK_{n+1}^{M}(F). Then one has H^{n+1}(G_{F} , ℤ/ℓ) = 0.

We deduce the above theorem from two results, one related to the Milnor K-theory of fields and the other to the cohomology of Galois groups. The first one is but a version of a lemma of Suslin [9], which was also used in [10]. We only rewrote it for the Milnor K-theory groups modulo ℓ. The second statement is essentially the “relative conjecture” of Bloch and Kato [3]. It is obvious for ℓ = 2 (when it holds for any pro-finite group). For odd primes ℓ we show that the desired cohomological sequence is exact provided that certain Bockstein homomorphisms vanish. This is the key point of this paper.

Proposition 2. Let m ⩾ 2 be an integer, G be a pro-finite group, and H ⊂ G be a normal subgroup such that Σ = G/H is a cyclic group of order m. Let T_{m^{2}} be a free module over ℤ/m^{2} endowed with an action of G, and let T_{m} = mT_{m^{2}} = T_{m^{2}}/mT_{m^{2}} be the corresponding G-module over ℤ/m. Assume that for a certain n ⩾ 0 both Bockstein homomorphisms

H^{n}(G, T_{m}) ⟶ H^{n+1}(G, T_{m}) and H^{n}(H, T_{m}) ⟶ H^{n+1}(H, T_{m})

vanish. Then the following sequence of cohomology groups is exact

H^{n}(G, T_{m}) ⟶^{res} H^{n}(H, T_{m})_{Σ} ⟶^{cor} H^{n}(G, T_{m})

⟶^{u ∪} H^{n+1}(G, T_{m}) ⟶^{res} H^{n+1}(H, T_{m})^{Σ} ⟶^{cor} H^{n+1}(G, T_{m}),

where u ∈ H^{1}(G, ℤ/m) is the class corresponding to the character G ⟶ Σ ≃ ℤ/m. (The letter Σ in the upper or lower index denotes the invariants or the coinvariants with respect to a natural action of the group Σ.)

In sections 3 and 5 of this paper we apply the same techniques to obtain further exact sequences of Galois cohomology for cyclic, biquadratic, and dihedral field extensions. In particular, our method proves the biquadratic exact sequences conjectured by Merkurjev–Tignol [7] and Kahn [4].

I would like to express my gratitude to Vladimir Voevodsky for posing the problem and for very helpful conversations and explanations. He also always urged me to write down this proof. I am grateful to Alexander Vishik for numerous very helpful discussions. Most of this work was done during my stay at the Institute for Advanced Study in Princeton and the rest at the Institute des Hautes Etudes Scientifiques in Paris, Max-Planck-Intitut für Mathematik in Bonn and the Independent University of Moscow. I wish to thank all these four institutions.

1. Suslin’s Lemma for Milnor K-theory modulo ℓ

The goal of this section is to deduce Theorem 1 from Proposition 2. This is the least elementary part of this paper, to the extent that the proof of Corollary 4 is based on existence of transfer homomorphisms on the Milnor K-theory groups. We will only use the transfer maps for extensions of degree ℓ of fields having no extensions of degree prime to ℓ. The existence of transfers for such field extensions was established in [1]; their basic properties are the base change and the projection formula.

Let us fix a prime number ℓ, a number n ⩾ 1, and a field F with char F ≠ ℓ, having no finite extensions of degree prime to ℓ. Introduce the notation k_{m}(K) = K_{m}^{M}(K)/ℓ for the Milnor K-theory groups modulo ℓ of a field K. For a field extension K ⊂ L we will denote by

i_{L/K}: k_{m}(K) ⟶ k_{m}(L) and tr_{L/K}: k_{m}(L) ⟶ k_{m}(K)

the inclusion and transfer homomorphisms.

Lemma 3. Assume that for any finite field extensions F ⊂ K ⊂ L with [L : K] = ℓ the sequence of Milnor K-theory groups

k_{n}(K) ⟶^{i} k_{n}(L)_{Σ} ⟶^{tr} k_{n}(K), Σ = Gal(L/K)

is exact. Further assume that [E : F] = ℓ and the map tr_{E/F} : k_{n}(E) ⟶ k_{n}(F) is surjective. Then the sequence

k_{n+1}(F) ⟶^{i} k_{n+1}(E)_{Σ} ⟶^{tr} k_{n+1}(F), Σ = Gal(E/F)

is also exact.

Proof. In order to prove that the map tr_{E/F} : k_{n+1}(E)_{Σ}/i_{E/F} k_{n+1}(F) ⟶ k_{n+1}(F) is injective, we will construct a surjective homomorphism

ϕ: k_{n+1}(F) ⟶ k_{n+1}(E)_{Σ}/i_{E/F} (k_{n+1}(F))

for which tr_{E/F} ∘ϕ = id. This is done as follows.

By our assumptions, the map tr_{E/F} : k_{n}(E)_{Σ}/i_{E/F} k_{n}(F) ⟶ k_{n}(F) is an isomorphism. Therefore, we can define a homomorphism

Φ: k_{1}(F) ⊗ k_{n}(F) ⟶ k_{n+1}(E)_{Σ}/i_{E/F} (k_{n+1}(F))

by the rule Φ(b ⊗ α) = \overline{bβ}, where β ∈ k_{n}(E) is such that tr_{E/F} β = α. Let us verify that Φ can be factorized through k_{n+1}(F). It suffices to show that Φ((1 − a) ⊗ α) = 0 whenever α ∈ k_{n}(F) is divisible by the class (a) ∈ k_{1}(F) of an element a ∈ F \ {0, 1}.

Let us first assume that a is not an ℓ-th power in E. Consider the field K = F (\sqrt[ℓ]{a}); let L = EK be the field generated by E and K over F. Then we have tr_{L/K} i_{L/E}β = i_{K/F} tr_{E/F} β = 0, since tr_{E/F} β = α is divisible by (a). By assumption, it follows that i_{L/E}β ∈ i_{L/K}k_{n}(K)+(1−σ)k_{n}(L), where σ is a generator of the group Gal(L/K) = Gal(E/F). Now we have

(1 − a) · β = tr_{L/E}((1 − \sqrt[ℓ]{a})) · β = tr_{L/E}((1 − \sqrt[ℓ]{a}) · i_{L/E}β) ⊂ tr_{L/E}(i_{L/K}k_{n+1}(K) + (1 − σ)k_{n+1}(L)) ⊂ i_{E/F} k_{n+1}(F) + (1 − σ)k_{n+1}(E),

hence Φ((1 − a) ⊗ α) = 0.

