Relation to Subgroups

Example 3.2.9 Let nowGbe a finite cyclic group of order n, generated by an element σ. Consider the maps Z[G] →Z[G] defined by

N : a7→

n−1

X

i=0

σⁱa and σ−1 : a7→σa−a.

One checks easily that ker(N) = Im (σ−1) and Im (N) = ker(σ−1). Hence we obtain a free resolution

· · ·→^N Z[G]−→^σ−1 Z[G]→^N Z[G]−→^σ−1 Z[G]→Z→0, the last map being induced by σ 7→1.

For a G-module A, define maps N : A→ A and σ−1 : A → A by the same formulae as above and put NA:= ker(N). Using the above resolution, one finds

H⁰(G, A) =A^G, H²ⁱ⁺¹(G, A) =NA/(σ−1)A and H²ⁱ⁺²(G, A) =A^G/N A (3) for i >0.

Remark 3.2.10 IfK|k is a finite Galois extension with cyclic Galois group Gas above, the above calculation showsH¹(G, K^×) =NK^×/(σ−1)K^×. The first group is trivial by Hilbert’s Theorem 90 and we get back the original form of the theorem, as established in Example 2.3.4 of the previous chapter.

3.3 Relation to Subgroups 77 Proof: Given anH-homomorphismλ :M→A, consider the mapm7→λm, whereλ_m ∈Hom_H(Z[G], A) is the map determined byg 7→λ(gm). The kind reader will check that we get an element of HomG(M,HomH(Z[G], A)) in this way, and that the two constructions are inverse to each other.

Applying the lemma to the terms of a projective resolution P_• of Z and passing to cohomology groups, we get:

Corollary 3.3.2 (Shapiro’s Lemma) Given a subgroup H of G and an H-module A, there exist canonical isomorphisms

Hⁱ(G, M_H^G(A))→^∼ Hⁱ(H, A) for all i≥0.

The case when H = {1} is particularly important. In this case an H-moduleA is just an abelian group; we denoteM_H^G(A) simply by M^G(A) and call it the co-induced module associated with A.

Corollary 3.3.3 The group Hⁱ(G, M^G(A)) is trivial for all i >0.

Proof: In this case the right hand side in Shapiro’s lemma is trivial (e.g.

because 0→Z→Z→0 gives a projective resolution ofZ).

Remarks 3.3.4

1. It is important to note that the construction of co-induced modules is functorial in the sense that every homomorphism A → B of abelian groups induces a G-homomorphism M^G(A) → M^G(B). Of course, a similar property holds for the modules M_H^G(A).

2. For a G-module A there is a natural injective map A→M^G(A) given by assigning to a∈A the homomorphismZ[G]→A of abelian groups induced by the mapping σ7→ σa.

3. If G is finite, the choice of a Z-basis of Z[G] induces a non-canonical isomorphism M^G(A)∼=A⊗^ZZ[G] for all abelian groups A.

Using Shapiro’s lemma we may define two basic maps relating the coho-mology of a group to that of a subgroup.

Construction 3.3.5 (Restriction maps)LetGbe a group,AaG-module and H a subgroup ofG. There are natural maps of G-modules

A→^∼ HomG(Z[G], A)→HomH(Z[G], A) =M_H^G(A),

the first one given by mapping a ∈ A to the unique G-homomorphism sending 1 to a and the second by considering a G-homomorphism as an H-homomorphism. Taking cohomology and applying Shapiro’s lemma we thus get maps

Res : Hⁱ(G, A)→Hⁱ(H, A)

for all i ≥ 0, called restriction maps. One sees that for i = 0 we get the natural inclusion A^G→A^H.

When the subgroup H has finite index, there is a natural map in the opposite direction.

Construction 3.3.6 (Corestriction maps) LetH be a subgroup ofG of finite index n and let A be aG-module.

Given an H-homomorphismφ : Z[G]→A, define a new map Z[G]→A by the assignment

φ^G_H : x7→

Xn j=1

ρjφ(ρ⁻_j¹x),

where ρ₁, . . . , ρ_n is a system of left coset representatives for H in G. This is manifestly a group homomorphism which does not depend on the choice of the ρj; indeed, if we replace the system of representatives (ρj) by another system (ρ_jτ_j) with some τ_j ∈H, we get ρ_jτ_jφ(τ_j⁻¹ρ⁻_j¹x) = ρ_jφ(ρ⁻_j¹x) for all j, the map φ being an H-homomorphism. Furthermore, the map φ^G_H is also aG-homomorphism, because we have for all σ ∈G

Xn j=1

ρjφ(ρ⁻_j¹σx) =σ Xn

j=1

(σ⁻¹ρj)φ((σ⁻¹ρj)⁻¹x)

!

