COHOMOLOGY OF FINITE AND PROFINITE GROUPS

MARK DICKINSON

1. Notation

Let G be a group and k a commutative ring. We let Λ denote the group algebra of G over k and I its augmentation ideal—the two-sided ideal of Λ generated by elements of the form g − 1 for g in G; these two objects fit into an exact sequence

0 → I → Λ → k → 0 (1)

of Λ-modules, where the map Λ → k sends every group element g to 1. This sequence is canonically split as a sequence of k-modules by the map k → Λ sending the identity element of k to that of Λ.

1.1. Modules. We work in the category of left Λ-modules; this is equivalent to the category of right Λ-modules, since any left Λ-module M can be regarded as a right Λ-module by setting mg = g^{−1}m for g in G and m in M and extending to an action of Λ by k-linearity. This category is an abelian category with enough injectives and enough projectives; that is, any module can be expresssed as a submodule of an injective module and as a quotient of a projective module. If M and M′ are any two Λ-modules then the k-modules M′ ⊗_{k} M and Hom_{k}(M′, M) are regarded as Λmodules via the ‘diagonal’ actions g(m′⊗_{k}m) = gm′⊗_{k}gm and (gf)(gm′) = gf(m′) respectively.

1.2. Functors. Given the above data there are six functors in play, appearing in two groups of three: firstly, we have the diagram

\xymatrix{\{ Λ-modules \} \ar[r]^{forgetful} & \{ k-modules \} \ar[l]_{induction} \ar[l]^{coinduction}}

in which the forgetful functor simply ‘forgets’ the group action, the induction functor sends a k-module X to the Λ-module Λ⊗_{k} X, and the coinduction functor sends X to the Λ-module Hom_{k}(Λ, X). The induction and coinduction functors are, respectively, left and right adjoints to the forgetful functor. A Λ-module isomorphic to a module of the form Λ ⊗_{k} X is said to be induced, and a module isomorphic to Hom_{k}(Λ, X) for some X is said to be coinduced.

The second set of functors is given in the diagram

\xymatrix{\{ Λ-modules \} \ar[r]^{coinvariants} \ar[r]_{invariants} & \{ k-modules \} \ar[l]_{trivial action}}

in which the ‘trivial-action’ functor turns a k-module X into a Λ-module by giving it trivial G-action; the invariants functor sends a Λ-module M to the module M^{G}, defined to be the set of elements of M on which G acts trivially; and the coinvariants functor sends M to the module denoted M_{G}, defined to be the quotient M/IM. It is the largest quotient of M on which G acts trivially. The coinvariants and invariants functors are left and right adjoints, respectively, to the trivial-action functor.

One can show that a module is induced if and only if it is of the form Λ⊗_{k} M for some Λ-module M. Indeed, let M be any Λ-module, and let M_{0} be the underlying k-module. Then there is a canonical isomorphism Λ⊗_{k} M → Λ⊗_{k} M_{0} of Λ-modules given by sending an element g ⊗_{k} m to the element g ⊗_{k} g^{−1}m. Similarly, a module is coinduced if and only if it is of the form Hom_{k}(Λ, M) for some Λ-module M; with M_{0} as above the canonical isomorphism Hom_{k}(Λ, M) → Hom_{k}(Λ, M_{0}) sends a map f to the map g ↦ g^{−1}f(g).

From the short exact sequence (1) above we obtain exact sequences

(2) 0 → Hom_{k}(k, M) → Hom_{k}(Λ, M) → Hom_{k}(I, M) → 0

and

0 → I ⊗_{k} M → Λ ⊗_{k} M → k ⊗_{k} M → 0 (3)

of Λ-modules, which are both split as sequences of k-modules. Since Hom_{k}(k, M) and k ⊗_{k} M can be identified with M, the first of these sequences shows that any Λ-module M can be embedded in a coinduced module while the second shows that any M is the quotient of an induced module. These two sequences are valuable in setting up the cohomology and homology theories.

1.3. Cohomology. Recall that the cohomology groups H^{∗}(G, M) of G with coefficients in M are defined by taking right-derived functors of the left-exact invariants functor

M ↦ M^{G} = { m ∈ M | gm = m for all g ∈ G }

while the homology groups H_{∗}(G, M) are defined by taking left-derived functors of the right-exact coinvariants functor

M ↦ M_{G} = M/IM.

These cohomology groups and homology groups are independent of the ring k in the sense that taking cohomology or homology commutes with the forgetful functors which forget the ring structure (thereby turning a k-module into an abelian group and a Λ-module into an abelian group with an action of G).

Recall also that the cohomology groups can be expressed in terms of cochains: let C^{i}(G, M) be the set of all set-maps G^{i} → M, and define a differential map δ_{i}: C^{i}(G, M) → C^{i+1}(G, M) for each i ≥ 0 by

(δ_{i}η)(g_{1}, …, g_{i+1}) = g_{1}η(g_{2}, …, g_{i+1}) +

∑_{j=1}^{i}(−1)^{j}η(g_{1}, …, g_{j}g_{j+1}, …, g_{i+1}) + (−1)^{i+1}η(g_{1}, …, g_{i}).

Then the ith cohomology group H^{i}(C^{∗}(G, M)) of the resulting cochain complex is naturally isomorphic to the cohomology group H^{i}(G, M).

