GALOIS MODULE STRUCTURE OF GALOIS COHOMOLOGY AND PARTIAL EULER-POINCARÉ

CHARACTERISTICS

NICOLE LEMIRE, JÁN MINÁČ, AND JOHN SWALLOW

Abstract. Let F be a field containing a primitive pth root of unity, and let U be an open normal subgroup of index p of the absolute Galois group G_{F} of F. Using the Bloch-Kato Conjecture we determine the structure of the cohomology group H^{n}(U, 𝔽_{p}) as an 𝔽_{p}[G_{F} /U]-module for all n ∈ ℕ. Previously this structure was known only for n = 1, and until recently the structure even of H^{1}(U, 𝔽_{p}) was determined only for F a local field, a case settled by Borevič and Faddeev in the 1960s. For the case when the maximal pro-p quotient T of G_{F} is finitely generated, we apply these results to study the partial Euler-Poincaré characteristics of χ_{n}(N) of open subgroups N of T. We show in particular that the nth partial Euler-Poincaré characteristic χ_{n}(N) is determined by only χ_{n}(T) and the conorm in H^{n}(T, 𝔽_{p}).

Let F be a field containing a primitive pth root of unity ξ_{p}. Let G_{F} be the absolute Galois group of F, U an open normal subgroup of G_{F} of index p, and G = G_{F} /U.

In the 1960s Z. I. Borevič and D. K. Faddeev classified the possible G-module structures of the first cohomology groups H^{1}(U, 𝔽_{p}) in the case F a local field [Bo]. Recently this result was extended from local fields to all fields F as above [MS1], and in this more general context the result has been further developed and applied in [MS2], [MS3],

Date: October 20, 2006. 2000 Mathematics Subject Classification. 12G05 (primary), 19D45 (secondary). Key words and phrases. Galois cohomology, Milnor K-theory, Euler-Poincaré

characteristic, Galois module, Bloch-Kato Conjecture.

Nicole Lemire was supported in part by Natural Sciences and Engineering Research Council of Canada grant R3276A01. Ján Mináč was supported in part by Natural Sciences and Engineering Research Council of Canada grant R0370A01, by a Distinguished Professorship for 2004–2005 at the University of Western Ontario, by the Mathematical Sciences Research Institute, Berkeley, and by the Institute for Advanced Study, Princeton. John Swallow was supported in part by National Security Agency grant MDA904-02-1-0061.

[MSS], and [MSS2]. It is important in the study of Galois cohomology to extend these results to all cohomology groups H^{n}(U, 𝔽_{p}), n ∈ ℕ.

The celebrated 1982 paper of Merkurjev-Suslin [MeSu] achieved substantial progress in the investigation of Galois cohomology, and in hindsight we know that these results were already sufficient to extend the results of Borevič and Faddeev to H^{2}(U, 𝔽_{p}) for any field F containing a primitive pth root of unity. Because of connections with Brauer groups, Galois embedding problems, and Galois pro-p-groups, a precise knowledge of the structure H^{2}(U, 𝔽_{p}) as an 𝔽_{p}[G]-module has significant applications: see [LLMS2], for example, for consequences characterizing Galois Demuškin pro-p-groups.

The main conjectures in contemporary Galois cohomology include the Bloch-Kato Conjecture, the natural generalization to Milnor Ktheory of Hilbert 90, and generalization of the exact sequences in Milnor K-theory contained in [MeSu] from the case n = 2 to any n ∈ ℕ. V. Voevodsky’s’s partially published proof of the Bloch-Kato Conjecture, based upon earlier work of A. S. Merkurjev, M. Rost, and A. A. Suslin, encompasses all of these. (See [R], [Su], [V1] and [V2].) In this paper we show that two results related to the Bloch-Kato Conjecture, contained in [V1] and [V2] and quoted precisely in section 2, are sufficient to determine the structure of all 𝔽_{p}[G]-Galois modules H^{n}(U, 𝔽_{p}) for all n ∈ ℕ. Moreover, this structure depends only on simple arithmetical invariants attached to the field extension E/F, where E is the fixed field of U in the separable closure F_{sep} of F. In particular, we use Hilbert 90 for Milnor K-theory in an essential way to prove that no cyclic 𝔽_{p}[G]-module of dimension j with 2 < j < p can occur as a summand of H^{n}(U, 𝔽_{p}). Because we rely only on the main conjectures mentioned, our results could already have been obtained conditionally in the early 1980s.

We have already used the results on the 𝔽_{p}[G]-structure of H^{n}(U, 𝔽_{p}) to obtain a generalization of Schreier’s formula to higher Galois cohomology groups [LLMS1], to obtain a new characterization of Demuškin groups [LLMS2], and to construct fields with free or trivial Galois cohomology modules [LMS]. Moreover, using ideas in this paper, we clarify when an analogue of Hilbert 90 is valid for Milnor k-theory and Galois cohomology [LMSS]. Our results here also lead to a characterization of all possible decompositions of H^{n}(U, F_{p}) into a sum of indecomposable 𝔽_{p}[G]-modules. In a special case, this has been achieved [LLMS2, Theorem 4]. An investigation of the moduli space of all decompositions of H^{n}(U, 𝔽_{p}) is a work in progress.

Here we use the results to produce remarkably simple formulas for the nth partial Euler-Poincaré characteristic of U. These formulas provide a new tool for studying the structure of Sylow p-subgroups of absolute Galois groups, about which little is currently known. Recall Artin-Schreier’s important observation that the only finite subgroups of absolute Galois groups are the trivial subgroup and ℤ/2ℤ, and Becker’s generalization of this result to the case of the Galois group of a maximal p-extension [Be]. These results lead naturally to the search for a classification of finitely generated subgroups of Sylow p-subgroups of absolute Galois groups. The question is settled for algebraic number fields and for fields of transcendence degree 1 over local fields [E1], [E2], [E3]. In the case of p = 2 for arbitrary fields F with |F^{×}/F^{×2}| ≤ 8 [JW1] partial progress was made, but there are still open questions about possible Galois pro-2 groups with three generators. Thus, we know only very few types of such groups, and while the elementary type conjecture asserts that there are no others, there is not much evidence for the validity of the conjecture. (See section 7 for additional comments and references on this conjecture.) It is significant, therefore, that we are able to derive results on the good behavior of partial Euler-Poincaré characteristics—and its relation to cohomological dimension—without appealing to the conjecture.

In the Appendix [BeLMS] written with Dave Benson, we use these results to provide examples of pro-p-groups which cannot be realized as absolute Galois groups.

1. Main Theorems

The main ingredient for our determination of the G-module structure of H^{n}(U, 𝔽_{p}) is Milnor K-theory. (See [Mi] and [FV, Chap. IX].) For i ≥ 0, let K_{i}F denote the ith Milnor K-group of the field F, with standard generators denoted by {f_{1}, …, f_{i}}, f_{1}, …, f_{i} ∈ F \ {0}. For α ∈ K_{i}F, we denote by ᾱ the class of α modulo p, and we use the usual abbreviation k_{n}F for K_{n}F/pK_{n}F. We denote by (f) the element in H^{1}(U, 𝔽_{p}) corresponding to f in F \ {0} and by (f_{1}, ..., f_{n}) the cup product (f_{1}) ⋯ (f_{n}) in H^{n}(U, 𝔽_{p}).

Let E be the fixed field of U in the separable closure F_{sep} of F. We write N_{E/F} for the norm map K_{n}E → K_{n}F, and we use the same notation for the induced map modulo p. We denote by i_{E} the natural homomorphism in the reverse direction. We also apply the same notation N_{E/F} and i_{E} for the corresponding homomorphisms between cohomology groups, although when convenient we replace these with the equivalent maps cor_{E/F} and res_{F,E}. The image of an element α ∈ K_{i}F in H^{i}(G_{F} , 𝔽_{p}) we also denote by α. We formulate our results in terms of Galois cohomology for intended applications, but we use Milnor Ktheory in our proof.

Our decomposition depends on four arithmetic invariants Υ_{1}, Υ_{2}, y, z, which we define as follows. Fix a ∈ F such that E = F (\sqrt[p]{a}), and let σ ∈ G satisfy \sqrt[p]{a}^{σ−1} = ξ_{p}. First, for an element ᾱ of k_{i}F, let

ann_{k_{n−1}F} ᾱ = ann (k_{n−1}F ⟶^{ᾱ·−} k_{n−1+i}F )

denote the annihilator of the product with ᾱ. When the domain of ᾱ is clear, we omit the subscript on the map and write simply ann ᾱ. We use analogous notation for the annihilator of α in H^{i}(U, 𝔽_{p}) when we work in Galois cohomology rather than in Milnor K-theory. Because we will often use the elements \overline{\{a\}}, \overline{\{ξ_{p}\}}, \overline{\{a, a\}}, and \overline{\{a, ξ_{p}\}}, we omit the bars for these elements. We also omit the bar in the element \overline{\{\sqrt[p]{a}\}} ∈ k_{n}E.