=σ Xn

j=1

ρjφ(ρ⁻_j¹x)

! , as theσρj form another system of left coset representatives.

The assignment φ 7→φ^G_H thus defines a well-defined map Hom_H(Z[G], A)→Hom_G(Z[G], A)∼=A,

so by taking cohomology and applying Shapiro’s lemma we get maps Cor : Hⁱ(H, A)→Hⁱ(G, A)

for all i≥0, called corestriction maps.

3.3 Relation to Subgroups 79 An immediate consequence of the preceding constructions is the following basic fact.

Proposition 3.3.7 Let G be a group, H a subgroup of finite index n in G and A a G-module. Then the composite maps

Cor◦Res : Hⁱ(G, A)→Hⁱ(G, A) are given by multiplication by n for all i≥0.

Proof: Indeed, ifφ:Z[G]→Ais aG-homomorphism, then for allx∈Z[G]

we have φ^G_H(x) =P

ρjφ(ρ⁻_j¹x) =P

ρjρ⁻_j¹φ(x) =nφ(x).

In the case H={1} we get:

Corollary 3.3.8 Let G be a finite group of order n. Then the elements of Hⁱ(G, A)have finite order dividingn for all G-modules Aand integersi >0.

Another basic construction is the following one.

Construction 3.3.9 (Inflation maps) LetGbe a group, andH a normal subgroup. Then for aG-moduleAthe submoduleA^H of fixed elements under H is stable under the action of G (indeed, for σ ∈ G, τ ∈ H and a ∈ A^H one has τ σa = σ(σ⁻¹τ σ)a =σa). Thus A^H carries a natural structure of a G/H-module.

Now take a projective resolution P_• of Z as a trivial G-module and a projective resolution Q_• of Z as a trivial G/H-module. Each Qi can be considered as a G-module via the projection G → G/H, so applying Lemma 3.1.7 with R = Z[G], B^• = Q_• and α = idZ we get a morphism P_• →Q_• of complexes of G-modules, whence also a map HomG(Q_•, A^H)→ HomG(P_•, A^H). Now since HomG(Qi, A^H) = HomG/H(Qi, A^H) for all i, the former complex equals HomG/H(Q_•, A^H), so by taking cohomology we get maps Hⁱ(G/H, A^H)→Hⁱ(G, A^H) which do not depend on the choices ofP_• and Q_• by the same argument as in the proof of Proposition 3.1.9. Com-posing with the natural map induced by the G-homomorphism A^H →A we finally get maps

Inf : Hⁱ(G/H, A^H)→Hⁱ(G, A), for all i≥0, called inflation maps.

Remark 3.3.10 Calculating the inflation maps in terms of the standard res-olution of Z, we see that inflating an i-cocycle Z[(G/H)ⁱ⁺¹]→A^H amounts to taking the lifting Z[Gⁱ⁺¹]→A^H induced by the projection G→G/H.

Similarly, one checks that the restriction of a cocycle Z[Gⁱ⁺¹] → A to a subgroup H is given by restricting it to a map Z[Hⁱ⁺¹]→A.

Remark 3.3.11 Given a normal subgroupH inGand a G-module A, with trivial H-action, the inflation map Inf : H²(G/H, A) → H²(G, A) has the following interpretation in terms of group extensions: given an extension 0 → A →^π E → (G/H) → 1, its class c(E) satisfies Inf(c(E)) = c(ρ^∗(E)), where ρ : G → G/H is the natural projection, and ρ^∗(E) is the pullback extension ρ^∗(E) defined as the subgroup of E ×G given by elements (e, g) satisfying π(e) = ρ(g). One verifies that ρ^∗(E) is indeed an extension of G by A, and the relation c(ρ^∗(E)) = Inf(c(E)) holds by the construction of inflation maps and that of the classc(E) in Example 3.2.6.

We now turn to the last basic construction relative to subgroups.

Construction 3.3.12 (Conjugation) Let P and A be G-modules and H anormal subgroup of G. For each σ∈G we define a map

σ_∗ : Hom_H(P, A)→Hom_H(P, A)

by setting σ_∗(φ)(p) := σ⁻¹φ(σ(p)) for each p∈ P and φ ∈HomH(P, A). To see that σ_∗(φ) indeed lies in HomH(P, A), we compute forτ ∈H

σ_∗(φ)(τ(p)) =σ⁻¹φ(στ(p)) =σ⁻¹φ(στ σ⁻¹σ(p)) =σ⁻¹στ σ⁻¹φ(σ(p)) =τ σ_∗(φ)(p), where we have used the normality ofHin the penultimate step. Asσ_∗⁻¹is ob-viously an inverse forσ_∗, we get an automorphism of the group HomH(P, A).