1.4. Subgroups. Suppose that H is a subgroup of G, with group algebra Γ; let M be a Γ-module, then Λ ⊗_{Γ} M and Hom_{Γ}(Λ, M) are Λ-modules and there are natural isomorphisms

(Λ ⊗_{Γ} M)_{G} ≅ M_{H}

and

(Hom_{Γ}(Λ, M))^{G} ≅ M^{H}.

These isomorphisms extend to isomorphisms

H_{∗}(H, M) ≅ H_{∗}(G, Λ ⊗_{Γ} M)

and

H^{∗}(H, M) ≅ H^{∗}(G, Hom_{Γ}(Λ, M))

of δ-functors; this follows since all the functors involved are universal, considered as functors from the category of Γ-modules to the category of k-modules: the universality of the functors on the left of the isomorphisms is clear, and the universality of the other two functors can be seen using Grothendieck’s criteria of effaceability and coeffaceability, since the exact functors Λ ⊗_{Γ} − and Hom_{Γ}(Λ, −) preserve projectives and injectives respectively.

Applying the above isomorphisms to the case where H is the trivial subgroup, we see that the cohomology groups of a coinduced module and the homology groups of an induced module vanish in positive degree.

2. Cohomology of finite groups

If G is finite then the canonical homomorphism Λ ⊗_{k} M → Hom_{k}(Λ, M) of Λmodules which sends an element g ⊗_{k} m to the map f given on group elements g′ by

f(g′) = \begin{cases}m & if g′ = g, and \\ 0 & otherwise \end{cases}

is an isomorphism. Thus the notions of induced module and coinduced module coincide.

2.1. Tate cohomology groups. We continue to assume that G is finite, and we let N denote the norm element ∑_{g∈G} g of the group algebra Λ. Multiplication by N is a Λ-endomorphism of M, and it is easy to check that its kernel contains IM while its image is contained in M^{G}, so that it induces a map N_{M}: M_{G} → M^{G} of Λ-modules. We will use this map to splice together the cohomology and homology long exact sequences arising from a short exact sequence. Let

0 → A → B → C → 0

be a short exact sequence of Λ-modules, then we have a commutative diagram

\xymatrix{H_{1}(G, C) \ar[r] & A_{G} \ar[r] \ar[d]^{N_{A}} & B_{G} \ar[r] \ar[d]^{N_{B}} & C_{G} \ar[r] \ar[d]^{N_{C}} & 0 \\ 0 \ar[r] & A^{G} \ar[r] & B^{G} \ar[r] & C^{G} \ar[r] & H^{1}(G, A)}

in which the rows are exact, being portions of the long exact sequences in homology and cohomology. Now the snake lemma allows us to turn this diagram into a long exact sequence

[figure: H_{1}(G, C); ker(N_{A}); ker(N_{B}); ker(N_{C}); coker(N_{A}); coker(N_{B}); coker(N_{C}); H^{1}(G, A)]

that connects the cohomology and homology long exact sequences. In the light of this construction, we define the Tate cohomology groups of a Λ-module M as follows:

Ĥ^{i}(G, M) = \begin{cases}H^{i}(G, M) & if i > 0, \\ coker(N_{M}) = M^{G}/NM_{G} & if i = 0 \\ ker(N_{M}) = _{N}M/IM & if i = −1, and \\ H_{−i−1}(G, M) & if i < −1. \end{cases}

Here _{N}M denotes those elements of M which are killed by N. To define Tate cohomology as a δ-functor we need the additional information giving the connecting maps for the long exact sequence associated to a short exact sequence; these connecting maps are given by the connecting maps in cohomology and homology along with the extra connecting map described above linking the degree −1 piece to the degree 0 piece.

Having made this definition, there are some things to check: firstly, one should check that this definition does yield a δ-functor; this just involves checking that the map from short exact sequences to long exact sequences is functorial. We’d also like to know that Tate cohomology behaves like a cohomological functor— that it vanishes on a suitable set of modules and that it has appropriate universal properties. But if M is induced then one can check that the Tate cohomology of M is zero in all dimensions; as a result, using the exact sequences of §1, the Tate cohomology is effaceable and coeffaceable in every dimension, which is to say that every module can be expressed as a quotient of an acyclic module and as a submodule of an acyclic module. It follows that the entire δ-functor is determined by its values in any one dimension, and that it is universal in the following sense: if i is any integer and T^{∗} is any other delta functor then each map of functors Ĥ^{i}(G, −) → T^{i} extends uniquely upwards to maps Ĥ^{j}(G, −) → T^{j} for each j ≥ i which are compatible with the δ-functor structure, while any map T^{i} → Ĥ^{i}(G, −) extends uniquely downwards to a δ-functor map T^{j} → Ĥ^{j}(G, −) for all j ≤ i.

Note that the cohomology and homology groups of G with coefficients in M can be recovered easily from the Tate cohomology groups.

2.2. Corestriction and Restriction. Let H be a subgroup of G, not necessarily normal, and let M be a module for G. We’d like to compare the two δ-functors H^{∗}(G, M) and H^{∗}(H, M), considered as functors from the category of Λ-modules to k-modules. Since any induced Λ-module is also induced when regarded as a module for H, both of these δ-functors are effaceable and coeffaceable in every dimension and so universal in the sense described above, hence any map between these two functors (in either direction) in any given dimension automatically extends both upwards and downwards to a unique map of δ-functors. We use this property to define a pair of maps between these two δ-functors.