Fix n ∈ ℕ and U an open normal subgroup of G_{F} of index p, and write E = Fix(U). Define invariants associated to E/F and n as follows:

d := dim_{𝔽_{p}} k_{n}F/N_{E/F} k_{n}E,  e := dim_{𝔽_{p}} N_{E/F} k_{n}E
Υ_{1} := dim_{𝔽_{p}} ann_{k_{n−1}F} {a, ξ_{p}}/ ann{a}, Υ_{2} := dim_{𝔽_{p}} k_{n−1}F/ ann{a, ξ_{p}}

y := \begin{cases}dim_{𝔽_{p}} (N_{E/F} k_{n}E) / \{a\} · k_{n−1}F, & p > 2 \\ dim_{𝔽_{2}} (N_{E/F} k_{n}E) / \{a\} · ann_{k_{n−1}F} \{a, −1\}, & p = 2 \end{cases}

z := \begin{cases}dim_{𝔽_{p}} (k_{n}F) / (\{ξ_{p}\} · k_{n−1}F + N_{E/F} k_{n}E) , & p > 2 \\ dim_{𝔽_{2}} (k_{n}F) / (\{a\} · k_{n−1}F + N_{E/F} k_{n}E) , & p = 2. \end{cases}

Our main results are then the following.

Theorem 1. If p > 2 and n ∈ ℕ then

H^{n}(U, 𝔽_{p}) ≃ X_{1} ⊕ X_{2} ⊕ Y ⊕ Z

where

(1) X_{1} is a trivial 𝔽_{p}[G]-module of dimension Υ_{1} and

X_{1} ∩ res_{F,E} H^{n}(G_{F} , 𝔽_{p}) = {0}

(2) X_{2} is a direct sum of Υ_{2} cyclic 𝔽_{p}[G]-modules of dimension 2 (3) Y is a free 𝔽_{p}[G]-module of rank y

(4) Z is a trivial 𝔽_{p}[G]-module of dimension z and

Z ⊂ res_{F,E} H^{n}(G_{F} , 𝔽_{p}).

Further we have

(5) Y^{G} = res_{F,E} cor_{E/F} H^{n}(U, 𝔽_{p})

(6) cor_{E/F} : X_{1} ⊕ X_{2} → (a) · H^{n−1}(G_{F} , 𝔽_{p}) is surjective

(7) Υ_{1} + Υ_{2} + y = e

(8) Υ_{2} + z = d.

Theorem 2. If p = 2 and n ∈ ℕ then

H^{n}(U, 𝔽_{2}) ≃ X_{1} ⊕ Y ⊕ Z

where

(1) X_{1} is a trivial 𝔽_{2}[G]-module of dimension Υ_{1} and

X_{1} ∩ res_{F,E} H^{n}(G_{F} , 𝔽_{2}) = {0}

(2) Y is a free 𝔽_{2}[G]-module of rank y

(3) Z is a trivial 𝔽_{2}[G]-module of dimension z and

Z ⊂ res_{F,E} H^{n}(G_{F} , 𝔽_{2}).

Further we have

(4) Y^{G} = res_{F,E} cor_{E/F} H^{n}(U, 𝔽_{2})

(5) cor_{E/F} : X_{1} → (a) · ann(a, −1) is an isomorphism

(6) Υ_{1} + y = e

(7) Υ_{2} + z = d.

Remark. The case n = 1 in the two theorems recovers the results of [MS1].

The isomorphism classes of 𝔽_{p}[G]-modules X_{1} ⊕ Z, X_{2}, and Y are determined by H^{n}(U, 𝔽_{p}), but the decomposition of H^{n}(U, 𝔽_{p}) into summands X_{1}, X_{2}, Y, and Z is not canonical. However, there is an equivalent, canonical version of our results, contained in the following statements. These statements describe (k_{n}E)^{G} and the intersection of (k_{n}E)^{G} with images of k_{n}E under successive powers of the augmentation ideal of 𝔽_{p}[G]. Here the augmentation ideal is generated by σ − 1, but the statements are independent of the choice of a generator σ of G as well as a primitive pth root of unity ξ_{p}. Assume first p > 2. (See Proposition 1 and Lemma 3.)

(1) for each i ≥ 3,

(σ − 1)^{i−1}k_{n}E ∩ (k_{n}E)^{G} = i_{E}N_{E/F} k_{n}E = (σ − 1)^{p−1}k_{n}E.

(2) (σ − 1)k_{n}E ∩ (k_{n}E)^{G} = i_{E}({ξ_{p}} · k_{n−1}F) + i_{E}N_{E/F} k_{n}E.

(3) 0 → ann{a} → k_{n−1}F ⟶^{{a}·−} k_{n}F ⟶^{i_{E}} (k_{n}E)^{G} ⟶^{N_{E/F}}

{a} · ann{a, ξ_{p}} → 0.

If p = 2 the exact sequence (3) is equivalent to Theorem 2. This equivalent, canonical reformulation of Theorems 1 and 2 follows from their proofs in sections 2, 3, and 4.

Now we turn to an application of the results above to Galois pro-p-groups. For a pro-p-group T with finite cohomology groups H^{i}(T, 𝔽_{p}) for 0 ≤ i ≤ n, the nth partial Euler-Poincaré characteristic χ_{n}(T) is defined as

χ_{n}(T) = ∑_{i=0}^{n}(−1)^{i} dim_{𝔽_{p}} H^{n}(T, 𝔽_{p}).

Generalizing a previous result of Šafarevič, Koch observed a very interesting criterion for cohomological dimension cd(G) being ≤ n, using the Euler-Poincaré characteristic of open subgroups of T. (See [Ko, Thm. 5.5].) For a strengthening of Koch’s result see [Sc].

Suppose now that T = G_{F} (p) is the maximal pro-p-quotient of an absolute Galois group G_{F} of a field F containing a primitive pth root of unity, and suppose T has finite rank. In this paper we show that for N an open subgroup of T of index p, the single condition that χ_{n}(N) = pχ_{n}(T) is equivalent to the condition that cor : H^{n}(N, 𝔽_{p}) → H^{n}(T, 𝔽_{p}) is surjective. Using this result, we strengthen Koch’s criterion in our case, showing that if χ_{n}(N) = pχ_{n}(T) for all open subgroups N of index p, then cd(T) ≤ n.

The essential part of section 7 is a formula for χ_{n}(N), where N is an open subgroup of T of index p. This formula depends only on p, χ_{n}(T), and the conorm in degree n. The connection with module structure takes particular notice of the maximal free submodules in the cohomology groups; we show that χ_{n}(N) differs from pχ_{n}(T) by a certain multiple of the degree n conorm, and if p > 2 then χ_{n}(N) is determined by the maximal free submodules in the cohomology groups together with this conorm. To see the role of the free submodules more clearly when the dimensions are finite, we set F^{i}(N) to be a maximal free 𝔽_{p}[T/N]-submodule of H^{i}(N, 𝔽_{p}) and define the nth partial free Euler-Poincaré characteristic χ_{n}^{𝔉}(N) by

χ_{n}^{𝔉}(N) = ∑_{i=0}^{n}(−1)^{i} dim_{𝔽_{p}} F^{i}(N).

(Observe that χ_{n}^{𝔉}(N) is independent of the choice of F^{i}(N) for each i.) Our theorem is as follows.

Theorem 3. Let p be a prime and F a field containing a primitive pth root of unity. Suppose T = Gal(F (p)/F) is a pro-p-group of finite rank, N is a subgroup of T of index p, and n ∈ ℕ.

(a) We have

pχ_{n}(T) − χ_{n}(N) = (−1)^{n}(p − 1) dim_{𝔽_{p}} \frac{H^{n}(T, 𝔽_{p})}{cor H^{n}(N, 𝔽_{p})}.

(b) Now additionally assume either that p > 2 or that p = 2 and −1 ∈ N_{E/F} (E), where E = Fix(N). Then

χ_{n}(N) = χ_{n}^{𝔉}(N) + (−1)^{n} dim_{𝔽_{p}} \frac{H^{n}(T, 𝔽_{p})}{cor H^{n}(N, 𝔽_{p})}.

Under the initial hypotheses of Theorem 3 we deduce the following corollaries.

Corollary 1. The following are equivalent.

(1) χ_{n}(N) = pχ_{n}(T)

(2) cor H^{n}(N, 𝔽_{p}) = H^{n}(T, 𝔽_{p})

Under the hypothesis either that p > 2 or that p = 2 and −1 ∈ N_{E/F} (E), then these are additionally equivalent to

(3) χ_{n}(N) = χ_{n}^{𝔉}(N).

Corollary 2 below shows in particular that if χ_{n}(N) = pχ_{n}(T) for all open subgroups N of T of index p, then the same relation holds for all open subgroups V of index p of any given open subgroup U of T.

Corollary 2. Let n ∈ ℕ. The following are equivalent:

(1) χ_{n}(N) = pχ_{n}(T) for all open subgroups N of T of index p

(2) cd(T) ≤ n

(3) χ_{n}(V) = pχ(U) for all open subgroups U of T and all open subgroups V of U of index p in U.

2. Bloch-Kato and Milnor K-theory

Our proofs rely on the following two results in Voevodsky’s work on the Bloch-Kato Conjecture. Because we apply Voevodsky’s results in the case when the base field contains a primitive pth root of unity we shall formulate Voevodsky’s results restricted to this case. The first is the Bloch-Kato Conjecture itself:

Theorem 4 ([V1, Lemma 6.11 and §7] and [V2, §6 and Thm. 7.1]).

(1) Let F be a field containing a primitive pth root of unity and m ∈ ℕ. Then the norm residue homomorphism

k_{m}F → H^{m}(G_{F} , μ_{p})

is an isomorphism.

(2) For any cyclic extension E/F of degree p, the sequence

K_{m}E ⟶^{σ−1} K_{m}E ⟶^{N_{E/F}} K_{m}F

is exact.