It remains to consider the case \sqrt[ℓ]{a} ∈ E. In this case we have ∑_{s=1}^{ℓ} σ^{s}(β) = i_{E/F} tr_{E/F} β = i_{E/F} α = 0 and therefore

(1 − a) · β = ∑_{s=1}^{ℓ} σ^{s}(1 − \sqrt[ℓ]{a}) · β = ∑_{s=1}^{ℓ}(1 − \sqrt[ℓ]{a}) · σ^{−s}(β) = 0

modulo (1 − σ)k_{n+1}(E).

Thus we have constructed the desired map ϕ. Since the field F has no finite extensions of degree less than ℓ, we have k_{n+1}(E) = k_{1}(F) · k_{n}(E) (see [1] and section 4); hence the map ϕ is surjective. It is obvious that tr_{E/F} ∘ϕ = id. □

Corollary 4. Assume that for any finite field extensions F ⊂ K ⊂ L with [L : K] = ℓ the following sequence of Milnor K-theory groups is exact

k_{n}(K) ⟶^{i} k_{n}(L)_{Σ} ⟶^{tr} k_{n}(K) ⟶^{u ·} k_{n+1}(K),

where u ∈ k_{1}(K) is the element corresponding to the cyclic extension K ⊂ L and Σ = Gal(L/K). Further assume that k_{n+1}(F) = 0. Then one has k_{n+1}(E) = 0 for any finite field extension F ⊂ E.

Proof. Proceeding by induction, we only have to consider the case when [E : F] = ℓ. Since k_{n+1}(F) = 0, it follows from our exact sequence that the transfer homomorphism tr_{E/F} : k_{n}(E) ⟶ k_{n}(F) is surjective. Therefore, the conditions of Lemma 3 are satisfied and we can conclude that k_{n+1}(E)_{Σ} = 0. Thus k_{n+1}(E) = (1 − σ)k_{n+1}(E) = (1 − σ)^{ℓ}k_{n+1}(E) = 0. □

Proof of Theorem 1. Let F ⊂ K ⊂ L be finite field extensions with [L : K] = ℓ. We will apply Proposition 2 for m = ℓ, the pro-finite group G = G_{K}, the subgroup H = G_{L}, the modules T_{ℓ^{2}} = μ_{ℓ^{2}}^{⊗n} and T_{ℓ} ≃ ℤ/ℓ over G, and the degree n. It follows from the assumptions of Theorem 1 that the required Bockstein homomorphisms vanish. From Proposition 2 we conclude that the conditions of Corollary 4 are satisfied for the field F. Thus we have k_{n+1}(E) = 0 for any field E finite over F.

Now assume that α is a nonzero element of H^{n+1}(G_{F} , ℤ/ℓ). It is clear that there exist finite field extensions F ⊂ K ⊂ L such that res_{K/F} α ≠ 0, but res_{L/F} (α) = 0 and [L : K] = ℓ. Using Proposition 2 again, we see that res_{K/F} α = u_{L/K} ∪ β for some β ∈ H^{n}(G_{K}, ℤ/ℓ). Since k_{n}(K) ≃ H^{n}(G_{K}, ℤ/ℓ) and k_{n+1}(K) = 0, it follows that res_{K/F} α = 0. This contradiction proves that H^{n+1}(G_{F} , ℤ/ℓ) = 0. □

2. The Six-Term Cohomological Exact Sequence

In this section we develop a rather general setting for exact sequences of cohomology under the assumptions of vanishing of Bockstein homomorphisms. The results below are actually valid for an arbitrary cohomological functor on an abelian category endowed with a central element with zero square, etc. We restrict ourselves to the cohomology of a pro-finite group for the sake of simplicity of exposition only.

Let us fix a pro-finite group G and a number m ⩾ 2. We will use the following notation: A_{2}, B_{2}, etc. will denote free modules over ℤ/m^{2} endowed with a discrete action of G and A_{1}, B_{1}, etc. will be the corresponding G-modules over ℤ/m, defined as A_{1} = mA_{2} = A_{2}/mA_{2}. For a G-module M, we will simply write H^{n}(M) = H^{n}(G, M). The Bockstein homomorphisms corresponding to exact triples

0 ⟶ M_{1} ⟶^{τ} M_{2} ⟶^{π} M_{1} ⟶ 0, τπ = m

will be denoted by β_{M}^{n}: H^{n}(M_{1}) ⟶ H^{n+1}(M_{1}).

Lemma 5. Let 0 → X_{2} ⟶^{i} Y_{2} ⟶^{p} Z_{2} → 0 be an exact triple of G-modules over ℤ/m^{2}, 0 → X_{1} ⟶ Y_{1} ⟶ Z_{1} → 0 be the corresponding exact triple of G-modules over ℤ/m, and ∂_{XZ}: H^{n}(Z_{1}) ⟶ H^{n+1}(X_{1}) be the induced homomorphism of pro-finite group cohomology.

Let h be a homotopy map on the complex X_{2} ⟶ Y_{2} ⟶ Z_{2} such that dh + hd = m · id, where d = (i, p). Then the induced map h_{1} is an endomorphism of degree −1 of the complex X_{1} ⟶ Y_{1} ⟶ Z_{1}. It follows that there is a map h_{XZ}: Z_{1} ⟶ X_{1} such that h_{XZ} ∘ p_{1} = h_{1} : Y_{1} ⟶ X_{1} and i_{1} ∘ h_{XZ} = −h_{1} : Z_{1} ⟶ Y_{1}.

Then the following equality holds: ∂_{XZ} = β_{X} ∘ H(h_{XZ}) − H(h_{XZ}) ∘ β_{Z}.

Proof. The desired equality of homomorphisms of cohomology is to be understood as the corollary to an identity in the G-module extension group Ext_{ℤ/m^{2}[G]}^{1}(Z_{1}, X_{1}), where ∂_{XZ} and the betas represent certain extension classes and h_{XZ} is just a map of modules. This identity in the group of extension classes is verified as follows.

The composition of the Bockstein extension X_{1} ⟶ X_{2} ⟶ X_{1} with h_{XZ}: Z_{1} ⟶ X_{1} is an extension of the form X_{1} ⟶ X_{2} ⊓_{X_{1}} Z_{1} ⟶ Z_{1}, while the composition of Z_{1} ⟶ Z_{2} ⟶ Z_{1} with h_{XZ} is X_{1} ⟶ X_{1} ⊔_{Z_{1}} Z_{2} ⟶ Z_{1} (where ⊓ and ⊔ denote the relative product and coproduct, respectively). We have to check that the difference of these extensions is isomorphic to X_{1} ⟶ Y_{1} ⟶ Z_{1}. The middle term of the desired difference can be defined as the homology module of the sequence

X_{1} ⟶^{(1,1)} (X_{2} ⊓_{X_{1}} Z_{1}) ⊕ (X_{1} ⊔_{Z_{1}} Z_{2}) ⟶^{(1,−1)} Z_{1}.

