COHOMOLOGICAL DIMENSION AND SCHREIER’S FORMULA IN GALOIS COHOMOLOGY

JOHN LABUTE, NICOLE LEMIRE^{†}, JÁN MINÁČ^{‡}, AND JOHN SWALLOW

Abstract. Let p be a prime and F a field containing a primitive pth root of unity. If p > 2 assume also that F is perfect. Then for n ∈ ℕ, the cohomological dimension of the maximal pro-p-quotient G of the absolute Galois group of F is n if and only if the corestriction maps H^{n}(H, 𝔽_{p}) → H^{n}(G, 𝔽_{p}) are surjective for all open subgroups H of index p. Using this result we derive a surprising generalization to dim_{𝔽_{p}} H^{n}(H, 𝔽_{p}) of Schreier’s formula for dim_{𝔽_{p}} H^{1}(H, 𝔽_{p}).

For a prime p, let F (p) denote the maximal p-extension of a field F. One of the fundamental questions in the Galois theory of p-extensions is to discover useful interpretations of the cohomological dimension cd(G) of the Galois group G = Gal(F (p)/F) in terms of the arithmetic of p-extensions of F. When cd(G) = 1, for instance, we know that G is a free pro-p-group [S1, §3.4], and when cd(G) = 2 we have important information on the G-module of relations in a minimal presentation [K, §7.3].

For a fixed n > 2, however, little is known about the structure of p-extensions when cd(G) = n. Now when n = 1 and G is finitely generated as a pro-p-group, we have Schreier’s well-known formula

(1)  h_{1}(H) = 1 + [G : H](h_{1}(G) − 1)

for each open subgroup H of G, where

h_{1}(H) := dim_{𝔽_{p}} H^{1}(H, 𝔽_{p}).

(See, for instance, [K, Example 6.3].)

Date: November 17, 2004. 2000 Mathematics Subject Classification. Primary 12G05, 12G10. Key words and phrases. cohomological dimension, Schreier’s formula, Galois the-

ory, p-extensions, pro-p-groups.

†Research supported in part by NSERC grant R3276A01. ‡Research supported in part by NSERC grant R0370A01, by the Mathematical

Sciences Research Institute, Berkeley, and by a 2004/2005 Distinguished Research Professorship at the University of Western Ontario.

Observe that from basic properties of p-groups it follows that for each open subgroup H of G there exists a chain of subgroups

G = G_{0} ⊃ G_{1} ⊃ ⋯ ⊃ G_{k} = H

such that G_{i+1} is normal in G_{i} and [G_{i} : G_{i+1}] = p for each i = 0, 1, …, k − 1. Since closed subgroups of free pro-p-groups are free [S1, Corollary 3, §I.4.2], Schreier’s formula (1) is equivalent to the seemingly weaker statement that the formula holds for all open subgroups H of G of index p:

(2)  h_{1}(H) = 1 + p(h_{1}(G) − 1).

We deduce a remarkable generalization of Schreier’s formula for each n ∈ ℕ, as follows. Let F^{×} denote the nonzero elements of a field F, and for c ∈ F^{×}, let (c) ∈ H^{1}(G, 𝔽_{p}) denote the corresponding class. For α ∈ H^{m}(G, 𝔽_{p}) abbreviate by ann_{n} α the annihilator

ann_{n} α = {β ∈ H^{n}(G, 𝔽_{p}) | α ∪ β = 0}.

Finally, set h_{n}(G) = dim_{𝔽_{p}} H^{n}(G, 𝔽_{p}).

Theorem 1. Suppose that ξ_{p} ∈ F and assume that F is perfect if p > 2. Suppose that h_{n}(G) < ∞. Let H be an open subgroup of G of index p, with fixed field F (\sqrt[p]{a}). Then

h_{n}(H) = a_{n−1}(G, H) + p(h_{n}(G) − a_{n−1}(G, H)),

where a_{n−1}(G, H) is the codimension of ann_{n−1}(a):

a_{n−1}(G, H) := dim_{𝔽_{p}} ( H^{n−1}(G, 𝔽_{p})/ ann_{n−1}(a) ).