The second result establishes an exact sequence connecting k_{m}F and k_{m}E for consecutive m. (We translate the statement of the original result to K-theory using the previous theorem.) In the following result a is chosen to satisfy E = F (\sqrt[p]{a}).

Theorem 5 ([V1, Def. 5.1 and Prop. 5.2]). Let F be a field containing a primitive pth root of unity with no extensions of degree prime to p. Then for any cyclic extension E/F of degree p and m ∈ ℕ, the sequence

k_{m−1}E ⟶^{N_{E/F}} k_{m−1}F ⟶^{{a}·−} k_{m}F ⟶^{i_{E}} k_{m}E

is exact.

We observe that we may remove the hypothesis that the field F has no extensions of degree prime to p. We give a proof of the following theorem in section 5.

Theorem 6 (Modification of Theorem 5). Let F be a field containing a primitive pth root of unity. Then for any cyclic extension E/F of degree p and m ∈ ℕ the sequence

k_{m−1}E ⟶^{N_{E/F}} k_{m−1}F ⟶^{{a}·−} k_{m}F ⟶^{i_{E}} k_{m}E

is exact.

3. Notation and Lemmas

We fix n ∈ ℕ and the cyclic extension E = F (\sqrt[p]{a}), and we write k_{n−1}F = ann{a} ⊕ V ⊕ W, where ann{a, ξ_{p}} = ann{a} ⊕ V. Observe that Υ_{1} = dim_{𝔽_{p}} V and Υ_{2} = dim_{𝔽_{p}} W. We denote by i_{E}: K_{n}F → K_{n}E the map induced by the inclusion of F in E. In what follows we will frequently refer to the element \sqrt[p]{a}, and so we abbreviate it by A. We recall that if p = 2 then {a, ξ_{p}} = {a, −1} = {a, a} ∈ k_{2}F, while if p > 2 then {a, a} = 0 ∈ k_{2}F. Finally, we will use the projection formula for taking the norms of standard generators of K_{i}F (see [FW, p. 81]).

Lemma 1. We have a vector space isomorphism

V ⊕ W ⟶^{{a}·−} {a} · k_{n−1}F

and, if p > 2, the compositum of the maps {ξ_{p}} · − and i_{E}

W ⟶^{{ξ_{p}}·−} {ξ_{p}} · W ⟶^{i_{E}} i_{E}({ξ_{p}} · W)

is a vector space isomorphism as well.

Proof. The first isomorphism follows from the fact that V ⊕ W is a complement in k_{n−1}F of the kernel of multiplication by {a}. For the second, assume p > 2. Suppose that w̄ ∈ W and ᾱ = {ξ_{p}} · w̄ ∈ ker i_{E}. Then by Theorem 6, ᾱ = {a} · c̄ for c ∈ K_{n−1}F. Since {a, a} = 0 we see that {a} · ᾱ = 0. But then w̄ ∈ ann{a, ξ_{p}} and so w̄ = 0. □

For γ ∈ K_{n}E, let l(γ) denote the dimension of the cyclic 𝔽_{p}[G]- submodule ⟨γ̄⟩ of k_{n}E generated by γ̄. Then we have

(σ − 1)^{l(γ)−1}⟨γ̄⟩ = ⟨γ̄⟩^{G} ≠ 0 and (σ − 1)^{l(γ)}⟨γ̄⟩ = 0.

We denote by N the map (σ − 1)^{p−1} on k_{n}E. Because (σ − 1)^{p−1} = 1 + σ + ⋯ + σ^{p−1} in 𝔽_{p}[G], we may use i_{E}N_{E/F} and N interchangeably on k_{n}E.

Lemma 2. Let p > 2. Suppose γ ∈ K_{n}E with l = l(γ) ≥ 2. Then if l ≥ 3,

(σ − 1)^{l−1}γ̄ ∈ i_{E}N_{E/F} k_{n}E,

and if l = 2

(σ − 1)γ̄ ∈ i_{E}({ξ_{p}} · k_{n−1}F) + i_{E}N_{E/F} k_{n}E.

Proof. If l = p, then (σ − 1)^{p−1}k_{n}E = i_{E}N_{E/F} k_{n}E shows the result in this case.

Suppose l < p. Then γ̄ ∈ ker(σ − 1)^{p−1} and so i_{E}N_{E/F} (γ̄) = 0. By Theorem 6, there exists b ∈ K_{n−1}F such that N_{E/F} γ̄ = {a} · b̄. Equivalently N_{E/F} γ = {a}·b+pf for some f ∈ K_{n}F. By the projection formula (see [FW, p. 81]),

N_{E/F} (γ − {A} · i_{E}(b) − i_{E}(f)) = 0.

Then by Theorem 4, there exists ω ∈ K_{n}E such that

(σ − 1)ω = γ − ({A} · i_{E}(b) + i_{E}f)

and hence, since l ≥ 2,

(σ − 1)^{l−1}γ̄ = (σ − 1)^{l}ω̄ + (σ − 1)^{l−2}i_{E}({ξ_{p}} · b).

If l(γ) ≥ 3 we deduce

(σ − 1)^{l−1}γ̄ = (σ − 1)^{l}ω̄

where l(ω) = l + 1. Set γ̄_{l+1} = ω and repeat the argument. We obtain γ_{k} ∈ K_{n}E of lengths l < k ≤ p with

(σ − 1)^{l−1}γ̄ = (σ − 1)^{k−1}γ̄_{k}.

Take ᾱ = γ̄_{p} to obtain

(σ − 1)^{l−1}γ̄ ∈ i_{E}N_{E/F} k_{n}E,

as required.

If l(γ) = 2 we have that

(σ − 1)γ̄ = (σ − 1)^{2}ω̄ + i_{E}({ξ_{p}} · b̄)

for some ω ∈ K_{n}E and some b ∈ K_{n−1}F. We see that l(ω) ≤ 3. If l(ω) < 3 then

(σ − 1)γ̄ ∈ i_{E}({ξ_{p}} · k_{n−1}F),

while if l(ω) = 3 then by the previous case we see that (σ − 1)^{2}ω̄ ∈ i_{E}N_{E/F} k_{n}E, so the result holds in either case. □ Proposition 1. If p > 2, then

(σ − 1)k_{n}E ∩ (k_{n}E)^{G} = i_{E}({ξ_{p}} · k_{n−1}F) + i_{E}N_{E/F} k_{n}E

and for 3 ≤ i ≤ p,

(σ − 1)^{i−1}k_{n}E ∩ (k_{n}E)^{G} = i_{E}N_{E/F} k_{n}E.

If p = 2 then

(σ − 1)k_{n}E ∩ (k_{n}E)^{G} = i_{E}N_{E/F} k_{n}E.

Proof. We have that for 2 ≤ i ≤ p,

i_{E}N_{E/F} k_{n}E = (σ − 1)^{p−1}k_{n}E ∩ (k_{n}E)^{G} ⊂ (σ − 1)^{i−1}k_{n}E ∩ (k_{n}E)^{G}.

The statement for p = 2 follows immediately. Now assume that p > 2. Note also that

i_{E}({ξ_{p}} · k_{n−1}F) = (σ − 1)({A} · k_{n−1}F) ⊂ (σ − 1)k_{n}E ∩ (k_{n}E)^{G}.

Thus i_{E}({ξ_{p}} · k_{n−1}F) + i_{E}N_{E/F} k_{n}E ∈ (σ − 1)k_{n}E ∩ (k_{n}E)^{G} and the reverse inclusion follows from Lemma 2.

Now we shall prove that if 3 ≤ i ≤ p, then

(σ − 1)^{i−1}k_{n}E ∩ (k_{n}E)^{G} = i_{E}N_{E/F} k_{n}E.

Because i_{E}N_{E/F} k_{n}E = (σ − 1)^{p−1}k_{n}E we see that

i_{E}N_{E/F} k_{n}E ⊂ (σ − 1)^{i−1}k_{n}E ∩ (k_{n}E)^{G}

for each 1 ≤ i ≤ p. In order to establish the reverse inclusion when 3 ≤ i ≤ p, consider (σ − 1)^{i−1}γ̄ ∈ (σ − 1)^{i−1}k_{n}E ∩ (k_{n}E)^{G}. We may assume without loss of generality that l(γ) = i, and then since p > 2, we have (σ − 1)^{i−1}γ̄ ∈ i_{E}N_{E/F} k_{n}E by Lemma 2, as desired. □

In the following lemma we elongate the exact sequence of Theorem 6.

Lemma 3. The following sequence is exact:

0 → ann{a} → k_{n−1}F ⟶^{{a}·−} k_{n}F ⟶^{i_{E}} (k_{n}E)^{G} ⟶^{N_{E/F}} {a}·ann{a, ξ_{p}} → 0.

Here the map ann{a} → k_{n−1}F is the natural inclusion.

Proof. We show first that N_{E/F} ((k_{n}E)^{G}) ⊂ {a} · ann{a, ξ_{p}}. Let ᾱ ∈ (k_{n}E)^{G} and β = N_{E/F} α. Since i_{E}(N_{E/F} ᾱ) = (σ − 1)^{p−1}ᾱ = 0 we have that β̄ = N_{E/F} ᾱ = {a} · b̄ for some b ∈ K_{n−1}F by Theorem 6.