Definition. The restriction map Res: Ĥ^{∗}(G, −) → Ĥ^{∗}(H, −) is defined to be the unique morphism of δ-functors whose dimension zero component is the natural transformation M^{G}/N_{G}M → M^{H}/N_{H}M induced by the inclusion map M^{G} → M^{H}. The corestriction map Cor: Ĥ^{∗}(H, −) → Ĥ^{∗}(G, −) is the unique morphism of δ-functors whose value in dimension −1 is the natural transformation N_{H} M/I_{H}M → _{N_{G}}M/I_{G}M induced by the inclusion map _{N_{H}} M → _{N_{G}}M.

It is left as an exercise for the reader to check that these definitions of Res and Cor agree with the standard restriction and corestriction maps defined in both homology and cohomology, in the situtation where H is a subgroup of finite index in G.

Proposition 1. Let H be a subgroup of G. The composition

Cor ∘ Res: Ĥ^{∗}(G, M) → Ĥ^{∗}(G, M)

of maps of δ-functors is multiplication by |G|/|H| in every dimension.

To prove this, check in dimension zero and use universality.

Corollary 2. Each Tate cohomology group Ĥ^{q}(G, M) is killed by |G|; if M is a finitely generated Λ-module then Ĥ^{q}(G, M) is finite.

Corollary 3. If multiplication by |G| is surjective on M then Ĥ^{q}(G, M) = 0 for every integer q.

In particular, if we choose a ring k and a finite group G such that |G| is an invertible element of k, then the entire cohomology theory vanishes. This is the situation of ordinary representation theory of finite groups, where one considers representations over a field of characteristic not dividing |G|.

2.3. Cup products. Let G be a finite group, and let A and B be Λ-modules.

Proposition 4. There is a unique family of homomorphisms

Ĥ^{p}(G, A) ⊗_{k} Ĥ^{q}(G, B) ⟶^{∪} Ĥ^{p+q}(G, A ⊗_{k} B)

with the following properties:

• The cup product is natural in A and B; that is, if A → A′ and B → B′ are two Λ-module maps, then the diagram

\xymatrix{Ĥ^{p}(G, A) ⊗_{k} Ĥ^{q}(G, B) \ar[r]^{∪} \ar[d] & Ĥ^{p+q}(G, A ⊗_{k} B) \ar[d] \\ Ĥ^{p}(G, A′) ⊗_{k} Ĥ^{q}(G, B′) \ar[r]^{∪} & Ĥ^{p+q}(G, A′ ⊗_{k} B′)}

commutes.

• When p and q are both equal to zero, the cup product is induced by the natural

map

A^{G} ⊗_{k} B^{G} → (A ⊗_{k} B)^{G}

of Λ-modules.

• If 0 → A′ → A → A′′ → 0 is a short exact sequence of Λ-modules such that the sequence 0 → A′ ⊗_{k} B → A ⊗_{k} B → A′′ ⊗_{k} B → 0 is also exact, then the diagram

\xymatrix{Ĥ^{p}(G, A′′) ⊗_{k} Ĥ^{q}(G, B) \ar[d]_{∪} \ar[r]^{δ⊗_{k}1} & Ĥ^{p+1}(G, A′) ⊗_{k} Ĥ^{q}(G, B) \ar[d]^{∪} \\ Ĥ^{p+q}(G, A′′ ⊗_{k} B) \ar[r]^{δ} & Ĥ^{p+q+1}(G, A′ ⊗_{k} B)}

whose horizontal maps are those arising from the long exact sequences, and whose vertical maps are given by cup product, commutes.

• Similarly, if

0 → B′ → B → B′′ → 0

is a short exact sequence of Λ-modules which remains exact when tensoring on the left with A, then the analogous diagram

\xymatrix{Ĥ^{p}(G, A) ⊗_{k} Ĥ^{q}(G, B′′) \ar[d]_{(−1)^{p}∪} \ar[r]^{1⊗_{k}δ} & Ĥ^{p}(G, A) ⊗_{k} Ĥ^{q+1}(G, B′) \ar[d]^{∪} \\ Ĥ^{p+q}(G, A ⊗_{k} B′′) \ar[r]^{δ} & Ĥ^{p+q+1}(G, A ⊗_{k} B′)}

also commutes. Note the sign in the left-hand vertical arrow.

The hypotheses for the last two conditions will always be satisfied when the short exact sequence in question is split as a sequence of k-modules. In particular the basic exact sequences in equations (2) and (3) of §1 are k-split; this allows us to use dimension shifting techniques to prove results involving the cup product.

As with the definition of a derived functor, the proof of uniqueness is straightforward using dimension shifting arguments and the proof of existence takes a little more work; see for example the article by Atiyah and Wall in Cassels and Fröhlich[1].

There are various formulas involving the cup product, all of which can be proved by the same technique: namely, prove in degree zero and then dimension shift. The two most important are those giving the associativity and graded commutativity. For a ∈ Ĥ^{p}(G, A), b ∈ Ĥ^{q}(G, B) and c ∈ Ĥ^{r}(G, C) we have

(a · b) · c = a · (b · c)

modulo the identification (A ⊗_{k} B) ⊗_{k} C ≅ A ⊗_{k} (B ⊗_{k} C) of Λ-modules and

a · b = (−1)^{pq}b · a

modulo the identification A ⊗_{k} B ≅ B ⊗_{k} A of Λ-modules. If H is a subgroup of G then we also have formulas involving the restriction and corestriction maps:

Res(a · b) = Res(a) · Res(b)

and

Cor(a · Res(b)) = Cor(a) · b.