Let us construct an isomorphism f from Y_{1} to this homology module. Given an element y_{1} ∈ Y_{1}, we choose its preimage y_{2} ∈ Y_{2} and set f(y_{1}) = (h(y_{2}), p_{1}(y_{1})) ⊕ (h_{1}(y_{1}), p(y_{2})). The rest is a straightforward computation. □

Theorem 6. Let G be a pro-finite group, m ⩾ 2 and n ⩾ 0 be some integers, and

0 ⟶ A_{2} ⟶ B_{2} ⟶ C_{2} ⟶ D_{2} ⟶ 0

be a 4-term exact sequence of free ℤ/m^{2}-modules with a discrete action of G in it. Suppose that we are given a homotopy map h in this exact sequence such that dh + hd = m. Assume that the Bockstein maps

β_{X}^{n}: H^{n}(G, X_{1}) ⟶ H^{n+1}(G, X_{1}), X_{1} = mX_{2} = X_{2}/m

vanish for all X = A, B, C, or D and the given n. Then there is a 6-term exact sequence of the form

H^{n}(B_{1} ⊕ D_{1}) ⟶^{(d_{1}, h_{1})} H^{n}(C_{1}) ⟶^{d_{1}} H^{n}(D_{1})

⟶^{ν} H^{n+1}(A_{1}) ⟶^{d_{1}} H^{n+1}(B_{1}) ⟶^{(h_{1}, d_{1})} H^{n+1}(A_{1} ⊕ C_{1}).

The middle arrow ν comes from a certain extension

0 ⟶ A_{1} ⟶ N ⟶ D_{1} ⟶ 0

of discrete G-modules over ℤ/m, whose compositions with the morphisms A_{1} ⟶ B_{1} and C_{1} ⟶ D_{1} are trivial. If the group G acts on the original exact quadruple of modules through its finite quotient group Γ, then N is a ℤ/m[Γ]-module.

Proof. Let L_{2} denote the image of the differential B_{2} ⟶ C_{2}. It is clear that L_{2} is a free module over ℤ/m^{2}. Now our 4-term exact sequence of G-modules consists of two exact triples: A_{2} ⟶ B_{2} ⟶ L_{2} and L_{2} ⟶ C_{2} ⟶ D_{2}. Reducing modulo m, we see that the map h_{1} : C_{1} ⟶ B_{1} defines a homomorphism from the exact triple L_{1} ⟶ C_{1} ⟶ D_{1} to A_{1} ⟶ B_{1} ⟶ L_{1}. The corresponding “intermediate” induced extension provides the desired exact triple A_{1} ⟶ N ⟶ D_{1}.

Now the 6-term sequence is constructed; it remains to check that it’s exact. Let us introduce notation for maps p: B_{2} ⟶ L_{2} and i: L_{2} ⟶ C_{2}; the same maps reduced modulo m will be be denoted by p_{1} and i_{1}. It is easy to see that the homotopy map h induces analogous homotopies on the exact triples A_{2} ⟶ B_{2} ⟶ L_{2} and L_{2} ⟶ C_{2} ⟶ D_{2}. Therefore, both triples satisfy the conditions of Lemma 5. We have connecting maps ∂_{AL}: H^{n}(L_{1}) ⟶ H^{n+1}(A_{1}) and ∂_{LD}: H^{n}(D_{1}) ⟶ H^{n+1}(L_{1}) in the pro-finite group cohomology; the cohomology maps induced by the module maps h_{AL}: L_{1} ⟶ A_{1} and h_{LD}: D_{1} ⟶ L_{1} will be denoted simply by h_{AL} and h_{LD}. According to Lemma 5, there are two identities: ∂_{LD} = β_{L} ∘ h_{LD} − h_{LD} ∘ β_{D} and ∂_{AL} = β_{A} ∘ h_{AL} − h_{AL} ∘ β_{L}. We also know that ν = ∂_{AL} ∘ h_{LD} = −h_{AL} ∘ ∂_{LD}.

Assume that x ∈ H^{n}(D_{1}) and ν(x) = 0. Let us prove that x ∈ d_{1}H^{n}(C_{1}). We have ∂_{AL}h_{LD}(x) = 0, hence h_{LD}(x) = p_{1}(z) for some z ∈ H^{n}(B_{1}). Since p_{1} is induced by a map of modules over ℤ/m^{2}, it commutes with betas. So it follows from our assumptions about vanishing of Bockstein homomorphisms that 0 = p_{1}β_{B}(z) = β_{L}p_{1}(z) = β_{L}h_{LD}(x) = β_{L}h_{LD}(x) − h_{LD}β_{D}(x) = ∂_{LD}(x). Thus x ∈ d_{1}H^{n}(C_{1}).

Next assume that w ∈ H^{n+1}(A_{1}) and d_{1}(w) = 0. Then w = ∂_{AL}(u) for some u ∈ H^{n}(L_{1}), hence w = β_{A}h_{AL}(u) − h_{AL}β_{L}(u) = h_{AL}β_{L}(u). The map i_{1} commutes with betas for the same reason as above, so according to our assumption about Bockstein maps we have i_{1}β_{L}(u) = β_{C}i_{1}(u) = 0. It follows that β_{L}(u) = ∂_{LD}(x) for some x ∈ H^{n}(D_{1}). Thus w = h_{AL}β_{L}(u) = h_{AL}∂_{LD}(x) = ν(−x).

Now assume that y ∈ H^{n}(C_{1}) and d_{1}(y) = 0. Let us prove that y ∈ d_{1}H^{n}(B_{1}) + h_{1}H^{n}(D_{1}). Since d_{1}(y) = 0, we have y = i_{1}(u) with u ∈ H^{n}(L_{1}). Then i_{1}β_{L}(u) = β_{C}i_{1}(u) = 0, hence β_{L}(u) = ∂_{LD}(x) for some x ∈ H^{n}(D_{1}). Therefore, ∂_{AL}(u) = β_{A}h_{AL}(u) − h_{AL}β_{L}(u) = −h_{AL}β_{L}(u) = −h_{AL}∂_{LD}(x) = ∂_{AL}h_{LD}(x). It follows that ∂_{AL}(u − h_{LD}(x)) = 0 and u − h_{LD}(x) = p_{1}(z) for some z ∈ H^{n}(B_{1}). Applying i_{1}, we obtain y = i_{1}p_{1}(z) + i_{1}h_{LD}(x) = d_{1}(z) − h_{1}(x).