Suppose p = 2. Since β̄ is in the image of N_{E/F}, we have by Theorem 6 that {a} · β̄ = {a, a} · b̄ = 0. Since {a, a} = {a, −1}, we have b̄ ∈ ann{a, −1}.

Now suppose that p > 2. Write β = {a} · b + pf for some f ∈ K_{n}F. Then by the projection formula (see [FW, p. 81]),

N_{E/F} (α − ({A} · i_{E}(b) + i_{E}(f))) = 0.

By Theorem 4, there exists ω ∈ K_{n}E such that

(σ − 1)ω = α − ({A} · i_{E}(b) − i_{E}(f))

and hence (σ − 1)^{2}ω̄ = {ξ_{p}} · i_{E}(b̄).

If (σ − 1)^{2}ω̄ = 0 then by Theorem 6, {ξ_{p}} · b̄ = {a} · h̄ for some h ∈ K_{n−1}F. Because {a, a} = 0, the right-hand side of the preceding equation is annihilated by {a}. Therefore b̄ ∈ ann{a, ξ_{p}}.

If (σ − 1)^{2}ω̄ ≠ 0 then l(ω) = 3 and Lemma 2 shows that

i_{E}(\{ξ_{p}\} · b̄) = i_{E}(N_{E/F} (\overline{λ}))

for some λ ∈ K_{n}E. By Theorem 6, we have \{ξ_{p}\}·b̄ = N_{E/F} (\overline{λ})+\{a\}·h̄ for some h ∈ K_{n−1}F. Now by Theorem 6 and the fact that {a, a} = 0, the right-hand side of the preceding equation is annihilated by {a}. Then b̄ ∈ ann{a, ξ_{p}}. Hence in all cases N_{E/F} ᾱ ∈ {a} · ann{a, ξ_{p}}.

Exactness at the first two terms is obvious, and exactness at the third term follows from Theorem 6.

For exactness at the fourth term, suppose

γ̄ ∈ (k_{n}E)^{G} and N_{E/F} γ̄ = 0.

Then N_{E/F} γ = pf for f ∈ K_{n}F. Let β = γ − i_{E}(f). Then N_{E/F} β = 0 and by Theorem 4 there exists α ∈ K_{n}E such that (σ − 1)α = β. If p = 2 then β̄ = i_{E}(N_{E/F} ᾱ) ∈ i_{E}k_{n}F and we are done. Thus assume p > 2.

Now if (σ − 1)ᾱ = β̄ = 0 we are done as then γ̄ = i_{E}(f̄). Hence assume (σ − 1)ᾱ ≠ 0. Then l(α) = 2 and by Lemma 2 we see that

β̄ = (σ − 1)ᾱ ∈ {ξ_{p}} · i_{E}k_{n−1}F + i_{E}N_{E/F} k_{n}F ⊂ i_{E}k_{n}F

and exactness at the fourth term is established.

Finally we show the exactness at the fifth term. Since

{a} · ann{a, ξ_{p}} = {a} · V

it is enough to show that each element {a} · v̄ where v̄ ∈ V can be written as N_{E/F} ᾱ for some ᾱ ∈ (k_{n}E)^{G}. Observe that

(σ − 1)({A} · i_{E}v̄) = {ξ_{p}} · i_{E}(v̄).

Also we have

N_{E/F} (\{A\} · i_{E}(v̄)) = \begin{cases}\{a\} · v̄ & if p > 2 \\ \{−a\} · v̄ & if p = 2. \end{cases}

Therefore it is enough to show that there exists an element γ̄ ∈ k_{n}E such that (σ − 1)γ̄ = {ξ_{p}} · i_{E}(v̄) and

N_{E/F} γ̄ = \begin{cases}0 & if p > 2 \\ \{−1\} · v̄ & if p = 2. \end{cases}

Indeed then we can set ᾱ = {A} · i_{E}(v̄) − γ̄.

Because v̄ ∈ ann{a, ξ_{p}} we see that {ξ_{p}} · i_{E}(v̄) ∈ ann{a}. By Theorem 6 there exists β̄ ∈ k_{n}E such that

{ξ_{p}} · v̄ = N_{E/F} β̄ and i_{E}(N_{E/F} β̄) = (σ − 1)^{p−1}β̄.

Then setting γ̄ = (σ − 1)^{p−2}β̄ we obtain our required element. The proof of our lemma is now complete. □

Next, we need a general lemma about 𝔽_{p}[G]-modules. The straightforward proof of this lemma is omitted.

Lemma 4. Let M_{i}, i ∈ I, be a family of 𝔽_{p}[G]-modules contained in a common 𝔽_{p}[G]-module N. Suppose that the 𝔽_{p}-vector subspace R of N generated by all M_{i}^{G} has the form R = ⊕_{i∈I}M_{i}^{G}. Then the 𝔽_{p}[G]-module M generated by M_{i}, i ∈ I, has the form M = ⊕_{i∈I}M_{i}.

Finally, we need a general structure proposition about 𝔽_{p}[G]-modules that shows that the structure of an 𝔽_{p}[G]-module X can be determined from the structure of X^{G}.

Proposition 2. Let X be an 𝔽_{p}[G]-module. Set L_{p} = (σ − 1)^{p−1}X and for 1 ≤ i < p, suppose that L_{i} is an 𝔽_{p}-complement of (σ − 1)^{i}X ∩ X^{G} in (σ − 1)^{i−1}X ∩ X^{G}.

Then there exist 𝔽_{p}[G]-modules X_{i}, i = 1, …, p, such that

(1) X = ⨁_{i=1}^{p} X_{i},

(2) X_{i}^{G} = L_{i} for i = 1, …, p,

(3) each X_{i} is a direct sum of dim_{𝔽_{p}}(L_{i}) cyclic 𝔽_{p}[G]-modules of length i, and

(4) for each i = 1, …, p, there exists an 𝔽_{p}-submodule Y_{i} of X_{i} with dim_{𝔽_{p}}(Y_{i}) = dim_{𝔽_{p}}(L_{i}) such that 𝔽_{p}[G]Y_{i} = X_{i}.

Proof. Since (σ − 1)^{p−1}X ⊂ X^{G}, we see that L_{p} = (σ − 1)^{p−1}X ∩ X^{G}. By reverse induction on i = p, p − 1, …, 1, we obtain

(σ − 1)^{i−1}X ∩ X^{G} = ⊕_{j=i}^{p}L_{j}.

Observe that because L_{i} is an 𝔽_{p}-complement of (σ − 1)^{i}X ∩ X^{G} in (σ − 1)^{i−1}X ∩ X^{G}, for each i = 1, …, p, there exists an 𝔽_{p}-submodule Y_{i} of X such that (σ − 1)^{i}Y_{i} = 0 and (σ − 1)^{i−1} : Y_{i} → L_{i} is an 𝔽_{p}- isomorphism. Set X_{i} to be the 𝔽_{p}[G]-submodule of X generated by Y_{i}. By Exclusion Lemma 4, the sum of the X_{k} is direct since the sum of the X_{k}^{G} = L_{k} is direct.

We need only show that X = ⊕_{k=1}^{p}X_{k}. Let J_{k} = ker(σ − 1)^{k} ⊂ X, k = 1, …, p. Then we have the filtration X^{G} = J_{1} ⊂ J_{2} ⊂ ⋯ ⊂ J_{p} = X. Let x ∈ X. Then (σ − 1)^{p−1}x ∈ L_{p}, and hence there exists y_{p} ∈ Y_{p} with (σ − 1)^{p−1}x = (σ − 1)^{p−1}y_{p}. Therefore x − y_{p} ∈ J_{p−1}.

More generally we show for all k ∈ {2, 3, …, p} that if x ∈ J_{k} then there exist x_{i} ∈ X_{i}, k ≤ i ≤ p, such that x − ∑_{i=k}^{p} x_{i} ∈ J_{k−1}. We have already shown our statement for k = p. Hence assume that x ∈ J_{k}, k < p. Then

(σ − 1)^{k−1}x ∈ (σ − 1)^{k−1}X ∩ X^{G} = ⊕_{i=k}^{p}L_{i}.

Hence there exist l_{i} ∈ L_{i}, i = k, …, p, such that

(σ − 1)^{k−1}x = ∑_{i=k}^{p} l_{i},

and there exist y_{i} ∈ Y_{i}, i = k, …, p, such that (σ − 1)^{i−1}y_{i} = l_{i}. Therefore

(σ − 1)^{k−1}x = (σ − 1)^{k−1} (∑_{i=k}^{p}(σ − 1)^{i−k}y_{i}) .

Setting x_{k} = y_{k} ∈ X_{k}, and x_{i} = (σ − 1)^{i−k}y_{i} ∈ X_{i}, i > k, we have

x − ∑_{i=k}^{p} x_{i} ∈ J_{k−1}

as required.

Therefore, using the fact that

J_{1} = X^{G} = ⊕_{i=1}^{p}L_{i} = ⊕_{i=1}^{p}X_{i}^{G} ⊂ ⊕_{i=1}^{p}X_{i},

as well as our statement above, we show by induction on k that J_{k} ⊂ ⊕_{i=1}^{p}X_{i} for all k = 1, 2, …, p. In particular J_{p} = X ⊂ ⊕_{i=1}^{p}X_{i}, which completes our proof. □

4. Proofs of Theorems 1 and 2

We determine now the structure of k_{n}E as an 𝔽_{p}[G]-module. We do so by invoking Proposition 1 to determine 𝔽_{p}-modules L_{i}, i = 1, …, p, such that

L_{p} = (σ − 1)^{p−1}k_{n}E = (σ − 1)^{p−1}k_{n}E ∩ (k_{n}E)^{G},

and such that L_{i} is an 𝔽_{p}-complement of (σ − 1)^{i}k_{n}E ∩ (k_{n}E)^{G} in (σ − 1)^{i−1}k_{n}E ∩ (k_{n}E)^{G}. In this way we may then apply Proposition 2.