See Cassels and Fröhlich for details.

The cup product has a particularly simple description in degree zero, which we now describe. Let a be an element of A^{G}, representing an element of Ĥ^{0}(G, A). Then a defines a homomorphism

α_{a}: B → A ⊗_{k} B

by sending b in B to a ⊗_{k} b. Since a has trivial G-action this is a homomorphism of Λ-modules and so applying the ith cohomology functor to it yields a homomorphism

α_{a}: Ĥ^{i}(G, B) → Ĥ^{i}(G, A ⊗_{k} B).

Now the assignment a ↦ α_{a} is k-linear, so that we can interpret the collection of all such maps as a Λ-homomorphism

A^{G} ⊗_{k} Ĥ^{i}(G, B) → Ĥ^{i}(G, A ⊗_{k} B).

One can check that this map is none other than the composition of the cup product map with the natural quotient map A^{G} → A^{G}/NA = Ĥ^{0}(G, A).

3. Cohomology of finite cyclic groups

We use the notation (G, φ) to denote a finite cyclic group G with given generator φ. In this section we give explicit descriptions of the Tate cohomology groups Ĥ^{i}(G, M) for a Λ-module M.

We let N or N_{G} denote the norm element of Λ as before, and we let D or D_{G} denote the element φ − 1 of Λ; D is a generator for the augmentation ideal I, considered as a Λ-module. Now consider the complex

K^{∗} = ⋯ → Λ ⟶^{D} Λ ⟶^{N} Λ ⟶^{D} Λ ⟶^{N} Λ → ⋯

of Λ-modules in which the maps are alternately multiplication by N and by D, the map from the degree zero part being the latter. For a Λ-module M define a set of functors T^{i}(G, −), one for each integer i, from the category of Λ-modules to the category of k-modules by first tensoring K^{∗} with M over Λ to give a sequence

⋯ → M ⟶^{D} M ⟶^{N} M ⟶^{D} M ⟶^{N} M → ⋯ (4)

(identifying the tensor product Λ ⊗_{Λ} M with M) and then taking cohomology:

T^{i}(G, M) = H^{i}(K^{∗} ⊗_{Λ} M).

More explicitly we have

T^{i}(G, M) = \begin{cases}M^{G}/NM & if i is even and \\ _{N}M/IM & if i is odd. \end{cases}

Given a short exact sequence 0 → A → B → C → 0 of Λ-modules we obtain a short exact sequence of complexes

0 → K^{∗} ⊗_{Λ} A → K^{∗} ⊗_{Λ} B → K^{∗} ⊗_{Λ} C → 0

and from this a boundary map in the usual way, making T^{∗}(G, −) a δ-functor.

Proposition 5. The two δ-functors T^{∗}(G, −) and Ĥ^{∗}(G, −) are isomorphic.

Proof. In degree zero the two functors T^{0}(G, −) and Ĥ^{0}(G, −) are naturally isomorphic, by definition. Note that T^{∗}(G, −) vanishes on induced modules, and so is universal in the sense previously described; the isomorphism in degree zero now extends uniquely up and down to an isomorphism of δ-functors by universality of both δ-functors. One can also check that the induced isomorphism in degree −1 is the obvious one.

Although the cohomology groups themselves do not depend on the choice of generator φ, the isomorphism just described does. To complete the description of the cohomology of finite cyclic groups, it remains to describe two things: functoriality in the group, and the cup product structure.

3.1. Cup products. In the case of a finite cyclic group, the cup product maps Ĥ^{i}(G, A) ⊗_{k} Ĥ^{j}(G, B) → Ĥ^{i+j}(G, A ⊗_{k} B) turn out to depend only on the values of i and j modulo two. Accepting this fact for the moment, we already have a good understanding of the cup product when one or other of the indices is equal to zero, and hence when either of the indices is even: if both i and j are even then by definition it is the map induced by the natural map

A^{G} ⊗_{k} B^{G} → (A ⊗_{k} B)^{G}

and if i is even and j is odd then it is the map induced by the natural map

A^{G} ⊗_{k} _{N}B → _{N}(A ⊗_{k} B),

from which we can compute the cup product when i is odd and j is even by graded commutativity. In this section we compute the cup product in the case when i and j are both odd and leave the reader to check that the product thus defined satisfies the required conditions.

let A be any Λ-module, then we showed in §1 that A fits into an exact sequence

0 → A → Hom_{k}(Λ, A) → Hom_{k}(I, A) → 0 (5)

of Λ-modules, which is canonically split as a sequence of k-modules. Tensoring this sequence with B we obtain a short exact sequence

0 → A ⊗_{k} B → Hom_{k}(Λ, A) ⊗_{k} B → Hom_{k}(I, A) ⊗_{k} B → 0 (6)

of k-modules. From the definition of the cup product these exact sequences give a commutative diagram

\xymatrix{Ĥ^{0}(G, Hom_{k}(I, A)) ⊗_{k} Ĥ^{1}(G, B) \ar[d]_{∪} \ar[r]^{δ⊗_{k}1} & Ĥ^{1}(G, A) ⊗_{k} Ĥ^{1}(G, B) \ar[d]^{∪} \\ Ĥ^{1}(G, Hom_{k}(I, A) ⊗_{k} B) \ar[r]^{δ} & Ĥ^{2}(G, A ⊗_{k} B)}

of k-modules, in which both horizontal maps are isomorphisms, since in both exact sequences the second module is induced. Now take elements a in _{N}A and b in _{N}B representing elements of _{N}A/IA = Ĥ^{1}(G, A) and _{N}B/IB = Ĥ^{1}(G, B) respectively. In order to trace the element a ⊗_{k} b around the diagram from top-right to bottomright we need explicit descriptions of the two boundary maps involved.