Finally, assume that t ∈ H^{n+1}(B_{1}) is such that d_{1}(t) = 0 and h_{1}(t) = 0. Then i_{1}p_{1}(t) = d_{1}(t) = 0, hence p_{1}(t) = ∂_{LD}(x) for some x ∈ H^{n}(D_{1}). We have ∂_{AL}h_{LD}(x) = −h_{AL}∂_{LD}(x) = −h_{AL}p_{1}(t) = −h_{1}(t) = 0, so h_{LD}(x) = p_{1}(z) with z ∈ H^{n}(B_{1}). It follows that p_{1}(t) = ∂_{LD}(x) = β_{L}h_{LD}(x) − h_{LD}β_{D}(x) = β_{L}h_{LD}(x) = β_{L}p_{1}(z) = p_{1}β_{B}(z) = 0 and therefore t ∈ d_{1}H^{n+1}(A_{1}). □

Remark. The middle arrow ν does not depend on the choice of a homotopy map h, provided that h satisfies the conditions of Theorem 6. Indeed, let us consider another homotopy map h′ = h + t, where dt + td = 0. Then the induced maps t_{AL} and t_{LD} commute with Bockstein homomorphisms, and according to the above computations we have ν′ − ν = t_{AL} ∘ ∂_{LD} = t_{AL} ∘ β_{L} ∘ h_{LD} − t_{AL} ∘ h_{LD} ∘ β_{D} = β_{A} ∘ t_{AL} ∘ h_{LD} − t_{AL} ∘ h_{LD} ∘ β_{D} = 0 on H^{n}(D_{1}). If A_{1} and D_{1} are permutational modules (as in all applications below), then t_{AL} ∘ h_{LD} can be lifted to a map D_{2} ⟶ A_{2} and it follows that the extension A_{1} ⟶ N ⟶ D_{1} does not depend on h either.

3. The Relative Conjecture of Bloch and Kato

In this section we will prove two theorems providing exact sequences for Galois cohomology of cyclic field extensions, both of them generalizations of Proposition 2. The first one will be essentially the “relative conjecture” of Bloch and Kato [3, Conjecture 3.1]. In particular, the proof of Theorem 1 will be completed.

Throughout this section, we will use the following notation: m ⩾ 2 is an integer, G is a pro-finite group, T_{m^{2}} is a free module over ℤ/m^{2} endowed with an action of G, and T_{m} = mT_{m^{2}} = T_{m^{2}}/mT_{m^{2}} is the corresponding G-module over ℤ/m. Furthermore, n ⩾ 0 is another integer, and it is assumed that the Bockstein homomorphisms

H^{n}(H, T_{m}) ⟶ H^{n+1}(H, T_{m}), H ⊂ G

vanish for all relevant open subgroups H in G, including G itself. In applications to Galois cohomology it will be always meant that G = G_{F} is the absolute Galois group of a field F and T_{m^{2}} = μ_{m^{2}}^{⊗n} is the cyclotomic module.

Proposition 7. Let H ⊂ G be a normal subgroup such that Σ = G/H is a cyclic group of an order k dividing m. Then the following sequence of cohomology groups is exact

H^{n}(G, T_{m}) ⟶^{m/k·res} H^{n}(H, T_{m})_{Σ} ⟶^{cor} H^{n}(G, T_{m})

⟶^{u ∪} H^{n+1}(G, T_{m}) ⟶^{res} H^{n+1}(H, T_{m})^{Σ} ⟶^{m/k·cor} H^{n+1}(G, T_{m}),

where u ∈ H^{1}(G, ℤ/m) is the class corresponding to the character G ⟶ Σ ↪ ℤ/m.

Proof. Consider the following very well known 4-term exact sequence of modules over the cyclic group Σ:

0 ⟶ ℤ ⟶^{1+···+σ^{k−1}} ℤ[Σ] ⟶^{1−σ} ℤ[Σ] ⟶^{σ^{i} ⟼1} ℤ ⟶ 0,

where σ is a generator of Σ. Let us construct a homotopy map h_{k} such that dh_{k} + h_{k}d = k. Set f(x) to be the polynomial for which k − (1 + ⋯ + x^{k−1}) = (1 − x)f(x). The leftmost component of h_{k} sends σ^{i} to 1 for all i, the middle part is given by the multiplication with f(σ) in the group ring ℤ[Σ], and the rightmost component sends 1 to 1+⋯+σ^{k−1}. For the purposes of our proof it suffices to put h = m/k ·h_{k}, tensor the above exact quadruple with T_{m^{2}}, and apply Theorem 6. □

The proof of Theorem 1 is now finished.

Proposition 8. Let H ⊂ G be a normal subgroup such that Σ = G/H is a cyclic group of an order k divisible by m. Let Π ⊂ Σ be the cyclic subgroup of order m and K ⊂ G be kernel of the projection G ⟶ Σ/Π. Then the following sequences of cohomology groups are exact

H^{n}(H)_{Σ} ⟶^{cor} H^{n}(K)_{Σ/Π} ⟶^{cor ∘ u∪} H^{n+1}(G) ⟶^{res} H^{n+1}(H)^{Σ} ⟶^{cor} H^{n+1}(K)^{Σ/Π}

and

H^{n}(K)_{Σ/Π} ⟶^{res} H^{n}(H)_{Σ} ⟶^{cor} H^{n}(G) ⟶^{u∪ ∘ res} H^{n+1}(K)^{Σ/Π} ⟶^{res} H^{n+1}(H)^{Σ},

where we write H^{i}(I) = H^{i}(I, T_{m}) for any subgroup I ⊂ G, and u ∈ H^{1}(K, ℤ/m) is the cohomology class corresponding to the character K ⟶ Π ≃ ℤ/m.