Proof of Theorem 1. Assume that p > 2, and set

L_{1} = X_{1} + Z, L_{2} = i_{E}({ξ_{p}} · W), L_{p} = i_{E}N_{E/F} k_{n}E,

where we choose 𝔽_{p}-complements X_{1}, W, and Z, as follows:

X_{1} ⊕ i_{E}k_{n}F  = (k_{n}E)^{G}
W ⊕ ann{a, ξ_{p}}  = k_{n−1}F
Z ⊕ (i_{E}({ξ_{p}} · k_{n−1}F) + i_{E}N_{E/F} k_{n}E)  = i_{E}k_{n}F.

Observe that X_{1} ∩ i_{E}k_{n}F = {0} and therefore X_{1} satisfies the second part of condition (1) in Theorem 1. Similarly, the second part of condition (4) follows from the definition of Z.

Further set L_{i} = 0 if i ≠ 1, 2, p. We claim that the 𝔽_{p}-modules L_{i} satisfy the hypotheses of Proposition 2.

First observe that L_{p} = i_{E}N_{E/F} k_{n}E = (σ − 1)^{p−1}k_{n}E satisfies the hypotheses of Proposition 2 for L_{p}. Next, by Proposition 1, if i ≥ 3 then

(σ − 1)^{i−1}k_{n}E ∩ (k_{n}E)^{G} = i_{E}N_{E/F} k_{n}E = (σ − 1)^{p−1}k_{n}E,

so that the L_{i} = 0, 3 ≤ i < p, satisfy the hypotheses for L_{i}. Moreover, again by Proposition 1,

(σ − 1)k_{n}E ∩ (k_{n}E)^{G} = i_{E}({ξ_{p}} · k_{n−1}F) + i_{E}N_{E/F} k_{n}E.

We show next that

L_{2} ⊕ L_{p} = i_{E}({ξ_{p}} · k_{n−1}F) + i_{E}N_{E/F} k_{n}E.

First let γ̄ ∈ i_{E}({ξ_{p}} · k_{n−1}F) ∩ i_{E}N_{E/F} k_{n}E. Then γ̄ = {ξ_{p}} · i_{E}(f̄) = i_{E}N_{E/F} ᾱ for some f ∈ K_{n−1}F and some α ∈ K_{n}E. By Theorem 6, {ξ_{p}} · f̄− N_{E/F} ᾱ = {a} · b̄ for some b ∈ k_{n−1}F. But then {a, ξ_{p}} · f̄ = 0, since N_{E/F} k_{n}E = ann{a} by Theorem 6 and {a, a} = 0. Therefore f̄ ∈ ann{a, ξ_{p}}. Conversely, if f̄ ∈ ann{ξ_{p}, a} then {ξ_{p}} · f̄ ∈ N_{E/F} k_{n}E. Because W is an 𝔽_{p}-complement of ann{a, ξ_{p}} in k_{n−1}F, we obtain that

i_{E}({ξ_{p}} · k_{n−1}F) + i_{E}N_{E/F} k_{n}E = i_{E}({ξ_{p}} · W) ⊕ i_{E}N_{E/F} k_{n}E
= L_{2} ⊕ L_{p}.

Finally, by construction of X_{1} and Z, observe that

(k_{n}E)^{G} = X_{1} ⊕ Z ⊕ L_{2} ⊕ L_{p}.

Therefore L_{1} = X_{1} ⊕ Z is an 𝔽_{p}-complement of (σ − 1)k_{n}E ∩ (k_{n}E)^{G} = L_{2} ⊕ L_{p} in (k_{n}E)^{G}, as desired.

Applying Proposition 2, we obtain 𝔽_{p}[G]-modules X_{2} and Y such that (X_{2})^{G} = L_{2}, Y^{G} = L_{p} = i_{E}N_{E/F} k_{n}E, X_{2} is a direct sum of dim_{𝔽_{p}} L_{2} cyclic 𝔽_{p}[G]-modules of length 2, and Y is a free 𝔽_{p}[G]-module of rank dim_{𝔽_{p}} L_{p}. By construction X_{1} and Z are trivial 𝔽_{p}[G]-modules.

We turn to the calculation of the dimensions of X_{1}, Z, L_{2}, and L_{p}. Note that for p > 2, ker(i_{E}) ⊂ N_{E/F} k_{n}E. Indeed, by Theorem 6, ker(i_{E}) = {a} · k_{n−1}F and N_{E/F} k_{n}E = ann{a}, and from N_{E/F} {A} = {a} and the projection formula (see [FW, p. 81]) we obtain ker(i_{E}) ⊂ N_{E/F} k_{n}E. Then we calculate

dim_{𝔽_{p}} L_{p} = dim_{𝔽_{p}} i_{E}N_{E/F} k_{n}E = dim_{𝔽_{p}} N_{E/F} k_{n}E/ ker(i_{E})
= dim_{𝔽_{p}} N_{E/F} k_{n}E/{a} · k_{n−1}F
= y;
dim_{𝔽_{p}} Z = dim_{𝔽_{p}} i_{E}k_{n}F/i_{E}({ξ_{p}} · k_{n−1}F) + i_{E}N_{E/F} k_{n}E
= dim_{𝔽_{p}} k_{n}F/({ξ_{p}} · k_{n−1}F + N_{E/F} k_{n}E)
= z; and
dim_{𝔽_{p}} L_{2} = dim_{𝔽_{p}} i_{E}({ξ_{p}} · W) = dim_{𝔽_{p}} W
= dim_{𝔽_{p}} k_{n−1}F/ ann{a, ξ_{p}}
= Υ_{2}.

(In the determination of dim_{𝔽_{p}} L_{2} we use both Lemma 1 and the definition of W.)

As for X_{1}, by Lemma 3 we have ker(N_{E/F})∩(k_{n}E)^{G} = i_{E}k_{n}F as well as N_{E/F} ((k_{n}E)^{G}) = {a} · ann{a, ξ_{p}}. Then X_{1}, as an 𝔽_{p}-complement of i_{E}k_{n}F in (k_{n}E)^{G}, is mapped isomorphically under N_{E/F} onto {a} · ann{a, ξ_{p}} = {a} · V. Therefore we calculate

dim_{𝔽_{p}} X_{1} = dim_{𝔽_{p}}{a} · ann{a, ξ_{p}}
= dim_{𝔽_{p}} ann{a, ξ_{p}}/ ann{a} = Υ_{1}.

Next we claim that N_{E/F} (X_{1} ⊕X_{2}) = {a}·k_{n−1}F. First observe that because i_{E}N_{E/F} (X_{1} ⊕ X_{2}) = (σ − 1)^{p−1}(X_{1} ⊕ X_{2}) = {0} we have

N_{E/F} (X_{1} ⊕ X_{2}) ⊂ ker i_{E} = {a} · k_{n−1}F.

Since {a} · k_{n−1}F = {a} · (V ⊕ W), it suffices to show

{a} · V ⊂ N_{E/F} (X_{1} ⊕ X_{2}) and {a} · W ⊂ N_{E/F} (X_{1} ⊕ X_{2}).

By the above, N_{E/F} maps X_{1} isomorphically onto {a} · V, so we have the first inclusion. For the second, let {a} · w̄ ∈ {a} · W. Then

N_{E/F} ({A} · i_{E}(w̄)) = {a} · w̄.

Observe that X_{2} = 𝔽_{p}[G]Y_{2}, where Y_{2} is an 𝔽_{p}-submodule of X_{2} mapped isomorphically under (σ − 1) onto i_{E}({ξ_{p}} · W). Therefore there exists γ̄ ∈ Y_{2} ⊂ X_{2} such that

(σ − 1)γ̄ = (σ − 1)({A} · i_{E}w̄).

Then {A} · i_{E}w̄ − γ̄ ∈ (k_{n}E)^{G}. Now

N_{E/F} ((k_{n}E)^{G}) = {a} · ann{a, ξ_{p}} = {a} · V = N_{E/F} (X_{1}).

Applying the projection formula, {a}·w̄−N_{E/F} (γ̄) ∈ N_{E/F} (X_{1}). Hence {a} · W ⊂ N_{E/F} (X_{1} ⊕ X_{2}), as required.

Lastly, we need to show the relations between the dimensions of these modules. From

y = dim_{𝔽_{p}} N_{E/F} k_{n}E/{a} · k_{n−1}F

and

dim_{𝔽_{p}}({a} · k_{n−1}F) = dim_{𝔽_{p}}{a} · (V ⊕ W)
= dim_{𝔽_{p}}(V ⊕ W) = Υ_{1} + Υ_{2}

(again using Lemma 1), we obtain that

Υ_{1} + Υ_{2} + y = dim_{𝔽_{p}} N_{E/F} k_{n}E = e.