It follows from the definition that σ_∗ is the identity for σ∈H.

Now we apply the above to a projective resolution P_• of the trivial G-module Z. Note that this is also a resolution by projective H-modules, because Z[G] is free as a Z[H]-module (a system of coset representatives yields a basis). The construction yields an automorphism σ_∗ of the com-plex HomH(P_•, A), i.e. an automorphism in each term compatible with the G-maps in the resolution. Taking cohomology we thus get automorphisms σⁱ_∗ :Hⁱ(H, A)→Hⁱ(H, A) in each degree i ≥0, and the same method as in Proposition 3.1.9 implies that they do not depend on the choice ofP_•. These automorphisms are trivial for σ ∈ H, so we get an action of the quotient G/H on the groups Hⁱ(H, A), called the conjugation action.

It is worthwhile to record an explicit consequence of this construction.

Lemma 3.3.13 Let

0→A→B →C →0

be a short exact sequence ofG-modules, andH a normal subgroup inG. The long exact sequence

0→H⁰(H, A)→H⁰(H, B)→H⁰(H, C)→H¹(H, A)→H¹(H, B)→ . . . is an exact sequence ofG/H-modules, where the groupsHⁱ(H, A)are equipped with the conjugation action defined above.

3.3 Relation to Subgroups 81 Proof: This follows immediately from the fact that the conjugation action as defined above induces an isomorphism of the exact sequence of complexes

0→HomH(P_•, A)→HomH(P_•, B)→HomH(P_•, C)→0 onto itself.

This lemma will be handy for establishing the following fundamental exact sequence involving inflation and restriction maps.

Proposition 3.3.14 Let G be a group, H a normal subgroup and A a G-module. There is a natural map τ : H¹(H, A)^G/H → H²(G/H, A^H) fitting into an exact sequence

0→H¹(G/H, A^H) −→^Inf H¹(G, A)−→^Res H¹(H, A)^G/H →^τ

→H²(G/H, A^H) −→^Inf H²(G, A).

We begin the proof by the following equally useful lemma.

Lemma 3.3.15 In the situation of the proposition we have

M^G(A)^H ∼=M^G/H(A) and H^j(H, M^G(A)) = 0 f or all j >0.

Proof: The first statement follows from the chain of isomorphisms M^G(A)^H = Hom(Z[G], A)^H ∼= Hom(Z[G/H], A) =M^G/H(A).

As for the second, the already used fact that Z[G] is free as aZ[H]-module implies that M^G(A) is isomorphic to a direct sum of copies of M^H(A).

But it follows from the definition of cohomology that H^j(H,L

M^H(A)) ∼= LH^j(H, M^H(A)), which is 0 by Corollary 3.3.3.

Proof of Proposition 3.3.14: Define C as the G-module fitting into the exact sequence

0→A→M^G(A)→C→0. (4)

This is also an exact sequence ofH-modules, so we get a long exact sequence 0→A^H →M^G(A)^H →C^H →H¹(H, A)→H¹(H, M^G(A)),

where the last group is trivial by the first statement of Lemma 3.3.15 and Corollary 3.3.3. Hence we may split up the sequence into two short exact sequences

0 → A^H →M^G(A)^H →B →0, (5)

0 → B →C^H →H¹(H, A)→0. (6)

Using Lemma 3.3.13 we see that these are exact sequences ofG/H-modules.

Taking the long exact sequence inG/H-cohomology coming from (5) we get 0→A^G→M^G(A)^G→B^G/H →H¹(G/H, A^H)→H¹(G/H, M^G(A)^H), where the last group is trivial by Lemma 3.3.15. So we have a commutative diagram with exact rows

0

y

0 −−−→ A^G −−−→ M^G(A)^G −−−→ B^G/H −−−→ H¹(G/H, A^H)→0

 y^id

 y^id

 y

0 −−−→ A^G −−−→ M^G(A)^G −−−→ C^G −−−→ H¹(G, A)→ 0

 y H¹(H, A)^G/H

 y H¹(G/H, B)

where second row comes from the long exactG-cohomology sequence of (4), and the column from the long exact sequence of (6). A diagram chase shows that we obtain from the diagram above an exact sequence

0→H¹(G/H, A^H)→^α H¹(G, A)→^β H¹(H, A)^G/H →H¹(G/H, B).