Recall that the Tate cohomology groups Ĥ^{∗}(G, A) were defined by taking the cohomology of the complex K^{∗} ⊗_{Λ} A; the standard diagram chase for defining the boundary map of the long exact sequence associated to a short exact sequence of complexes then gives the following recipe for computing the boundary maps for the short exact sequence (5) above. If i is even, then an element of Ĥ^{i}(G, Hom_{k}(I, A)) is represented by an element f of Hom_{k}(I, A) on which G acts trivially; in other words, an element of Hom_{Λ}(I, A). The image of such an element under the connecting map δ is given by choosing a lifting f̃ of f to Hom_{k}(Λ, A), applying the boundary map D to this lifting and then taking the (unique) preimage in A. The fact that the sequence (5) is k-split gives us a natural way of making a lifting to Hom_{k}(Λ, A): namely, lift f to f̃ defined on a group element G by f̃(g) = f(g − 1); it also gives a natural way of taking the preimage in A: namely, evaluate at the identity 1. Thus we have

δf = (Df̃)(1)
= (φf̃)(1) − f̃(1)
= φf̃(φ^{−1}) − f̃(1)
= φf(φ^{−1} − 1) − f(0)
= f(1 − φ).

A similar calculation applies if i is odd: let f be an element of Hom_{k}(I, A) which is killed by the boundary map N. Then we have

δf = (Nf̃)(1)
= (f̃+ φf̃+ ⋯ + φ^{|G|−1}f̃)(1)
= f̃(1) + φf̃(φ^{−1}) + ⋯ + φ^{|G|−1}f̃(φ^{1−|G|})
= 0 + f(1 − φ) + ⋯ + f(1 − φ^{|G|−1})

The connecting maps for the long exact sequence associated to the sequence (6) are entirely analogous. Now we can find the image of a ⊗_{k} b. The preimage of a under δ is some Λ-map f: I → A satisfying f(1 − φ) = a. Now f ⊗_{k} b in the top left k-module maps across to a ⊗_{k} b then down to a · b. Going around the diagram in the other direction, it maps down to the element represented by f ⊗_{k} b (since the cup product is just the map induced by the identity if either degree is even) and then across to

δ(f ⊗_{k} b) = (N(f̃⊗_{k} b))(1)

= ∑_{0≤j<|G|}(φ^{j}f̃⊗_{k} φ^{j}b)(1)

= ∑_{0≤j<|G|} f(1 − φ^{j}) ⊗_{k} φ^{j}b

= ∑_{0≤j<|G|}( ∑_{0≤i<j} φ^{i}f(1 − φ)) ⊗_{k} φ^{j}b

= ∑_{0≤i<j<|G|} φ^{i}a ⊗_{k} φ^{j}b

Thus we have, finally,

a · b = ∑_{0≤i<j<|G|} φ^{i}a ⊗_{k} φ^{j}b.

This completes the computation of the cup product for cyclic groups.

3.2. Functoriality. We now describe the maps associated to a change of cyclic groups; these maps exist only for the original cohomology functors H^{i}(G, −) and not for the Tate cohomology functors. Suppose α: (G, φ) → (H, ψ) is a homomorphism of finite cyclic groups which maps the generator φ of G to the generator ψ of H; in particular, α is surjective. Let M′ be a module for H, M a module for G and f: M′ → M a Λ-homomorphism, regarding M′ as a module for G in the natural way. We’d like an explicit description of the natural functoriality maps

H^{i}(H, M′) → H^{i}(G, M)

in terms of the description we gave above of the cohomology of cyclic groups. There is a map of complexes of Λ-modules

[figure: 0; M′; D_{H}; f; M′; N_{H}; f; M′; D_{H}; \frac{|G|}{|H|} f; M′; N_{H}; \frac{|G|}{|H|} f; M′; \frac{|G|^{2}}{|H|^{2}} f; 0; M; D_{G}; M; N_{G}; M; D_{G}; M; N_{G}; M]

where the vertical maps in degrees 2i and 2i+1 are both f multiplied by (|G|/|H|)^{i}. One checks that the diagram commutes, so that this is a map of complexes, and that it therefore provides a map H^{i}(H, M′) → H^{i}(G, M) of cohomology groups, natural in M and M′. It is easy to check that the collection of all such maps gives a morphism of δ-functors which agrees with the natural functoriality map in degree zero, and therefore must be equal to this map in all degrees, using universality properties.

4. Cohomology of profinite groups

In this section we outline the cohomology theory of profinite groups. First though, we recall the definition and a few basic properties of profinite groups.