Proof. The first of the desired exact sequences holds due to the following exact quadruple Q_{Σ} of permutational Σ-modules

ℤ ⟶ ℤ[Σ] ⊕ ℤ ⟶ ℤ[Σ] ⊕ ℤ[Σ/Π] ⟶ ℤ[Σ/Π],

where the first map is (1 + ⋯ + σ^{k−1}, −m), the third map is (pr_{Π}, −1 + σ̄), and the middle arrow is given by the matrix U with components U_{11} = 1 − σ, U_{12} = 0, U_{21} = pr_{Π}, and U_{22} = 1 + ⋯ + σ̄^{k/m−1}. Here σ is a generator of Σ, by σ̄ we denote its image in Σ/Π, and pr_{Π} is the projection map ℤ[Σ] ⟶ ℤ[Σ/Π]. It is not difficult to construct a homotopy map h with dh + hd = m. Using this exact quadruple, one argues as in the proof of Proposition 7 above. To obtain the second exact sequence of cohomology, one has to use the dual exact quadruple of Σ-modules Hom_{ℤ}(Q_{Σ}, ℤ). □

4. The Bass–Tate Lemma and Applications to Annihilator Ideals

The next theorem provides a more sophisticated version of the classical technique known as the “Bass–Tate lemma” [1].

Theorem 9. Let E/F be a separable extension of fields of degree k. Assume that the field F has no nontrivial finite extensions of degrees less or equal to k/2. Then K_{⩾1}^{M}(E) = ⨁_{i=1}^{∞} K_{n}^{M}(E) is a quadratic module over the ring K_{∗}^{M}(F), i. e., a graded module generated in degree 1 with relations in degree 2.

Lemma 10. Let E = F [t] be a finite extension of fields of degree k, where t ∈ E is a generator. Then any element of the field E can be represented as a fraction of polynomials f(t)/g(t), where f and g are two polynomials of degree ⩽ k/2 each.

Proof. Let V ⊂ E be the vector subspace of all elements of the form f(t), where deg f ⩽ k/2. Then 2 dim V ⩾ k + 1, thus for any element e ∈ E one has V ∩ eV ≠ 0. This simple argument was communicated to the author by Alexander Vishik. □

Proof of Theorem 9. Consider the field of rational functions F (x) in a transcendental variable x. Let D ⊂ F (x)^{∗} be the multiplicative subgroup generated by polynomials of degree ⩽ 1. Let R_{∗} = ⨁_{n=0}^{∞} R_{n} be the graded anti-commutative ring generated by R_{1} = D with relations {f, 1 − f} = 0 for all f, 1 − f ∈ D and {f, −f} = 0 for all f ∈ D. Arguing as in [1, Theorem I.5.1], it is not hard to show that R_{∗} is a free K_{∗}^{M}(F)-module with a basis consisting of 1 and {x − a} for all a ∈ F. Hence the K_{∗}^{M}(F)-module R_{⩾1} is the direct sum of K_{⩾1}^{M}(F) and a free module, so it is quadratic.

Now let us choose an element t ∈ E such that E = F [t]. Let π(x) ∈ F [x] be the irreducible equation of the element t. Obviously, there is a homomorphism p_{π}: R_{∗} ⟶ K_{∗}^{M}(E) assigning {f mod π} to {f} for any f ∈ D. According to Lemma 10, this homomorphism is surjective. Moreover, the ideal J_{∗} = ker p_{π} ⊂ R_{∗} is generated by its first-degree component J_{1}. This follows from the fact that for any e ∈ E \ {0, 1} one can find f ∈ D ⊂ F (x)^{∗} such that 1 − f ∈ D and π(f) = e. Since R_{∗} is anti-commutative, we have

J_{∗} = R_{∗} · J_{1} = (K_{∗}^{M}(F) + K_{∗}^{M}(F) · R_{1}) · J_{1} = K_{∗}^{M}(F) · (J_{1} + R_{1}J_{1}).

Therefore, the K_{∗}^{M}(F)-module J_{∗} is generated by J_{1} and J_{2}, while the module R_{⩾1} is quadratic. Thus the quotient module K_{⩾1}^{M}(E) = R_{⩾1}/J_{∗} is also quadratic. □

For reader’s convenience, the statement of Bass–Tate lemma proper is formulated here, together with an improvement by Karim Becher [2].

Proposition 11. Let E/F be a separable field extension of degree k = ℓ^{s}, where ℓ is a prime number. Assume that either (a) ℓ = 2, or (b) the field F has no finite extensions of degree different from powers of ℓ and less or equal to k/2, or (c) E/F is a tower of extensions of degree ℓ such that the largest proper subfield of E in this tower has no nontrivial finite extensions of degree ⩽ ℓ/2. Then the Milnor ring K_{∗}^{M}(E) is generated in degree ⩽ s as a module over K_{∗}^{M}(F).

Proof. The case (a) is covered by the result of [2]. The case (b) follows from Lemma 10 and [1, Lemma I.5.2]. In the case (c) one argues by induction in s, applying (b) to extensions of degree ℓ. □

From now on and for the rest of this section we will assume that the Milnor–Kato conjecture holds for all relevant fields, primes, and degrees. Under this assumption we will deduce corollaries from Proposition 7, Proposition 8, and Theorem 9.

Corollary 12. Let ℓ be a prime number, m be a power of ℓ, and F be a field containing a primitive root of unity of degree m. Consider the ring H^{∗}(G_{F} , ℤ/m). Let u ∈ H^{1}(G_{F} , Z/m) be an element of additive order k = ℓ^{s}. Assume that the maximal proper subfield of E over F has no nontrivial finite extensions of degree ⩽ ℓ/2. Then the ideal Ann(u) in the ring H^{∗}(G_{F} , ℤ/m) is generated by elements of degree less or equal to s. In particular, if m = ℓ is a prime and F has no nontrivial extensions of degree ⩽ ℓ/2, then Ann(u) is generated in degree 1.

Proof. Let E/F be the cyclic extension of degree k corresponding to the element u. According to Proposition 7, we have an exact sequence of H^{∗}(G_{F} , ℤ/m)-modules

H^{∗}(G_{E}, ℤ/m) ⟶^{cor} H^{∗}(G_{F} , ℤ/m) ⟶^{u ∪} H^{∗}(G_{F} , ℤ/m).

But H^{∗}(G_{E}, ℤ/m) is generated in degree ⩽ s as a module over H^{∗}(G_{F} , ℤ/m); this follows from the Milnor–Kato conjecture and Proposition 11. □

Corollary 13. Let ℓ be a prime number, m and k be powers of ℓ, and F be a field containing a primitive root of unity of degree m. Let L/F be a cyclic extention of degree k. Consider the kernel J of the ring homomorphism H^{∗}(G_{F} , ℤ/m) ⟶ H^{∗}(G_{L}, ℤ/m). Assume that the maximal proper subfield of E over F has no nontrivial finite extensions of degree ⩽ ℓ/2. Then the ideal J in the ring H^{∗}(G_{F} , ℤ/m) is generated by elements of degree less or equal to s + 1, where k/m = ℓ^{s} if k is divisible by m and s = 0 if m is divisible by k.