The proof of Theorem 1 brings additional insight into the structure of Schreier’s formula; in fact, it makes Schreier’s formula transparent for any n ∈ ℕ. In section 1, we derive several interpretations for the statement cd(G) = n. First, we prove in Theorem 2 that if F contains a primitive pth root of unity ξ_{p} and F is perfect if p > 2, then cd(G) ≤ n if and only if the corestriction maps cor : H^{n}(H, 𝔽_{p}) → H^{n}(G, 𝔽_{p}) are surjective for all open subgroups H of G of index p. As a corollary, we show that the corresponding cohomology groups H^{n+1}(H, 𝔽_{p}) are all free as 𝔽_{p}[G/H]-modules if and only if cd(G) ≤ n, under the additional hypothesis that F = F^{2} +F^{2} when p = 2. Finally, we show in Theorem 3 that if G is finitely generated, then cd(G) ≤ n if and only if a single corestriction map, from the Frattini subgroup Φ(G) = G^{p}[G, G] of G, is surjective. In section 2 we prove Theorem 1.

For basic facts about Galois cohomology and maximal p-extensions of fields, we refer to [K] and [S1]. In particular, we work in the category of pro-p-groups.

1. When is cd(G) = n?

As a consequence of recent results of Rost and Voevodsky on the Bloch-Kato conjecture, we have the following interesting translation of the statement cd(G) ≤ n for a given n ∈ ℕ.

Theorem 2. Suppose that ξ_{p} ∈ F and assume that F is perfect if p > 2. Then for each n ∈ ℕ we have cd(G) ≤ n if and only if

cor : H^{n}(H, 𝔽_{p}) → H^{n}(G, 𝔽_{p})

is surjective for every open subgroup H of G of index p.

Proof. Suppose that F satisfies the conditions of the theorem, and let G_{F (p)} be the absolute Galois group of F (p).

Observe that since F contains ξ_{p}, the maximal p-extension F (p) is closed under taking pth roots and hence H^{1}(G_{F (p)}, 𝔽_{p}) = {0}. By the Bloch-Kato conjecture, proved in [V1, Theorem 7.1], the subring of the cohomology ring H^{⋆}(G_{F (p)}, 𝔽_{p}) consisting of elements of positive degree is generated by cup-products of elements in H^{1}(G_{F (p)}, 𝔽_{p}) . Hence H^{n}(G_{F (p)}, 𝔽_{p}) = {0} for n ∈ ℕ. Then, considering the Lyndon-Hochschild-Serre spectral sequence associated to the exact sequence

1 → G_{F (p)} → G_{F} → G → 1,

we have that

(3)  inf : H^{⋆}(G, 𝔽_{p}) → H^{⋆}(G_{F} , 𝔽_{p})

is an isomorphism.

Now suppose that cor : H^{n}(H, 𝔽_{p}) → H^{n}(G, 𝔽_{p}) is surjective for all open subgroups H of G of index p. Let K be the fixed field of such a subgroup H. Then K = F (\sqrt[p]{a}) for some a ∈ F^{×}. From Voevodsky’s theorem [V1, Proposition 5.2], modified in [LMS1, Theorem 5] and translated to G from G_{F} via the inflation maps (3) above, we obtain the following exact sequence:

(4) H^{n}(H, 𝔽_{p}) ⟶^{cor} H^{n}(G, 𝔽_{p}) ⟶^{−∪(a)} H^{n+1}(G, 𝔽_{p}) ⟶^{res} H^{n+1}(H, 𝔽_{p}).

Therefore res : H^{n+1}(G, 𝔽_{p}) → H^{n+1}(H, 𝔽_{p}) is injective for every open subgroup H of G of index p.

Now consider an arbitrary element

α = (a_{1}) ∪ ⋯ ∪ (a_{n+1}) ∈ H^{n+1}(G, 𝔽_{p}),

where a_{i} ∈ F^{×} and (a_{i}) is the element of H^{1}(G, 𝔽_{p}) associated to a_{i}, i = 1, 2, …, n + 1. Suppose that (a_{1}) ≠ 0, and set K = F (\sqrt[p]{a_{1}}) and H = Gal(F (p)/K). We have 0 = res(α) ∈ H^{n+1}(H, 𝔽_{p}). Since res is injective, α = 0. Again by the Bloch-Kato conjecture [V1, Theorem 7.1], we know that H^{n+1}(G, 𝔽_{p}) is generated by the elements α above. Hence H^{n+1}(G, 𝔽_{p}) = {0} and therefore cd(G) ≤ n. (See [K, page 49].)