Furthermore, since

z = dim_{𝔽_{p}} \frac{i_{E}k_{n}F}{i_{E}(\{ξ_{p}\} · k_{n−1}F) + i_{E}N_{E/F} k_{n}E}

= dim_{𝔽_{p}} \frac{i_{E}k_{n}F}{i_{E}(\{ξ_{p}\} · W) ⊕ i_{E}N_{E/F} k_{n}E}

and Υ_{2} = dim_{𝔽_{p}} i_{E}({ξ_{p}} · W), we use ker(i_{E}) ⊂ N_{E/F} k_{n}E to deduce

Υ_{2} + z = dim_{𝔽_{p}} k_{n}F/N_{E/F} k_{n}E = d.

The proof is complete. □

Proof of Theorem 2. Assume p = 2, and set

L_{1} = X_{1} + Z
L_{2} = i_{E}N_{E/F} k_{n}E,

where we choose 𝔽_{2}-complements X_{1} and Z, as follows:

X_{1} ⊕ i_{E}k_{n}F  = (k_{n}E)^{G}
Z ⊕ i_{E}N_{E/F} k_{n}E  = i_{E}k_{n}F.

Observe that X_{1} ∩ i_{E}k_{n}E = {0} and therefore X_{1} satisfies the second part of condition (1) of Theorem 2. Similarly, the second part of condition (3) follows from the definition of Z. Note also that by construction the sum X_{1} + Z is direct.

Since by Proposition 1

(σ − 1)k_{n}E ∩ (k_{n}E)^{G} = i_{E}N_{E/F} k_{n}E,

we see that X_{1} + Z is an 𝔽_{2}-complement of (σ − 1)k_{n}E ∩ (k_{n}E)^{G} in (k_{n}E)^{G}. Therefore L_{1} and L_{2} satisfy the hypotheses of Proposition 2, and there exist a free 𝔽_{2}[G]-module Y with Y^{G} = i_{E}N_{E/F} k_{n}E and trivial 𝔽_{2}[G]-modules X_{1} and Z such that

k_{n}E = X_{1} ⊕ Y ⊕ Z.

We next determine the rank of Y and the dimensions of X_{1} and Z. We claim first that

ker(i_{E}) ∩ N_{E/F} k_{n}E = {a} · ann{a, −1}.

Indeed, by Theorem 6, ker(i_{E}) = {a} · k_{n−1}F and N_{E/F} k_{n}E = ann{a}. Then, we deduce from {a} · f̄ ∈ ker(i_{E}) ∩ N_{E/F} k_{n}E and the fact that {a, a} = {a, −1} for p = 2 that

{a, −1} · f̄ = {a, a} · f̄ = 0.

Conversely, suppose f̄ ∈ ann{a, −1}. Then because {a, a} = {a, −1} we have

{a} · f̄ ∈ {a} · k_{n−1}F ∩ ann{a} = ker(i_{E}) ∩ N_{E/F} k_{n}E.

We then calculate

dim_{𝔽_{2}} L_{2} = dim_{𝔽_{2}} i_{E}N_{E/F} k_{n}E
= dim_{𝔽_{2}} N_{E/F} k_{n}E/{a} · ann{a, −1} = y

and

dim_{𝔽_{2}} Z = dim_{𝔽_{2}} i_{E}k_{n}F/i_{E}N_{E/F} k_{n}F
= dim_{𝔽_{2}} k_{n}F/(N_{E/F} k_{n}E + {a} · k_{n−1}F) = z.

As for X_{1}, by Lemma 3, we observe that ker(N_{E/F}) ∩ (k_{n}E)^{G} = i_{E}k_{n}F and N_{E/F} ((k_{n}E)^{G}) = {a} · ann{a, −1}. Hence X_{1}, as an 𝔽_{2}- complement of i_{E}k_{n}F in (k_{n}E)^{G}, is mapped isomorphically under N_{E/F} onto {a} · ann{a, −1} = {a} · V. Therefore

dim_{𝔽_{2}} X_{1} = dim_{𝔽_{2}}{a} · ann{a, −1}
= dim_{𝔽_{2}} ann{a, −1}/ ann{a} = Υ_{1}.

Now we establish the relations between these dimensions. Since

y = dim_{𝔽_{2}} N_{E/F} k_{n}E/{a} · ann{a, −1},

we have

Υ_{1} + y = dim_{𝔽_{p}} N_{E/F} k_{n}E = e.

Finally, we calculate

d = dim_{𝔽_{2}} k_{n}F/N_{E/F} k_{n}E
= dim_{𝔽_{2}} k_{n}F/(N_{E/F} k_{n}E + {a} · k_{n−1}F)+
dim_{𝔽_{2}}(N_{E/F} k_{n}E + {a} · k_{n−1}F)/N_{E/F} k_{n}E
= z + dim_{𝔽_{2}}{a} · k_{n−1}F/{a} · ann{a, −1}
= z + dim_{𝔽_{2}} k_{n−1}F/ ann{a, −1} = z + Υ_{2}.

The proof is complete. □

5. Proof of Theorem 6

For the case p = 2 we have the long exact sequence of Galois cohomology groups due to Arason [A, Satz 4.5]. Suppose then that p > 2, and assume first that F is perfect. Let S be any Sylow p-subgroup of G_{F} = Gal(F_{sep}/F), and set L to be the fixed field of S. Because F is perfect, the separable closure F_{sep} is identical to the algebraic closure F̄, and hence each finite extension of L has degree a power of p. In particular, all of the hypotheses of Theorem 5 are valid for the field L in place of F. Furthermore, ([L : F], p) = 1. (Here we use basic properties of supernatural numbers and Sylow p-subgroups. See [Se, Chapter 1].) Therefore if E = F (\sqrt[p]{a}) is a cyclic extension of F of degree p, so is EL = L(\sqrt[p]{a}) over L. By Theorem 5 we see that the sequence

k_{m−1}EL ⟶^{N_{EL/L}} k_{m−1}L ⟶^{{a}·−} k_{m}L ⟶^{i_{EL}} k_{m}EL

is exact for each m ∈ ℕ.

We claim that i_{L}: k_{m}F → k_{m}L is injective. Indeed, suppose that i_{L}(α) = 0 for some α ∈ k_{m}F. Then there exists a finite subextension M/F of L/F such that i_{M}(α) = 0. Then

0 = N_{M/F} (i_{M}(α)) = [M : F]α.

(See [FV, Thm. XI.3.8].) Because [M : F] is coprime with p, we see that α = 0 and i_{L} is injective as asserted. Similarly we have that i_{EL}: k_{m}E → k_{m}EL is injective.

We then have the following commutative diagram:

\xymatrix{k_{m−1}EL \ar[r]^{N_{EL/L}} & k_{m−1}L \ar[r]^{\{a\}·−} & k_{m}L \ar[r]^{i_{LE}} & k_{m}EL \\ k_{m−1}E \ar[r]^{N_{E/F}} \ar[u]_{i_{EL}} & k_{m−1}F \ar[r]^{\{a\}·−} \ar[u]_{i_{L}} & k_{m}F \ar[r]^{i_{E}} \ar[u]_{i_{L}} & k_{m}E \ar[u]_{i_{EL}}}

The fact that the first square is commutative follows from [BT, p. 383]. The commutativity of the remaining part of the diagram is clear. Because the vertical maps are injective, we see that the bottom row of the diagram is a complex: the composition of any two consecutive maps is the zero map. We now establish exactness at the second and third terms of the complex.

Let α ∈ k_{m−1}F such that {a} · α = 0. Then {a} · i_{L}(α) = 0 and therefore there exists an element β ∈ k_{m−1}EL such that N_{EL/L}(β) = i_{L}(α). Let M/F be a finite extension such that β is defined over EM. Then N_{EM/M}(β) = i_{M}(α) (see [BT, p. 383]), and we have

N_{EM/F} (β) = N_{M/F} (N_{EM/M}(β)) = N_{M/F} (i_{M}(α)) = [M : F]α.

Thus N_{E/F} (N_{EM/E}(β)) = N_{EM/F} (β) = [M : F]α.

Because ([M : F], p) = 1 we see that α ∈ N_{E/F} (k_{m−1}E). Therefore we have established the exactness of our complex at k_{m−1}F.

Now assume that α ∈ k_{m}F such that i_{E}(α) = 0 ∈ k_{m}E. Then arguing as above, we see that there exist a finite extension M/F and β ∈ k_{m−1}M such that {a} · β = i_{M}(α) ∈ k_{m}M.

Applying N_{M/F} and using the projection formula we see that

{a} · N_{M/F} (β) = N_{M/F} (i_{M}(α)) = [M : F]α.

Because [M : F] is coprime with p, α ∈ {a} · N_{M/F} (γ) for a suitable element γ ∈ k_{m−1}M. Hence we see that our complex is also exact at k_{m}F and the full complex is exact.

Now if F is not perfect, let F_{pc} denote the perfect closure of F (see, for instance, [Ka, pp. 69–70]). Since finite purely inseparable extensions of F are of qth-power degree, where q is the characteristic of F, and q ≠ p since ξ_{p} ∈ F, we obtain ([F_{pc} : F], p) = 1. The argument above establishes the theorem for the perfect field F_{pc}, and a similar transfer argument descends from F_{pc} to F. □

6. From H^{i}(G_{F} , 𝔽_{p}) to H^{i}(G_{F} (p), 𝔽_{p})

In the previous sections, we worked with cohomology groups of absolute Galois groups. In this section we observe that, assuming the Bloch-Kato Conjecture, we may replace these groups with cohomology groups of maximal pro-p-quotients of absolute Galois groups.