Here we have to identify the maps α and β with inflation and restriction maps, respectively. For α, this follows by viewing A^H and B as G-modules via the projection G→G/H and considering the commutative diagram

B^G/H −−−→^id B^G −−−→ C^G



y y y

H¹(G/H, A^H) −−−→^λ H¹(G, A^H) −−−→ H¹(G, A)

where the composite of the maps in the lower row is by definition the inflation map. Hereλis simply given by viewing a 1-cocycleG/H →A^H as a 1-cocycle G→ A^H, and the diagram commutes by the functoriality of the long exact cohomology sequence. As for β, its identification with a restriction map follows from the commutative diagram

C^G −−−→ H¹(G, A)

 y

 y^Res C^H −−−→ H¹(H, A)

3.3 Relation to Subgroups 83 where the left vertical map is the natural inclusion.

Now the remaining part of the required exact sequence comes from the commutative diagram

H¹(H, A)^G/H −−−→ H¹(G/H, B) −−−→ H¹(G/H, C^H) −−−→^Inf H¹(G, C)



y^∼= y^∼=

H²(G/H, A^H) −−−→^Inf H²(G, A) where the top row, coming from (6), is exact atH¹(G/H, B), and the vertical isomorphisms are induced by the long exact sequences coming from (5) and (4), using again thatM^G(A) andM^G(A)^H have trivial cohomology. Commu-tativity of the diagram relies on a compatibility between inflation maps and long exact sequences which is proven in the same way as the one we have just considered for H¹. Finally, the exactness of the sequence of the proposition atH²(G/H, B^H) comes from the exactness of the row in the above diagram, together with the injectivity of the inflation mapH¹(G/H, C^H)→H¹(G, C) that we have already proven (for A in place ofC).

Remark 3.3.16 The map τ of the proposition is called the transgression map. For an explicit description of τ in terms of cocycles, see Neukirch-Schmidt-Wingberg [1], Proposition 1.6.5.

Proposition 3.3.17 In the situation of the previous proposition, let i > 1 be an integer and assume moreover that the groups H^j(H, A) are trivial for 1≤j ≤i−1. Then there is a natural map

τi,A :Hⁱ(H, A)^G/H →Hⁱ⁺¹(G/H, A^H) fitting into an exact sequence

0→Hⁱ(G/H, A^H) −→^Inf Hⁱ(G, A)−→^Res Hⁱ(H, A)^{G/H τ}−→^i,A

→Hⁱ⁺¹(G/H, A^H) −→^Inf Hⁱ⁺¹(G, A).

Proof: Embed A into the co-induced module M^G(A) and let CA be the cokernel of this embedding. The G-module M^G(A) is an H-module in par-ticular, and the assumption thatH¹(H, A) vanishes implies the exactness of the sequence 0→A^H →M^G(A)^H →C_A^H →0 by the long exact cohomology sequence. This is a short exact sequence ofG/H-modules, so taking the as-sociated long exact sequence yields the first and fourth vertical maps in the commutative diagram

0 −−−→ H^j−1(G/H, C_A^H) −−−→^Inf H^j−1(G, CA) −−−→^Res H^j−1(H, CA)^G/H →



y y y

0 −−−→ H^j(G/H, A^H) −−−→^Inf H^j(G, A) −−−→^Res H^j(H, A)^G/H →

τ_j−1,CA

−−−−→ H^j(G/H, C_A^H) −−−→^Inf H^j(G, CA)

 y

 y

τj,A

−−−→ H^j+1(G/H, A^H) −−−→^Inf H^j+1(G, A)

where the other vertical maps come from long exact sequences associated with 0 → A → M^G(A) → C_A → 0, and the maps τ_j,A and τ_j−1,CA are yet to be defined. The second and fifth vertical maps are isomorphisms because H^j(G, M^G(A)) = 0 forj >0 according to Corollary 3.3.3. Moreover, Lemma 3.3.15 shows that the groups H^j(G/H, M^G(A)^H) and H^j(H, M^G(A)) are also trivial for j > 0, hence the first and fourth vertical maps and the map H^j−1(H, CA)→H^j(H, A) inducing the third vertical map are isomorphisms as well. In particular, the assumption yields that H^j(H, C_A) = 0 for all 1 ≤ j < i−1. By induction starting from the case i = 1 proven in the previous proposition, we may thus assume that the map τi−1,CA has been defined and the upper row is exact for j = i. We may then define τ_i,A by identifying it to τi−1,CA via the isomorphisms in the diagram, and from this obtain an exact lower row.

Remarks 3.3.18

1. The proposition is easy to establish using the Hochschild-Serre spectral sequencefor group extensions (see e.g. Shatz [1] or Weibel [1]).

2. The argument proving partb) above is an example of a very useful tech-nique called dimension shifting, which consists of proving statements about cohomology groups by embedding G-modules into co-induced modules and then using induction in long exact sequences. For other examples where this technique can be applied, see the exercises.

Dans le document CENTRAL SIMPLE ALGEBRAS AND GALOIS COHOMOLOGY (Page 76-84)