Conversely, if cd(G) ≤ n then from exact sequence (4) we conclude that cor : H^{n}(H, 𝔽_{p}) → H^{n}(G, 𝔽_{p}) is surjective for open subgroups H of G of index p. □

Using conditions obtained in [LMS2] for H^{n}(H, 𝔽_{p}) to be a free 𝔽_{p}[G/H]-module, we obtain the following corollary. We observe the convention that {0} is a free 𝔽_{p}[G/H]-module.

Corollary. Suppose that ξ_{p} ∈ F and assume that F is perfect if p > 2. If p = 2 assume also that F = F^{2} + F^{2}. Then for each n ∈ ℕ, we have that H^{n+1}(H, 𝔽_{p}) is a free 𝔽_{p}[G/H]-module for every open subgroup H of G of index p if and only if cd(G) ≤ n.

Observe that the condition F = F^{2} + F^{2} is satisfied in particular when F contains a primitive fourth root of unity i: for all c ∈ F^{×}, c = ((c + 1)/2)^{2} + ((c − 1)i/2)^{2}.

Proof. Assume that F is as above, n ∈ ℕ, and that H^{n+1}(H, 𝔽_{p}) is a free 𝔽_{p}[G/H]-module for every open subgroup H of G of index p. If p > 2, then it follows from [LMS2, Theorem 1] that the corestriction maps cor : H^{n}(H, 𝔽_{p}) → H^{n}(G, 𝔽_{p}) are surjective for all such subgroups H.

If p = 2, then we consider open subgroups H of index 2 with corresponding fixed fields K = F (\sqrt{a}). From [LMS2, Theorem 1] we obtain that ann_{n}(a) = ann_{n} ((a) ∪ (−1)). It follows from the hypothesis F = F^{2}+F^{2} that (c)∪(−1) = 0 ∈ H^{2}(G, 𝔽_{2}) for each c ∈ F^{×} and in particular for c = a. Hence ann_{n}(a) = H^{n}(G, 𝔽_{2}). But then from exact sequence (4) above, we deduce that cor : H^{n}(H, 𝔽_{2}) → H^{n}(G, 𝔽_{2}) is surjective.

Since our analysis holds for all open subgroups H of index p, by Theorem 2 we conclude that cd(G) ≤ n.

Assume now that cd(G) ≤ n. Then by Serre’s theorem in [S2] we find that cd(H) ≤ n for every open subgroup H of G. Hence H^{n+1}(H, 𝔽_{p}) = {0} which, by our convention, is a free 𝔽_{p}[G/H]-module, as required.

□

Remark. When p = 2 and F ≠ F^{2}+F^{2}, the statement of the corollary may fail. Consider the case F = ℝ. Then the only subgroup H of index 2 in G = ℤ/2ℤ is H = {1}. Then for all n ∈ ℕ, H^{n+1}(H, 𝔽_{2}) = {0} and is free as an 𝔽_{2}[G/H]-module. However, cd(G) = ∞.

Under the additional assumption that G is finitely generated, we show that the surjectivity of a single corestriction map is equivalent to cd(G) ≤ n.

Theorem 3. Suppose that ξ_{p} ∈ F and assume that F is perfect if p > 2. Suppose that G is finitely generated. Then for each n ∈ ℕ we have cd(G) ≤ n if and only if

cor : H^{n}(Φ(G), 𝔽_{p}) → H^{n}(G, 𝔽_{p})

is surjective.

Proof. Because G is finitely generated, the index [G : Φ(G)] is finite, and we may consider a suitable chain of open subgroups

G = G_{0} ⊃ G_{1} ⊃ ⋯ ⊃ G_{k} = Φ(G)

such that [G_{i} : G_{i+1}] = p for each i = 0, 1, …, k − 1.

By Serre’s theorem in [S2], cd(H) = cd(G) for every open subgroup H of G. Hence if cd(G) ≤ n we may iteratively apply Theorem 2 to the chain of open subgroups to conclude that

cor : H^{n}(Φ(G), 𝔽_{p}) → H^{n}(G, 𝔽_{p})

is surjective.

