an element of the coset σUM and, as UM was chosen arbitrarily, this implies that σ is in the closure of H in Gal (K|L). But H is closed by assumption, whence the claim. Finally, the assertion about finite extensions follows from the above in view of Corollary 4.1.8.
Remark 4.1.11 The group Gal (K|k) contains many nonclosed subgroups if K|k is an infinite extension. For instance, cyclic subgroups are usually nonclosed; as a concrete example, one may take the cyclic subgroup ofZb gen-erated by 1. In fact, a closed subgroup of a profinite group is itself profinite, but it can be shown that an infinite profinite group is always uncountable.
Thus none of the countable subgroups in a profinite group are closed.
4.2 Cohomology of Profinite Groups
LetG= lim
← Gα be a profinite group. In this section we attach toG another system of cohomology groups, different from that of the previous chapter for infinite G, which reflects the profiniteness of G and which is more suitable for applications.
By a (discrete) continuous G-module we shall mean a G-module A such that the stabiliser of each a ∈ A is open in G. Unless otherwise stated, we shall always regardA as equipped with the discrete topology; continuous G-modules are then precisely the ones for which the action ofG(equipped with its profinite topology) is continuous. If Gα = G/Uα is one of the standard quotients ofG, the submodule AUα is naturally a Gα-module. The canonical surjection φαβ : Gβ → Gα between two of the standard quotients induces inflation maps Infβα : Hi(Gα, AUα)→Hi(Gβ, AUβ) for alli≥0. Furthermore, the compatibility conditionφαγ =φαβ◦φβγ implies that the groupsHi(Gα, A) together with the maps Infβα form a direct system in the following sense.
Construction 4.2.1 A (filtered) direct system of abelian groups (Bα, ψαβ) consists of:
• a directed partially ordered set (Λ,≤);
• for each α∈Λ an abelian group Bα;
• for each α ≤ β a homomorphism ψαβ : Bα → Bβ such that we have equalities ψαγ =ψβγ ◦ψαβ forα ≤β ≤γ.
The direct limit of the system is defined as the quotient of the direct sum L
α∈ΛBα by the subgroup generated by elements of the form bβ−ψαβ(bα).
It is denoted by lim
→ Bα. Direct limits of abelian groups with additional structure (e.g. rings or modules) are defined in an analogous way.
Also, given direct systems (Bα, ψαβ) and (Cα, ραβ) indexed by the same directed set Λ, together with maps λα : Bα → Cα satisfying λβ ◦ ψαβ = ραβ◦λα for all α ≤β, we have an induced mapλ : lim
→ Bα →lim
→ Cα, called the direct limit of the maps λα.
We can now define:
Definition 4.2.2 Let G= lim
← Gα be a profinite group and A a continuous G-module. For all integers i ≥ 0, we define the i-th continuous cohomology groupHconti (G, A) as the direct limit of the direct system (Hi(Gα, AUα),Infβα) constructed above. In the case when G = Gal (ks|k) for some separable closureks of a field k, we also denote Hconti (G, A) byHi(k, A) and call it the i-th Galois cohomology group of k with values in A.
Example 4.2.3 ConsiderZwith trivial action by a profinite groupG. Then Hcont1 (G,Z) = 0. Indeed, by definition this is the direct limit of the groups H1(G/U,Z) = Hom(G/U,Z) for U open and normal in G, which are trivial, as theG/U are finite and Z is a torsion free abelian group.
Remark 4.2.4 It follows from the definition that Hcont0 (G, A) = H0(G, A) for all continuous G-modules A and that Hconti (G, A) = Hi(G, A) if G is finite.
However, for i >0 andGinfinite the two groups are different in general.
Take, for instance, p = 1, G = Zb and A = Q with trivial Z-action. Thenb Hcont1 (Z,b Q) = lim
→ Hom(Z/nZ,Q) = 0, because Q is torsion free.