Definition. A profinite group is a topological group whose underlying topological space is compact and totally disconnected

Recall that a topological space X is totally disconnected, by definition, if given any two distinct points x and y of the space there is a subset X which is simultaneously open and closed, and which contains x but not y. Equivalently, each connected component of the space is a singleton. The space X is compact if when given any family of closed sets whose intersection is contained in the union of a family of opens, there is a finite subfamily of the closed sets whose intersection is contained in the union of some finite subfamily of the opens. The combination of these two topological properties results in a space whose topology is determined completely by its Boolean algebra of ‘clopen’ sets—those sets which are simultaneously open and closed. (More precisely, there is a contravariant equivalence of categories between the category of all compact totally disconnected topological spaces and the category of Boolean algebras; the equivalence in each direction is given by taking homomorphisms into the ‘schizophrenic’ object 2 = {0, 1}, regarded as a two-point discrete topological space or as a Boolean algebra as appropriate.)

Let G be a profinite group and H a subgroup. One can show that H with its subspace topology is profinite if and only if it is closed; that in the case when H is normal, the quotient G/H with its quotient topology is profinite if and only if H is closed, and that H is open if and only if it is closed and finite index in G. This should be enough justification for considering only the closed subgroups of G.

Examples. • Any finite discrete group is a profinite group • Any limit (projective limit, product, etc.) of profinite groups in the category

of topological groups is again a profinite group. This follows essentially from Tychonoff’s theorem; the inverse limit can be expressed as a closed subgroup of a product of compact and totally disconnected groups, and so is compact and totally disconnected.

• Let G be any topological group; then we can form the profinite completion Ĝ

of G:

Ĝ = \underleftarrow{lim}_{H}G/H

where the limit is taken over all subgroups H such that G/H is a finite discrete group; equivalently, such that H is closed and finite index in G. This operation of profinite completion is most commonly applied to a group with the discrete topology—an example of this is the profinite completion ℤ̂ of the integers ℤ.

• Similarly, one can form the pro-p-completion of a topological group G for any prime p, yielding for instance the p-adic integers ℤ_{p} from ℤ with the discrete topology.

• The absolute Galois group of a field with its Krull topology is profinite. Let K be any field, and K_{s} a separable algebraic closure. Then a basic theorem of Galois theory (see for example Cassels and Fröhlich [3]) identifies the group of K-automorphisms of K_{s} with the inverse limit

\underleftarrow{lim}_{L} Gal(L/K)

of Galois groups, as L runs over all finite Galois extensions of K. Considering this inverse limit as an inverse limit of finite discrete topological groups gives us a profinite group; the closed subgroups are then in one to one correspondence with the extensions (finite or infinite) of K contained in K_{s}; the open subgroups correspond to finite extensions, and normal subgroups correspond to normal extensions.

Here is the essential theorem on profinite groups.

Theorem 6. Let G be a profinite group, then

• The system U of open normal subgroups of G forms a system of neighbour-

hoods of the identity element 1; more precisely, – any open neighbourhood of 1 contains some subgroup in U, and

– the intersection ⋂_{U∈U} U of all open normal subgroups of G is {1}.

• Let H be any system of closed normal subgroups of G, satisfying the condition that any intersection of a finite number of subgroups in H is contained in some element of H. Then the natural continuous map

α: G/ ⋂_{H∈H} H → \underleftarrow{lim}_{H∈H}G/H

is a homeomorphism.

Corollary 7. Any profinite group is isomorphic to its profinite completion.

Corollary 8. Any profinite group is isomorphic to an inverse limit of finite groups.

Thus the common definition of a profinite group as an inverse limit of finite discrete groups in the category of topological groups agrees with the definition given above.

It is not hard to show that the two statements of the first part of the theorem are equivalent, via purely topological considerations: suppose that the intersection of all open normal subgroups is trivial, and consider an open neighbourhood U of the identity. Each open subgroup is closed, so that U is a family of closed sets whose intersection is contained in U, and by compactness we can take a finite subfamily whose intersection is contained in U. This intersection is a finite intersection of open normal subgroups, hence is itself an open normal subgroup. This proves one direction. For the reverse, suppose that any open neighbourhood of 1 contains some open normal subgroup; then given any element g of G not equal to the identity the complement of this element is an open neighbourhood of 1, and so contains an open normal subgroup. Thus the intersection of all open normal subgroups contains only 1.

We now prove the first part of the theorem. Let U be an open neighbourhood of the identity. Firstly, topological considerations show that U contains some open and closed neighbourhood of the identity (consider all open and closed neighbourhoods: their intersection is contained in U and so by compactness some one of them is contained in U). Now consider the subgroup

H = { g ∈ G | gU = U }

of G. I claim that H is open. To see this note that one can express H as the intersection of the subspace

S = { g ∈ G | gU ⊂ U }

with its inverse. Let μ: G×G → G be the multiplication map of G, and π_{1} : G×G → G the projection map onto the first factor. Note that π_{1} preserves open and closed sets and that the subspace S can be written as

S = G − π_{1}(G × U ∩ μ^{−1}(G − U))

which is immediately seen to be open and closed. Hence H is an open subgroup of G which is contained in U, and by intersecting this open subgroup with its (finitely many) conjugates by elements of G/H gives an open normal subgroup as required.