Assume now that cor : H^{n}(Φ(G), 𝔽_{p}) → H^{n}(G, 𝔽_{p}) is surjective. For each open subgroup H of G of index p we have a commutative diagram of corestriction maps

\xymatrix{H^{n}(Φ(G), 𝔽_{p}) \ar[r] \ar[dr] & H^{n}(H, 𝔽_{p}) \ar[d] \\  & H^{n}(G, 𝔽_{p})}

since Φ(G) ⊂ H. We obtain that cor : H^{n}(H, 𝔽_{p}) → H^{n}(G, 𝔽_{p}) is surjective, and by Theorem 2 we deduce that cd(G) ≤ n, as required.

□

2. Schreier’s Formula for H^{n}

We now prove Theorem 1. Suppose that cd(G) = n, and let H be an open subgroup of G of index p. By Theorem 2, the corestriction map cor : H^{n}(H, 𝔽_{p}) → H^{n}(G, 𝔽_{p}) is surjective.

Let K = F (\sqrt[p]{a}) be the fixed field of H. Since H^{n+1}(G, 𝔽_{p}) = \{0\} by hypothesis, we conclude that ann_{n−1} ((a) ∪ (ξ_{p})) = H^{n−1}(G, 𝔽_{p}). Then by [LMS1, Theorem 1], we obtain the decomposition

H^{n}(H, 𝔽_{p}) = X ⊕ Y,

where X is a trivial 𝔽_{p}[G/H]-module and Y is a free 𝔽_{p}[G/H]-module. Moreover

x := rank_{𝔽_{p}} X = dim_{𝔽_{p}} H^{n−1}(G, 𝔽_{p})/ ann_{n−1}(a) = a_{n−1}(G, H), and
y := rank Y = dim_{𝔽_{p}} H^{n}(G, 𝔽_{p})/(a) ∪ H^{n−1}(G, 𝔽_{p}).

Therefore h_{n}(H) = dim_{𝔽_{p}} H^{n}(H, 𝔽_{p}) = x + py.

Now, considering the exact sequence

0 → \frac{H^{n−1}(G, 𝔽_{p})}{ann_{n−1}(a)} ⟶^{−∪(a)} H^{n}(G, 𝔽_{p}) → \frac{H^{n}(G, 𝔽_{p})}{(a) ∪ H^{n−1}(G, 𝔽_{p})} → 0,

we see that dim_{𝔽_{p}} H^{n}(H, 𝔽_{p}) is equal to the sum of the dimension x of the kernel and p times the dimension y of the cokernel, and the theorem follows.

Observe that we have established a more general formula than the formula displayed in Theorem 1, since we have not assumed that h_{n}(G) is finite.

When n = 1, ann_{n−1}(a) = {0} so that a_{n−1}(G, H) = 1. Therefore when G is finitely generated we recover Schreier’s formula (2):

h_{1}(H) = 1 + p(h_{1}(G) − 1).

References

[K] H. Koch, Galois theory of p-extensions. Berlin: Springer-Verlag, 2002.

[LMS1] N. Lemire, J. Mináč, and J. Swallow. Galois module structure of Galois cohomology. ArXiv:math.NT/0409484 (2004).

[LMS2] N. Lemire, J. Mináč, and J. Swallow. When is Galois cohomology free or trivial? ArXiv:math.NT/0410617 (2004).

[S1] J.-P. Serre. Galois cohomology. Heidelberg: Springer-Verlag, 1997.

[S2] J.-P. Serre. Sur la dimension cohomologique des groupes profinis. Topology 3 (1965), 413–420.

[V1] V. Voevodsky. Motivic cohomology with ℤ/2-coefficients. Publ. Inst. Hautes Études Sci., No. 98 (2003), 59–104.

[V2] V. Voevodsky. On motivic cohomology with ℤ/l coefficients. K-theory preprint archive 639 (2003). www.math.uiuc.edu/K-theory/0639/.

Department of Mathematics and Statistics, McGill University, Burnside Hall, 805 Sherbrooke Street West, Montreal, Quebec H3A 2K6 CANADA

  E-mail address: labute@math.mcgill.ca

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