On the other hand, H1(bZ,Q) is the group of Z-module homomorphisms Zb → Q. But as Q is a divisible abelian group (i.e. the equation nx = y is solvable inQfor alln ∈Z), one knows that a homomorphismC →Qfrom a subgroupC of an abelian groupB extends to a homomorphismB →Q (see e.g. Weibel [1], p. 39; note that the proof of this fact uses Zorn’s lemma).
Applying this with C =Z, B = Zb and the natural inclusion Z → Q we get a nontrivial homomorphism Zb →Q.
Convention 4.2.5 From now on, all cohomology groups of a profinite group will be understood to be continuous, and we drop the subscript cont from the notation.
We now come to a basic property of the cohomology of profinite groups.
Proposition 4.2.6 For a profinite group G and a continuous G-module A the groups Hi(G, A) are torsion abelian groups for all i >0. Moreover, if G is a pro-p-group, then they are p-primary torsion groups.
4.2 Cohomology of Profinite Groups 105 Proof: This follows from the definition together with Corollary 3.3.8.
Corollary 4.2.7 Let V be a Q-vector space equipped with a continuous ac-tion by a profinite group G. Then Hi(G, V) = 0 for all i >0.
Proof: It follows from the construction of cohomology that in this case the groups Hi(G, V) are Q-vector spaces; since for i > 0 they are also torsion groups, they must be trivial.
Recall that Corollary 3.3.8 was obtained as a consequence of a statement about restriction and corestriction maps. We now adapt these to the profinite situation.
Construction 4.2.8 Let G be a profinite group, H a closed subgroup and A a continuous G-module. Define continuous restriction maps
Res : Hi(G, A)→Hi(H, A) as the direct limit of the system of usual restriction maps
Hi(G/Uα, AUα)→Hi(H/(H∩Uα), AUα), where the Uα are the standard open normal subgroups ofG.
In the case when H is open in G, one defines continuous corestriction maps Cor : Hi(H, A) → Hi(G, A) in a similar way. Finally, when H is a closed normal subgroup in G, one defines inflation maps
Inf : Hi(G/H, AH)→Hi(G, A) as the direct limit of the system of inflation maps
Hi((G/Uα)/(H∩Uα), AH∩Uα)→Hi(G/Uα, AUα).
Manifestly, in the case of a finite G we get back the previous restriction, corestriction and inflation maps.
Remark 4.2.9 In the above situation, one may define the module MHG(A) to be the direct limit lim
→ HomH/(H∩Uα)(Z[G/Uα], AUα), where the Uα are the standard open normal subgroups of G. We have a continuous G-action de-fined by g(φα(xα)) = φα(xαgα), where gα is the image of g in G/Uα; one checks that this action is well defined and continuous. (Note that in the spirit of the convention above we employ the notationMHG(A) for anotherG-module as before; the one defined in Chapter 3 is not continuous in general.) Then we
have MGG(A)∼=A and the Shapiro isomorphism Hi(G, MHG(A))∼= Hi(H, A) holds with a similar proof as in the non-continuous case. In particular, one has the vanishing of the cohomology Hi(G, MG(A)) of (continuous) co-induced modules fori >0. One may then also define the continuous restric-tion and corestricrestric-tion maps using this Shapiro isomorphism, by mimicking the construction of Chapter 3.
As in the non-continuous case, we have:
Proposition 4.2.10 LetGbe a profinite group,H an open subgroup of index n and A a continuous G-module. Then the composite maps
Cor◦Res : Hi(G, A)→Hi(G, A)
are given by multiplication by n for all i > 0. Consequently, the restriction Hi(G, A)→Hi(H, A)is injective on the prime-to-ntorsion part ofHi(G, A).
Proof: Each element of Hi(G, A) comes from some Hi(G/Uα, AUα), and Proposition 3.3.7 applies. The second statement follows because the multiplica-tion-by-n map is injective on the subgroup of elements of order prime to n.
A refined version of the last statement is the following.
Corollary 4.2.11 Let G be a profinite group, p a prime number and H a closed subgroup such that the image of H in each finite quotient of G has order prime top. Then for each continuous G-moduleA the restriction map Hi(G, A)→Hi(H, A)is injective on the p-primary torsion part ofHi(G, A).