Proof. The case when k is divisible by m follows from Proposition 8 in the way analogous to the proof of Corollary 12. Namely, one has to use the exact sequence

H^{n}(G_{E}, ℤ/m) ⟶^{cor ∘ u∪} H^{n+1}(G_{F} , ℤ/m) ⟶^{res} H^{n+1}(G_{L}, ℤ/m),

where F ⊂ E ⊂ L is the intermediate extension of degree k/m over F and u ∈ H^{1}(G_{E}, ℤ/m) is the class of the extension L/E. The case when m is divisible by k follows directly from Proposition 7. □

The next corollary is the main result of this section.

Corollary 14. Let ℓ be a prime number and F be a field containing a primitive root of unity of degree ℓ. Assume that F has no nontrivial finite extensions of degree ⩽ ℓ/2 (note that for ℓ = 2 or ℓ = 3 this always holds). Then for any element u ∈ H^{1}(G_{F} , ℤ/ℓ) the ideal Ann(u) in the cohomology ring H^{∗}(G_{F} , ℤ/ℓ) is a quadratic module over the this ring, i. e., it is isomorphic to a graded module with generators and relations, where generators are in degree 1 and relations in degree 2.

Proof. Let E/F be the cyclic extension of degree ℓ corresponding to the class u and Σ be its Galois group. Consider the spectral sequence related to the maximal (ℓ-term) filtration of the G_{F} -module ℤ/ℓ[Σ]. It follows from the exact sequence

H^{n}(G_{E}, ℤ/ℓ) ⟶^{cor} H^{n}(G_{F} , ℤ/ℓ) ⟶^{u ∪} H^{n+1}(G_{F} , ℤ/ℓ) ⟶^{res} H^{n+1}(G_{E}, ℤ/ℓ)

that this spectral sequence degenerates at the term E_{2}^{∗,∗}. In other words, there is a filtration V^{1} ⊃ ⋯ ⊃ V^{ℓ} on the H^{∗}(G_{F} , ℤ/ℓ)-module H^{∗}(G_{E}, ℤ/ℓ) with associated quotient modules of the form V^{1}/V^{2} ≃ ker(u∪), V^{i}/V^{i+1} ≃ ker(u∪)/ im(u∪) for i = 2, …, ℓ − 1, and V^{ℓ} ≃ coker(u∪). From this filtration it follows that the H^{∗}(G_{F} , ℤ/ℓ)-modules H^{⩾1}(G_{E}, ℤ/ℓ) and Ann(u) are quadratic simultaneously. (They are also simultaneously Koszul, see [8].) Now it remains to apply Theorem 9. Alternatively, one can argue as in the proof of Corollary 19 below. □

Conjecture 15. Let ℓ be a prime number and F be a field containing a primitive root of unity of degree ℓ. Then for any element u ∈ H^{1}(G_{F} , ℤ/ℓ) the H^{∗}(G_{F} , ℤ/ℓ)-module Ann(u) ⊂ H^{∗}(G_{F} , ℤ/ℓ) is Koszul. For any cyclic extension E/F of degree ℓ, the H^{∗}(G_{F} , ℤ/ℓ)-module H^{⩾1}(G_{E}, ℤ/ℓ) is Koszul (see [8] for the definition).

5. Biquadratic and Dihedral Field Extensions

Using Theorem 6 and assuming the Milnor–Kato conjecture, one can construct some exact sequences of Galois cohomology for any finite field extension whose Galois group admits a “nontrivial enough” exact quadruple of permutational representations over it. Unfortunately, examples of finite groups with such exact quadruples of representations that I am aware of are very few. Almost all of them are dihedral groups and their subgroups. The latter include cyclic groups and the biquadratic group ℤ/2 × ℤ/2. The goal of this section is to construct those exact quadruples and deduce some corollaries, including the conjectures of Merkurjev–Tignol [7] and Kahn [4] about biquadratic field extensions. Throughout this section we will assume that the Milnor–Kato conjecture holds for ℓ = 2 and the characteristic of our fields is not equal to 2. We will sometimes use the notation H^{i}(F) = H^{i}(G_{F} , ℤ/2).

Let Δ = Δ_{k} be a dihedron of order k, i. e., a regular polygon with k vertices and k edges. We will denote by Δ′ the set of all vertices of Δ and by Δ′′ the set of all its edges. Let Γ = Γ_{k} be the group of all automorphisms (symmetries) of Δ and Σ ⊂ Γ be the subgroup of orientation-preserving automorphisms (rotations). Then Σ is a cyclic group of order k and a normal subgroup of order 2 in Γ.

Assume that k is divisible by 4. Let Π be the (only) subgroup of order 2 in Σ. Then the following quadruple of permutational representations of Γ is exact

(Q_{Γ})  ℤ ⟶ ℤ[Δ′] ⊕ ℤ ⟶ ℤ[Δ′′] ⊕ ℤ[Δ′/Π] ⟶ ℤ[Δ′′/Π],

where the arrows are defined by the formulas below. Choose an arbitrary generator σ ∈ Σ; let σ̄ be its image in Σ/Π. Further, let σ^{±1/2} and σ̄^{±1/2} be the uniquely defined maps Δ′ ⟶ Δ′′ ⟶ Δ′ and Δ′/Π ⟶ Δ′′/Π ⟶ Δ′/Π whose squares are σ^{±1} and σ̄^{±1}, respectively. Denote by pr_{Π} the projection maps modulo Π and by #X the sum of all elements of a set X. Then the first map in this sequence is (#Δ′, −2), the third one is (pr_{Π}, − σ̄^{−1/2} − σ̄^{1/2}), and the middle arrow is given by the matrix U with components U_{11} = σ^{−1/2} + σ^{1/2}, U_{12} = #Δ′′, U_{21} = pr_{Π}, and U_{22} = #Δ′/Π.