To prove the second part of the theorem, first note that the domain of α is compact while the target is Hausdorff; hence any continuous bijection will be a homeomorphism and we need only check that α is a bijection. Let x be any element of the right hand side, then x can be thought of as a vector (x_{H})_{H} indexed by the subgroups H of H where each x_{H} is a coset of H and x_{H} is contained in x_{H′} whenever H is contained in H′. Then an element g of G satisfies α(g) = x if and only if g is contained in each of the cosets x_{H}; thus in order to show that α is a bijection it is equivalent to show that the intersection of the cosets x_{H} contains precisely one element. This intersection is an intersection of an inverse system of non-empty closed subsets of G, and so is non-empty and closed, by compactness, hence α is surjective. Now suppose that α(g) = 1 for some element g of G. Then g is contained in the identity coset H for all subgroups H of H, that is, it is contained in the intersection of all these subgroups, so that α is also injective.

4.1. Discrete modules. Let G be a profinite group. In order to do homological algebra we need an abelian category to work in. The appropriate category turns out to be the category of discrete modules.

Definition. A discrete module for G is a G-module M such that each element of M is fixed by some open subset of G, or equivalently by some open normal subgroup of G. That is,

M = ⋃_{U∈U} M^{U}.

Equivalently, a discrete module is a discrete topological abelian group with a continuous G-action G × M → M.

Proposition 9. The category of discrete modules for G is an abelian category. It has enough injectives.

Proof. To check that this category is abelian is straightforward. To prove that it has enough injectives, note that the functor

M ↦ ⋃_{U∈U} M^{U}

from the category of all G-modules to the category of discrete G-modules is a right adjoint to the inclusion functor, and as with any right adjoint to an exact functor, it preserves injectives.

Remark. The category of discrete modules for G does not have enough projectives, in general. It is also worth pointing out that the category of topological abelian groups with a continuous G-action is not an abelian category, since it is not true that any injection is the kernel of some map, which prevents one from completing an injection to an exact sequence. One category that one might consider using is the category of compact Hausdorff topological abelian groups with a continuous G-action; this is an abelian category, in fact dual to the category of discrete Gmodules via Pontryagin duality. It thus has enough projectives, but not enough injectives.

We can now define the cohomology of a profinite group. We give three equivalent definitions.

Theorem 10. Let G be a profinite group. Then the following three sets of data define canonically isomorphic δ-functors

• The functors M ↦ H^{i}(G, M) = \underrightarrow{lim}_{U∈U}H^{i}(G/U, M^{U}) in which the maps of the direct limit are the inflation maps.

• The right derived functors of the left exact invariants functor M ↦ M^{G} from the category of discrete G-modules to the category of abelian groups.

• For a G-module M define the complex of continuous cochains: this is a cochain complex C^{∗}(G, M) such that C^{i}(G, M) is the abelian group of continuous maps from G^{i} to M (with the discrete topology on M). Define the boundary map δ_{i}: C^{i}(G, M) → C^{i+1}(G, M) as described in section 1. Then take the δ-functor given by taking cohomology of this complex: H^{i}(G, M) = H^{i}(C^{∗}(G, M)).

Each of the above qualifies for the title of the cohomology of G. Note that according to the first description above each of the cohomology groups in strictly positive degree is a torsion group, since the cohomology groups of a finite quotient G/U of G are killed by the order of G/U.

Proof. First we show the equivalence of the first and third definitions given above. This follows from the description of the cohomology of finite groups in terms of cochains, and from the fact that taking direct limits (over some fixed indexing category) in abelian groups is an exact functor, and so commutes with taking cohomology of a complex. Since a continuous map to a discrete topological space is just a locally constant map, we have

C^{i}(G, M) = \underrightarrow{lim}_{U∈U}C^{i}(G/U, M^{U})

and taking cohomology gives

H^{i}(C^{∗}(G, M)) = H^{i}(\underrightarrow{lim}_{U∈U}C^{∗}(G, U)) = \underrightarrow{lim}_{U∈U}H^{i}(G/U, M^{U}).

To show that the first definition is equivalent to the second it is enough to show that it is universal, and so by Grothendieck’s criterion of effaceability it is enough to show that it vanishes on injectives in degree one and higher. But the inflation map from G/U-modules to discrete G-modules is exact and has right adjoint M ↦ M^{U}, which therefore preserves injectives. Hence if J is an injective discrete G-module then J^{U} is an injective G/U-module for each open normal subgroup U, hence H^{i}(G/U, J^{U}) = 0, and so the direct limit of these is zero, as required.

These three definitions of cohomology of a profinite group complement each other neatly: the first allows us to relate the cohomology of profinite groups to the cohomology of finite groups; the third gives a description suitable for computation, and the second ensures that the theory has all the appropriate formal properties expected of a derived functor.

5. Computing the cohomology of ℤ̂

In this section we attempt to compute the cohomology of the profinite completion of the integers, using the definitions above and what we know about the cohomology of finite cyclic groups. Recall that ℤ̂ is defined by an inverse limit

ℤ̂ = \underleftarrow{lim}_{N}ℤ/Nℤ

of finite cyclic groups, with the index N running over the multiplicative system of positive integers. It is a topological ring, whose identity element 1 provides a topological generator of the underlying profinite abelian group.

Let M be a discrete ℤ̂-module; then M can be thought of as an abelian group along with an automorphism β of that abelian group with the property that all of the orbits of β are finite; that is, for each element m of M there is some integer N such that β^{N}m = m. We let M^{N} denote the submodule of M consisiting of all those elements which are preserved by β^{N}, so that M = ⋃_{N} M^{N}.