Proof: Assume that an element of Hi(G, A) of order prime to p maps to 0 in Hi(H, A). It comes from an element of some H1(Gα, AUα) of which we may assume, up to replacing Uα by a larger subgroup, that it maps to 0 in Hi(H/(H∩Uα), AUα). By the proposition (applied to the finite groupG/Uα) is must then be 0.
The main application of the above corollary will be to pro-p-Sylow sub-groupsof a profinite group G. By definition, these are subgroups ofG which are pro-p-groups for some prime number p and whose images in each finite quotient of G are of index prime top.
Proposition 4.2.12 A profinite group G possesses pro-p-Sylow subgroups for each prime number p, and any two of these are conjugate in G.
The proof uses the following well-known lemma.
Lemma 4.2.13 An inverse limit of nonempty finite sets is nonempty.
4.2 Cohomology of Profinite Groups 107 Proof: The proof works more generally for compact topological spaces.
Given an inverse system (Xα, φαβ) of nonempty compact spaces, consider the subsetsXλµ ⊂Q
Xα consisting of the sequences (xα) satisfyingφλµ(xµ) =xλ
for a fixed pair λ ≤ µ. These are closed subsets of the product, and their intersection is precisely lim
← Xα. Furthermore, the directedness of the index set implies that finite intersections of the Xλµ are nonempty. Since Q
Xα is compact by Tikhonov’s theorem, it ensues that lim
← Xα is nonempty.
Proof of Proposition 4.2.12: Write G as an inverse limit of a system of finite groupsGα. For eachGα, denote by Sα the set of itsp-Sylow subgroups (for the classical Sylow theorems, see e.g. Lang [3]). These form an inverse system of finite sets, hence by the lemma we may find an element S in the limit lim
← Sα. ThisS corresponds to an inverse limit of p-Sylow subgroups of the Gα and hence gives a pro-p-Sylow subgroup of G. If P and Q are two pro-p-Sylow subgroups of G, their images in eachGα are p-Sylow subgroups there and hence are conjugate by somexα∈Gα by the finite Sylow theorem.
WritingXαfor the set of possiblexα’s, we get again an inverse system of finite sets, whose nonempty inverse limit contains an element xwithx−1P x=Q.
Corollary 4.2.11 now implies:
Corollary 4.2.14 IfP is a pro-p-Sylow subgroup of a profinite group G, the restriction maps Res : Hi(G, A) →Hi(P, A) are injective on the p-primary torsion part of Hi(G, A) for all i >0 and continuous G-modulesA.
To conclude this section, we mention another construction from the previ-ous chapter which carries over without considerable difficulty to the profinite case, that of cup-products.
Construction 4.2.15 Given a profinite groupGand continuous G-modules A and B, define the tensor product A⊗B as the tensor product of A and B over Z equipped with the continuous G-action induced by σ(a ⊗ b) = σ(a)⊗σ(b). In the previous chapter we have constructed for all i, j ≥0 and all open subgroups U of G cup-product maps
Hi(G/U, AU)×Hj(G/U, BU)→Hi+j(G/U, AU⊗BU), (a, b)7→a∪b satisfying the relation
Inf(a∪b) = Inf(a)∪Inf(b)
for the inflation map arising from the quotient mapG/V →G/U for an open inclusion V ⊂U. Note that by the above definition of the G-action we have
a natural map AU ⊗ BU → (A ⊗B)U, so by passing to the limit over all inflation maps of the above type we obtain cup-product maps
Hi(G, A)×Hj(G, B)→Hi+j(G, A⊗B), (a, b)7→a∪b for continuous cohomology.
It follows immediately from the non-continuous case that this cup-product is also associative and graded-commutative, and that moreover it satisfies compatibility formulae with restriction, corestriction and inflation maps as in Proposition 3.4.10. It also satisfies the exactness property of Proposi-tion 3.4.8, but for this we have to establish first the long exact cohomology sequence in the profinite setting. We treat this question in the next section.