Moreover, there exists a chain homotopy h such that dh + hd = 2. Of the three maps constituting h, the leftmost one is given by the pair (pr_{1}, − k/2 − 1), where pr_{1} : ℤ[X] → ℤ sends every x ∈ X to 1. To define the remaining two maps, one needs some preparatory work. Let f(x) be a Laurent polynomial solving the equation (x^{−1/2} + x^{1/2})^{2}f(x) = 1 + ∑_{i=−k/2+1}^{k/2−1} x^{i}. Set ψ = (σ^{−1/2} + σ^{1/2})f(σ) ∈ σ^{1/2}ℤ[Σ] and ψ̄ = ψ mod Π ∈ σ̄^{1/2}ℤ[Σ/Π]. The rightmost piece of h is equal to (1+π, σ̄^{1/2}#Σ/Π− ψ̄), where π is the only nontrivial element of Π. The middle arrow is given by the matrix H with components H_{11} = ψ, H_{12} = 1 + π, H_{21} = pr_{1}, and H_{22} = 0.

Further useful exact quadruples can be obtained by restricting Q_{Γ} to a dihedral subgroup of index 2. The group Γ_{k/2} can be embedded into Γ_{k} in two ways; namely, under the first embedding the action of Γ_{k/2} on Δ′_{k} is transitive, while under the second embedding the action of G_{k/2} on Δ′′_{k} is. These embeddings correspond to the two essentially different ways of defining a regular polygon with k/2 vertices in terms of the given polygon Δ_{k}: one can make edges of the new polygon out of every second edge of the original one, or one can take every second vertice of the original polygon to be the new polygon’s vertices. Taking the restriction of Q_{Γ} with respect to the first of these emdeddings, we obtain an exact quadruple of Γ_{k/2}-modules

(Q_{Γ}^{+}) ℤ ⟶ ℤ[Δ^{+}] ⊕ ℤ ⟶ ℤ[Δ′′] ⊕ ℤ[Δ′′] ⊕ [Δ^{+}/Π] ⟶ ℤ[Δ′/Π] ⊕ ℤ[Δ′′/Π],

where Δ′ and Δ′′ denote the sets of all vertices and edges of the regular polygon Δ_{k/2}, while Δ^{+} denotes the principal homogeneous space for the dihedral group Γ_{k/2} which is defined as the set of all pairs (an edge of Δ_{k/2}; one of its two vertices).

Now let M/F be an extension of fields with a dihedral Galois group Γ. Then the category of intermediate field extensions K/F embeddable into M is equivalent to the category of sets endowed with a transitive action of Γ. Let L′ and L′′ be the intermediate fields corresponding to the Γ-sets Δ′ and Δ′′, let K′ and K′′ be the fields corresponding to Δ′/Π and Δ′′/Π, and E ⊃ F be the normal extension of F corresponding to the subgroup Π ⊂ Γ.

Proposition 16. For any dihedral field extension M/F of degree [M: F] divisible by 8, there is an exact sequence of Galois cohomology of the form

H^{n}(L′′) ⊕ H^{n}(K′) ⟶^{(cor_{L′′/K′′}, cor_{E/K′′} ∘ res_{E/K′})} H^{n}(K′′) ⟶^{cor_{K′′/F} ∘ u_{L′′/K′′} ∪} H^{n+1}(F)

⟶^{res_{L′/F}} H^{n+1}(L′) ⟶^{(cor_{M/L′′} ∘ res_{M/L′}, cor_{L′/K′})} H^{n+1}(L′′) ⊕ H^{n+1}(K′).

For any dihedral extension M/F of degree divisible by 4, there is an exact sequence

H^{n}(L′)⊕H^{n}(L′′)⊕H^{n}(E) ⟶^{cor^{4}} H^{n}(K′)⊕H^{n}(K′′) ⟶^{cor_{K^{(i)}/F} ∘ u_{L^{(i)}/K^{(i)}} ∪} H^{n+1}(F)

⟶^{res} H^{n+1}(M) ⟶^{cor^{3}} H^{n+1}(L′) ⊕ H^{n+1}(L′′) ⊕ H^{n+1}(E).

Proof. The proposition follows from Theorem 6 applied to the above exact quadruples Q_{Γ} and Q_{Γ}^{+}, respectively. There are also dual exact sequences of cohomology, corresponding to the dual exact quadruples Hom_{ℤ}(Q_{Γ}, ℤ) and Hom_{ℤ}(Q_{Γ}^{+}, ℤ), but we will not write them down here. □

Remark. There is yet another exact quadruple of permutational representations of a dihedral group Γ, whose restrictions to the subgroup of rotations and dihedral subgroups of index 2 are also of some interest. Let Δ_{k} be the dihedron of an arbitrary order k and Π ⊂ Σ ⊂ Γ_{k} be a cyclic subgroup of order m. Then there is a self-dual exact quadruple of the form

ℤ[Δ′/Π] ⟶ ℤ[Δ′] ⊕ ℤ ⟶ ℤ[Δ′′] ⊕ ℤ ⟶ ℤ[Δ′′/Π]

if k/m is even (and one should replace Δ′′ with Δ′ if k/m is odd). The first map in this sequence is (#Π, − pr_{1}), the third map is (pr_{Π}, − #Δ′′/Π), and the middle arrow is given by the matrix U with components U_{11} = σ^{−(k/m−1)/2} + ⋯ + σ^{(k/m−1)/2}, U_{12} = #Δ′′, U_{21} = pr_{1}, and U_{22} = m in the above notation. It is not difficult to find a homotopy map h with dh + hd = m.

Let Θ = {1, a, b, c} denote an abelian group isomorphic to ℤ/2×ℤ/2. The group Θ can be embedded into Γ_{4} in two ways: as above, under the first embedding the action of Θ on Δ′ is transitive, while under the second embedding the action of Θ on Δ′′ is. Taking the restrictions on Θ of the exact quadruple Q_{Γ} for Γ = Γ_{4}, we obtain (after one simple cancellation in the second case) two exact quadruples of permutational representations of Θ. They have the form

(Q_{Θ})  ℤ ⟶ ℤ[Θ] ⊕ ℤ ⟶ ℤ[Θ/a] ⊕ ℤ[Θ/b] ⊕ ℤ[Θ/c] ⟶ ℤ^{⊕3}/ℤ

and

ℤ ⟶ ℤ[Θ/a] ⊕ ℤ[Θ/b] ⟶ ℤ[Θ] ⊕ ℤ ⟶ ℤ[Θ/c],

where we denote for simplicity by a the subgroup {1, a} etc. All maps in these sequences can be defined directly in the rather obvious way.