Let F (p) be the compositum of all finite Galois p-power extensions of F in a fixed separable closure F_{sep} of F, G_{F} = Gal(F_{sep}/F), and T = Gal(F (p)/F) = G_{F} (p). Since F contains ξ_{p} we see that F (p) is closed under taking pth roots and hence H^{1}(G_{F (p)}, 𝔽_{p}) = {0}. By the Bloch-Kato Conjecture (see Theorem 4) the subring of the cohomology ring H^{∗}(G_{F (p)}, 𝔽_{p}) consisting of elements of positive degree is generated by H^{1}(G_{F (p)}, 𝔽_{p}). Hence for each i ∈ ℕ, H^{i}(G_{F (p)}, 𝔽_{p}) = {0}. Then, considering the Lyndon-Hochschild-Serre spectral sequence associated to the exact sequence

1 → G_{F (p)} → G_{F} → T → 1,

we have that inf : H^{i}(T, 𝔽_{p}) → H^{i}(G_{F} , 𝔽_{p}) is an isomorphism for all i ∈ ℕ ∪ {0}.

Recall that E = F (\sqrt[p]{a}), and let N = Gal(F (p)/E). By the same argument, we have that inf : H^{i}(N, 𝔽_{p}) → H^{i}(G_{E}, 𝔽_{p}) is an isomorphism. A straightforward argument with cochains shows that this isomorphism is T/N-equivariant.

We denote by (a) ∈ H^{1}(T, 𝔽_{p}) the element corresponding to a and by (a, ξ_{p}) the cup product (a) · (ξ_{p}) in H^{2}(T, 𝔽_{p}). From Theorems 4, 6, and the above argument we see that the following sequence is exact for i ≥ 0:

H^{i}(N, 𝔽_{p}) ⟶^{cor} H^{i}(T, 𝔽_{p}) ⟶^{−·(a)} H^{i+1}(T, 𝔽_{p}) ⟶^{res} H^{i+1}(N, 𝔽_{p}). (1)

Hence cor H^{i}(N, 𝔽_{p}) = ann_{H^{i}(T,𝔽_{p})}(a). Observe that, by naturality, the inflation map inf induces the following isomorphisms:

cor H^{i}(N, 𝔽_{p}) = ann_{H^{i}(T,𝔽_{p})}(a) →^{inf} cor H^{i}(G_{E}, 𝔽_{p}) = ann_{H^{i}(G_{F} ,𝔽_{p})}(a),

ann_{H^{i}(T,𝔽_{p})}(a, ξ_{p}) →^{inf} ann_{H^{i}(G_{F} ,𝔽_{p})}(a, ξ_{p}).

From now on we shall freely use the isomorphisms related to the cohomology groups H^{i}(T, 𝔽_{p}), H^{i}(G_{F} , 𝔽_{p}) and the cohomology groups of their open subgroups observed in this section.

7. Partial Euler-Poincaré Characteristics

In this section we determine dim_{𝔽_{p}} H^{i}(N, 𝔽_{p}) for all open subgroups of T. In the case when T is finitely generated we use this result to calculate the nth partial Euler-Poincaré characteristic of N. We show that a very simple relation exists between the nth partial Euler-Poincaré characteristics of N and T, and we then sharpen Koch’s well-known criterium for the cohomological dimension of T. These results help clarify the structure of finitely generated subgroups of Sylow p-subgroups of absolute Galois groups, a class about which we know comparatively little.

Recall that the elementary type conjecture asserts a classification of finitely generated subgroups of Sylow p-subgroups of absolute Galois groups. A version of this conjecture was first introduced in the context of Witt rings of quadratic forms by Marshall [M1] (see also [M2]), and Jacob and Ware then considered the conjecture in terms of Galois groups G_{F} (2) [JW1], [JW2]. Efrat [E1] extended the conjecture to possible groups G_{F} (p): each finitely generated group G_{F} (p) may be constructed from groups ℤ/2ℤ, ℤ_{p}, and G_{F} (p) for F a local field, using some specific semidirect products and free products in the category of pro-p-groups. (For a precise definition, see [E1] and [E3]. Another variant of the elementary type conjecture may be found in [Ku]. For a more recent investigation of the elementary type conjecture, see [En].)

Although the elementary type conjecture attempts a far-reaching generalization of the work of Artin-Schreier and Becker [Be], at the moment there is little evidence for the conjecture in such generality, although the conjecture holds for algebraic number fields and fields of transcendence degree 1 over local fields [E1], [E2], [E3] and for G_{F} (2) when |F^{×}/F^{×2}| ≤ 8 [JW1]. The significance of this work on the nth partial Euler-Poincaré characteristic of finitely generated subgroups and their connections with cohomological dimension lies in the fact that we do not use the elementary type conjecture to achieve the new results. As a consequence, the formulas provide a new tool for the investigation of the validity of the elementary type conjecture. (For further connections among the elementary type conjecture, Demuškin groups, and Galois modules, see [LLMS2].)

For the remainder of the section we will use without mention the fact that cor H^{i}(N, 𝔽_{p}) = ann_{H^{i}(T,𝔽_{p})}(a), which follows from Theorem 6. Moreover, from this point on, we will write ann(a) for ann_{H^{i}(T,𝔽_{p})}(a) and ann(a, ξ_{p}) for ann_{H^{i}(T,𝔽_{p})}(a, ξ_{p}). For each i ≥ 0 we set

a_{i} = dim_{𝔽_{p}} ann(a)
d_{i} = dim_{𝔽_{p}} H^{i}(T, 𝔽_{p})/ ann(a)
h_{i} = dim_{𝔽_{p}} H^{i}(T, 𝔽_{p}) = a_{i} + d_{i}.

Proposition 3. Let N be an open subgroup of T = Gal(F (p)/F) of index p. (We do not assume that T has finite rank.)

(a) We have

dim_{𝔽_{p}} H^{i}(N, 𝔽_{p}) = d_{i−1} + d_{i} + p dim_{𝔽_{p}} \frac{cor H^{i}(N, 𝔽_{p})}{(a) · H^{i−1}(T, 𝔽_{p})}.

(b) Assume further either that p > 2 or that p = 2 and a is a sum of two squares in F. Then

dim_{𝔽_{p}} H^{i}(N, 𝔽_{p}) = d_{i−1} + d_{i} + py.

(c) Assume further either that p > 2 and a_{i} is finite, that p = 2, a is a sum of two squares in F, and a_{i} is finite, or that p = 2 and h_{i−1} and h_{i} are finite. Then

dim_{𝔽_{p}} H^{i}(N, 𝔽_{p}) = d_{i−1} + d_{i} + p(a_{i} − d_{i−1}).

Proof. Assume first that p > 2. Observe that

d_{i−1} = dim_{𝔽_{p}} \frac{H^{i−1}(T, 𝔽_{p})}{ann(a, ξ_{p})} + dim_{𝔽_{p}} \frac{ann(a, ξ_{p})}{ann(a)} = Υ_{2} + Υ_{1}.

By Theorem 1(8), we have d_{i} = Υ_{2} + z. Then from Theorem 1 we conclude

dim_{𝔽_{p}} H^{i}(N, 𝔽_{p}) = Υ_{1} + 2Υ_{2} + py + z
= (Υ_{1} + Υ_{2}) + (Υ_{2} + z) + py
= d_{i−1} + d_{i} + py.

In the case when a_{i} < ∞ we deduce

dim_{𝔽_{p}} H^{i}(N, 𝔽_{p}) = d_{i−1} + d_{i} + p(a_{i} − d_{i−1}).

Now assume that p = 2 and that a is a sum of two squares in F. In this case, using Theorem 2, we have:

dim_{𝔽_{2}} H^{i}(N, 𝔽_{2}) = Υ_{1} + 2y + z.

Because a is a sum of two squares in F, we see that −1 ∈ N_{E/F} (E^{×}). Then (a) · (a) = (a) · (−1) = 0 in H^{2}(T, 𝔽_{2}) and we obtain that ann(a, −1) = H^{i−1}(T, 𝔽_{2}) and Υ_{1} = dim_{𝔽_{2}} H^{i−1}(T, 𝔽_{2})/ ann(a) = d_{i−1}. Furthermore, in this case (a) · H^{i−1}(T, 𝔽_{2}) ⊂ cor H^{i}(N, 𝔽_{2}) and therefore

z = dim_{𝔽_{2}} \frac{H^{i}(T, 𝔽_{2})}{cor H^{i}(N, 𝔽_{2})} = d_{i}.

Thus we recover the same formula as above:

dim_{𝔽_{2}} H^{i}(N, 𝔽_{2}) = d_{i−1} + d_{i} + 2y.

In the case when a_{i} < ∞ we deduce

dim_{𝔽_{2}} H^{i}(N, 𝔽_{2}) = d_{i−1} + d_{i} + 2(a_{i} − d_{i−1}).