Proposition 11. Let M be a ℤ̂-module as described above. Then the cohomology groups of M can be described as follows:

• H^{0}(ℤ̂, M) ≅ M ℤ̂

• H^{1}(ℤ̂, M) ≅ { m ∈ M | (1 + β + ⋯ + β^{N−1})m = 0 for some N }/(β − 1)M • H^{2}(ℤ̂, M) ≅ (M ℤ̂ ⊗_{ℤ} ℚ/ℤ)/ Norm(M)

• H^{i}(ℤ̂, M) = 0 for all i > 2.

where Norm: M → M ℤ̂ ⊗_{ℤ} ℚ/ℤ is a map given by mapping an element m contained in M^{N} to the element (1 + β + ⋯ + β^{N−1})m ⊗_{ℤ} 1/N. One can easily check that this definition does not depend on the choice of N made.

There are two special cases of interest: the case where module M is a torsion module, and the case when M has trivial ℤ̂-action.

Corollary 12. • Suppose that the ℤ̂-module M is torsion. Then H^{1}(ℤ̂, M) =

M/(β − 1)M and H^{2}(ℤ̂, M) = 0.

• Suppose that M has trivial ℤ̂-action. Then H^{1}(ℤ̂, M) is the torsion submodule of M and H^{2}(ℤ̂, M) = M ⊗_{ℤ} ℚ/ℤ.

Proof of proposition 11. We use the description of the cohomology of a profinite group in terms of the cohomology of its finite discrete quotients, along with the calculations made in §3 of the cohomology of finite cyclic groups. The result for H^{0}(ℤ̂, M) is immediate from the definition. For each integer i ≥ 0 we have

H^{i}(ℤ̂, M) = \underrightarrow{lim}_{N}H^{i}(ℤ/Nℤ, M^{N})

where the maps in the direct system are the inflation maps. Recall that we computed in section 3.2 what these inflation maps were. In particular, in degree one we have

H^{1}(ℤ̂, M) = \underrightarrow{lim}_{N} \frac{\{ m ∈ M | (1 + β + ⋯ + β^{N−1})m = 0 \}}{(β − 1)M^{N}}

with the inflation maps being the maps induced by the identity endomorphism of M. Therefore

H^{1}(ℤ̂, M) = \frac{⋃_{N}\{ m ∈ M | (1 + β + ⋯ + β^{N−1})m = 0 \}}{⋃_{N}(β − 1)M^{N}}

= \frac{\{ m ∈ M | (1 + β + ⋯ + β^{N−1})m = 0 for some N \}}{(β − 1)M}

as claimed. In degree two we have

H^{2}(ℤ̂, M) = \underrightarrow{lim}_{N} \frac{M ℤ̂}{(1 + β + ⋯ + β^{N−1})M^{N}}

where if N divides N′ then the map from the Nth term to the N′th in the direct limit is the map induced by the multiplication by N′/N on M. Now one can define a map from this direct limit to (M ℤ̂ ⊗_{ℤ} ℚ/ℤ)/ Norm(M) by sending an element m ∈ M ℤ̂ of the Nth term of the direct limit to the element represented by m⊗_{ℤ}1/N, and it is left to the reader to check that this map is in fact an isomorphism.

It remains to show that the cohomology groups vanish in dimension 3 and higher; we already know that H^{2}(ℤ̂, M) vanishes when M is torsion, and by a dimension shifting argument it follows that H^{i}(ℤ̂, M) vanishes for all i ≥ 2 when M is torsion. (In the terminology of Serre[4], ℤ̂ has cohomological dimension 1.) We show that H^{3}(ℤ̂, M) vanishes for any ℤ̂-module M; it follows again by a dimension shifting argument that H^{i}(ℤ̂, M) vanishes for all i ≥ 3. We remarked earlier that all the higher cohomology groups of a profinite group are torsion, and so it is enough to show that for each positive integer N the N-torsion of H^{3}(G, M) vanishes. Fix such an integer N; the multiplication by N map on M gives rise to two exact sequences

0 → M[N] → M → NM → 0

and

0 → NM → M → M/NM → 0

of ℤ̂-modules, where M[N] denotes the N-torsion of M. The first module of the first sequence and the last module of the second are both torsion, so that their cohomology groups vanish in degree 2 and higher. The long exact sequences associated to these short exact sequences then give injections H^{3}(ℤ̂, M) → H^{3}(ℤ̂, NM) and H^{3}(ℤ̂, NM) → H^{3}(ℤ̂, M) whose composition is just the multiplication by N map. Hence the N-torsion of H^{3}(ℤ̂, M) is trivial. This completes the proof of the proposition.

Remark. This proof works more generally to show that if a profinite group has cohomological dimension d then it has strict cohomological dimension at most d+1; that is, its cohomology groups vanish in dimension d + 2 and higher.

References

1. Michael Francis Atiyah and C.T.C. Wall, Cohomology of groups, In Cassels and Fröhlich [2].

2. J.W.S. Cassels and A. Fröhlich (eds.), Algebraic number theory, Academic Press, 1967.

3. K. Gruenberg, Profinite groups, In Cassels and Fröhlich [2].

4. Jean-Pierre Serre, Cohomologie galoisienne, fifth ed., Lecture Notes in Mathematics, no. 5, Springer-Verlag, 1994.