We are interested in the exact sequence of cohomology corresponding to the first of these quadruples. Let L/F be a biquadratic field extension with Galois group Θ. We will consider Θ as a two-dimensional vector space over the field ℤ/2. Notice that there is a canonical isomorphism between Θ and the dual vector space Θ^{∗}, given by the only bilinear form on Θ with the property that (θ, θ) = 0 for all θ ∈ Θ. The three intermediate fields F ⊂ K_{θ} ⊂ L bijectively correspond to nonzero elements a, b, c of the group Θ in a natural way. The six-term exact sequence of Galois cohomology corresponding to the exact quadruple Q_{Θ} has the form

H^{n}(L) ⊕ H^{n}(F)^{⊕3} ⟶^{(cor, res^{3})} ⨁_{θ=a,b,c} H^{n}(K_{θ}) ⟶^{cor^{3}} Θ ⊗_{ℤ/2} H^{n}(F)

⟶^{∪} H^{n+1}(F) ⟶^{res} H^{n+1}(L) ⟶^{cor^{3}} ⨁_{θ=a,b,c} H^{n+1}(K_{θ}),

where the middle arrow Θ⊗H^{n}(F) ⟶ H^{n+1}(F) is defined in terms of the embedding Θ ≃ Hom(Θ, ℤ/2) ⟶ Hom(G_{F} , ℤ/2) ≃ H^{1}(F). This 6-term exact sequence is a part of one of the 7-term exact sequences of Merkurjev–Tignol and one of the 8-term exact sequences of Kahn [7, 4] for a biquadratic extension of fields.

Corollary 17. Both 8-term sequences of Bruno Kahn [4] for Galois cohomology in a biquadratic field extension L/F are exact, provided that the first Bockstein homomorphism H^{n}(E, ℤ/2) ⟶ H^{n+1}(E, ℤ/2) vanishes in the relevant degree n for the five fields E between F and L (namely, F, L, and all three intermediate fields).

Proof. The middle 6-term part of the first of Kahn’s 8-term sequences is written down right above; its exactness follows from Theorem 6 applied to the exact quadruple Q_{Θ} tensored with μ_{4}^{⊗n}. The middle part of the second sequence can be obtained from the dual exact quadruple Hom_{ℤ}(Q_{Θ}, ℤ). Exactness at the remaining terms follows from exactness at the middle terms due to Merkurjev–Tignol and Kahn’s results. □

Corollary 18. Let L/F be a field extension of degree 4 and M/F be its Galois closure. Suppose that the degree [M : F] is a power of 2. Then the kernel J of the restriction map H^{∗}(G_{F} , ℤ/2) ⟶ H^{∗}(G_{L}, ℤ/2) is generated in degrees 1 and 2 as an ideal in H^{∗}(G_{F} , ℤ/2). Moreover, the kernel of the restriction map H^{∗}(G_{F} , ℤ/2) ⟶ H^{∗}(G_{M}, ℤ/2) is also generated in degrees 1 and 2.

Proof. There are only three cases: either L/F is a cyclic extension, or a biquadratic one, or it has the form L′/F for a dihedral extension M/F with Galois group Γ_{4}. The first case follows from Corollary 13, the second from Corollary 17, and the third from Proposition 16 (both statements). Note that in the biquadratic case the ideal J is actually generated by a two-dimensional subspace in H^{1}(G_{F} , ℤ/2). □

Corollary 19. For any field F and any two-dimensional subspace Θ ⊂ H^{1}(G_{F} , ℤ/2) the kernel of the multiplication map Θ ⊗ H^{∗}(G_{F} , ℤ/2) ⟶ H^{∗+1}(G_{F} , ℤ/2) is a quadratic module over the algebra H^{∗}(G_{F} , ℤ/2).

Proof. Clearly, any two-dimensional subspace in H^{1}(F) corresponds to a biquadratic field extension L/F, so we can apply the 6-term exact sequence of Corollary 17. The kernel in question is isomorphic to the quotient module of ⨁_{θ} H^{⩾1}(K_{θ}) by the image of the module H^{⩾1}(L) ⊕ H^{⩾1}(F)^{⊕3}. Since the first module is quadratic (by Theorem 9) and the second module is generated in degree 1 and 2 (by Proposition 11), the quotient module is quadratic. □

Conjecture 20. For any field F, the ideal generated by any two-dimensional subspace Θ ⊂ H^{1}(G_{F} , ℤ/2) in the cohomology algebra H^{∗}(G_{F} , ℤ/2) is a Koszul module (see [8]) over that algebra.

Remark. It is possible to extend the first statement of Corollary 18 to arbitrary field extensions of degree 4 using the following exact quadruple of permutational representations of the symmetric group 𝕊_{4}

ℤ ⟶ ℤ[X_{4}] ⊕ ℤ ⟶ ℤ[X_{6}] ⊕ ℤ ⟶ ℤ[X_{3}],

where X_{4} is a four-element set on which 𝕊_{4} acts in the standard way, X_{6} is the set of all two-element subsets of X_{4}, and X_{3} is the quotient of X_{6} modulo the involution ∗ sending a subset to its complement. The first map in this sequence is (#X_{4}, −2), the third one sends a two-element subset to its class modulo ∗ and 1 to − #X_{3}, and the middle arrow is given by the matrix U with component maps U_{11} : ℤ[X_{4}] ⟶ ℤ[X_{6}] sending each element of X_{4} to the sum of three two-element subsets containing it, U_{12} = pr_{1}, U_{21} = #X_{6}, and U_{22} = 2. It is easy to construct a homotopy map h with dh + hd = 2. In fact, one can generalize the 6-term biquadratic exact sequences to arbitrary field extensions of degree 4 using this exact quadruple!

References

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[3] S. Bloch, K. Kato. P-adic étale cohomology. Publ. Math. IHÉS 63, p. 107–152, 1986.

[4] B. Kahn. On the cohomology of biquadratic extensions. Comment. Math. Helvetici 69, p. 120– 136, 1994.

[5] K. Kato. Galois cohomology of complete discrete valuation fields. In Algebraic K-theory, Part II (Oberwolfach, 1980), Lecture Notes in Math. 967, p. 215–238, 1982.

[6] J. Milnor. Algebraic K-theory and quadratic forms. Inventiones Math. 9, p. 318–344, 1970.

[7] A. S. Merkurjev, J.-P. Tignol. Galois cohomology of biquadratic extensions. Comment. Math. Helvetici 68, p. 138-169, 1993.

[8] L. Positselski. Koszul property and Bogomolov’s conjecture. Preprint, K-theory electronic preprint archives, http://www.math.uiuc.edu/K-theory/0296 , August 1998.

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[10] V. Voevodsky. The Milnor conjecture. Preprint, December 1996.

Independent University of Moscow and MPI für Mathematik, Bonn

 E-mail address: posic@mccme.ru, posic@mpim-bonn.mpg.de