Finally assume that p = 2 and that a is not necessarily a sum of two squares in F. Then from Theorem 2

dim_{𝔽_{2}} H^{i}(N, 𝔽_{2}) = Υ_{1} + 2y + z

= dim_{𝔽_{2}} \frac{ann(a, −1)}{ann(a)} + 2 dim_{𝔽_{2}} \frac{cor H^{i}(N, 𝔽_{2})}{(a) · ann(a, −1)}

+ dim_{𝔽_{2}} \frac{H^{i}(T, 𝔽_{2})}{(a) · H^{i−1}(T, 𝔽_{2}) + cor H^{i}(N, 𝔽_{2})}. (2)

Now assume that h_{i−1} and h_{i} are finite. Then equation (2) may be simplified, as follows. Observe that

dim_{𝔽_{2}} \frac{(a) · H^{i−1}(T, 𝔽_{2}) + cor H^{i}(N, 𝔽_{2})}{cor H^{i}(N, 𝔽_{2})}

= dim_{𝔽_{2}} \frac{(a) · H^{i−1}(T, 𝔽_{2})}{(a) · H^{i−1}(T, 𝔽_{2}) ∩ cor H^{i}(N, 𝔽_{2})}

We claim that

(a) · H^{i−1}(T, 𝔽_{2}) ∩ cor H^{i}(N, 𝔽_{2}) = (a) · ann(a, −1).

The right-hand side is contained in the left-hand side by Theorem 2(5). Now suppose that for some h ∈ H^{i−1}(T, 𝔽_{2}), (a) · h = cor s for some s ∈ H^{i}(N, 𝔽_{2}). Then (a, −1) · h = (a, a) · h = 0 since cor H^{i}(N, 𝔽_{2}) = ann(a) and, since p = 2, (a, a) = (a, −1). Then h ∈ ann(a, −1) and our claim is proved. Therefore

dim_{𝔽_{2}} \frac{(a) · H^{i−1}(T, 𝔽_{2}) + cor H^{i}(N, 𝔽_{2})}{cor H^{i}(N, 𝔽_{2})} = dim_{𝔽_{2}} \frac{(a) · H^{i−1}(T, 𝔽_{2})}{(a) · ann(a, −1)}

= dim_{𝔽_{2}} \frac{H^{i−1}(T, 𝔽_{2})}{ann(a, −1)}.

Now in equation (2) we add and subtract two copies of

dim_{𝔽_{2}} \frac{(a) · H^{i−1}(T, 𝔽_{2})}{(a) · ann(a, −1)} = dim_{𝔽_{2}} \frac{H^{i−1}(T, 𝔽_{2})}{ann(a, −1)}.

We then obtain

dim_{𝔽_{2}} H^{i}(N, 𝔽_{2}) = (dim_{𝔽_{2}} \frac{ann(a, −1)}{ann(a)} + dim_{𝔽_{2}} \frac{H^{i−1}(T, 𝔽_{2})}{ann(a, −1)}) +
(dim_{𝔽_{2}} \frac{H^{i}(T, 𝔽_{2})}{cor H^{i}(N, 𝔽_{2})})

+ 2( dim_{𝔽_{2}} cor H^{i}(N, 𝔽_{2}) − dim_{𝔽_{2}}(a) · H^{i−1}(T, 𝔽_{2}))

= d_{i−1} + d_{i} + 2(a_{i} − d_{i−1}).
□

Remark. We observe that if p = 2 and h_{i−1} and h_{i} are finite, the last formula of the proposition also follows from Arason’s long exact sequence [A, Satz 4.5].

8. Proofs of Theorem 3 and Corollary 2

Proof of Theorem 3. Let T = G_{F} (p) be of finite rank and N an open subgroup of index p. Hence the nth partial Euler-Poincaré characteristic χ_{n}(T) is well-defined. For 0 ≤ i ≤ n, let a_{i}, d_{i}, and h_{i} be defined as in section 7. Because dim_{𝔽_{p}} F^{×}/F^{×p} < ∞ implies that dim_{𝔽_{p}} E^{×}/E^{×p} < ∞ by Theorems 1 and 2 in the case n = 1, we see that a_{i}, d_{i}, h_{i}, and χ_{i}(N) are all well-defined for all i ∈ ℕ.

By Proposition 3 we calculate

χ_{n}(N) = ∑_{i=0}^{n}(−1)^{i} dim_{𝔽_{p}} H^{i}(N, 𝔽_{p})

= 1 + ∑_{i=1}^{n}(−1)^{i}(d_{i−1} + d_{i} + p(a_{i} − d_{i−1}))

= (−1)^{n}d_{n} + p (∑_{i=1}^{n}(−1)^{i}a_{i}) − p (∑_{i=1}^{n}(−1)^{i}d_{i−1}) .

Since h_{i} = a_{i} + d_{i}, we have

pχ_{n}(T) − χ_{n}(N) = p (1 + ∑_{i=1}^{n}(−1)^{i}(a_{i} + d_{i})) + (−1)^{n+1}d_{n}

− p (∑_{i=1}^{n}(−1)^{i}a_{i}) + p (∑_{i=1}^{n}(−1)^{i}d_{i−1})

= p(−1)^{n}d_{n} + (−1)^{n+1}d_{n}
= (−1)^{n}(p − 1)d_{n}.

We have therefore shown item (a).

Now if p > 2 then by Theorem 1(7) and the fact that F^{0}(N) = {0} we have

χ_{n}^{𝔉}(N) = p ∑_{i=1}^{n}(−1)^{i}(a_{i} − d_{i−1}).

Moreover, if p = 2 and −1 ∈ N_{E/F} (E), then (a, −1) = 0 in H^{2}(T, 𝔽_{2}) and ann_{H^{n−1}(T,𝔽_{2})}(a, −1) = H^{n−1}(T, 𝔽_{2}). By Theorem 2(6) we obtain that the formula for the rank of Y has the same shape as the formula in the case p > 2:

rank_{𝔽_{2}[G]} Y = dim_{𝔽_{2}} cor H^{n}(N, 𝔽_{2})/(a) · H^{n−1}(T, 𝔽_{2})
= a_{n} − Υ_{1}
= a_{n} − d_{n−1}.

Therefore in this case we have

χ_{n}^{𝔉}(N) = 2 ∑_{i=1}^{n}(−1)^{i}(a_{i} − d_{i−1})

as well. In either case p > 2 or p = 2 and −1 ∈ N_{E/F} (E), we then calculate

χ_{n}(N) = 1 + ∑_{i=1}^{n}(−1)^{i}(d_{i−1} + d_{i} + p(a_{i} − d_{i−1}))
= (−1)^{n}d_{n} + χ_{n}^{𝔉}(N),

showing equation (b). □

Corollary 1 is an immediate consequence of Theorem 3.

Proof of Corollary 2.

(1) ⟹ (2): Assume that χ_{n}(N) = pχ_{n}(T) for all subgroups N of T of index p. Then by Corollary 1, the corestriction map cor_{n} : H^{n}(N, 𝔽_{p}) → H^{n}(T, 𝔽_{p}) is surjective for all such subgroups N. By exact sequence (1) of section 6, res_{n+1} : H^{n+1}(T, 𝔽_{p}) → H^{n+1}(N, 𝔽_{p}) is injective for all such subgroups N.

By the Bloch-Kato Conjecture (see Theorem 4), H^{n+1}(T, 𝔽_{p}) is generated by elements (a_{1}) · (a_{2}) ⋯ (a_{n+1}), a_{i} ∈ F^{×}, (a_{i}) ∈ H^{1}(T, 𝔽_{p}). We claim that these elements are equal to zero. If (a_{1}) = 0, then (a_{1}) ⋯ (a_{n+1}) = 0. If (a_{1}) ≠ 0, then let E = F (\sqrt[p]{a_{1}}) be the cyclic extension of degree p and set N = Gal(F (p)/E). Then (a_{1}) ⋯ (a_{n+1}) lies in the kernel of res_{n+1} : H^{n+1}(T, 𝔽_{p}) → H^{n+1}(N, 𝔽_{p}). Since res_{n+1} is injective, (a_{1}) ⋯ (a_{n+1}) = 0. Thus H^{n+1}(T, 𝔽_{p}) = {0}, as required.

(2) ⟹ (3) follows from the fact that for each open subgroup U of T, cd(U) = cd(T) ≤ n (see [NSW, Prop. 3.3.5]) and hence for each open subgroup V of U of index p, cor: H^{n}(V, 𝔽_{p}) → H^{n}(U, 𝔽_{p}) is surjective. (See [NSW, Prop. 3.3.8].) Then by Corollary 1 we have χ_{n}(V) = pχ_{n}(U).

(3) ⟹ (1) follows by setting U = T. □

Acknowledgements

We are grateful to M. Rost and to J.-P. Serre for correspondence concerning the Bloch-Kato Conjecture and to C. Weibel for his counsel concerning the formulation of the main theorems. We thank A. Adem, D. J. Benson, M. Bush, J. F. Carlson, S. Chebolu, D. Eisenbud, M. Fried, F. Hajir, H. Koch, M. Kolster, J. Labute, T. Lawson, H. Lenstra, C. Maire, T. Nguyen-Quang-Do, Z. Reichstein, M. Rost, A. Schultz, J.-P. Serre, R. T. Sharifi, J.-P. Tignol, C. Weibel, T. Weigel, and P. Zalesskii for their interesting comments, suggestions, and encouraging remarks concerning this paper. Finally we thank the referee for the very interesting and useful comments and suggestions related to this paper. In particular the Appendix [BeLMS] is a result of his/her suggestion.

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Department of Mathematics, Middlesex College, University of Western Ontario, London, Ontario N6A 5B7 CANADA

E-mail address: nlemire@uwo.ca

E-mail address: minac@uwo.ca

Department of Mathematics, Davidson College, Box 7046, Davidson, North Carolina 28035-7046 USA

E-mail address: joswallow@davidson.edu