CHAPTER 5 Lecture 5

1. Group Cohomology

1.1. Basic Facts. Let G be a profinite group and let M be a discrete submodule of G in the usual sense that

M = ⋃_{H open subgroup of G} M^{H}.

Equivalently we could take the union over all open normal subgroups. This condition merely means that G acts on each element of M through a finite quotient. We recall that the cohomology groups of G with coefficients in M are abelian groups defined by

H^{j}(G, M) = \underrightarrow{lim}_{H◁G}H^{j}(G/H, M^{H})

where the limit is over all open normal subgroups H of G and the maps of the directed system are the inflation maps. (Note that open subgroups of G automatically have finite index, so that all of the groups G/H are finite. See [Se-LF, Chapter 7] for the definitions of the standard maps of cohomology groups.) There are various other equivalent definitions of these cohomology groups; for example,

H^{j}(G, M) = Ext_{ℤ[G]}^{j}(ℤ, M) = \underrightarrow{lim}_{H◁G} Ext_{ℤ[G/H]}^{j}(ℤ, M^{H})

where here one can simply consider Ext as it is usually defined for rings. The equivalence of these definitions stems from the fact that (for profinite G and discrete M; things are more complicated in general) these H^{j}(G, M) are the right derived functors of the left exact functor

M ↦ M^{G},

where M^{G} is the submodule of G-invariants of M. That is,

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

In particular, this shows that

H^{0}(G, M) = M^{G},

which we note is independent of all topological considerations.

0Last modified February 23, 1998

1.2. The First Cohomology Group. We now turn to the first cohomology group, which is the one which we will primarily be concerned with. In this case, since on H^{1} inflation is injective (see below), we can actually realize H^{1}(G, M) as a union

H^{1}(G, M) = ⋃_{H◁G} H^{1}(G/H, M^{H}),

the union as usual over open, normal subgroups of G.

There are other somewhat more concrete descriptions of H^{1}(G, M) as well. The most standard is to realize it as equivalence classes of crossed homomorphisms. We define a crossed homomorphism to be a continuous map

h : G → M

such that

h(g_{1}g_{2}) = h(g_{1}) + g_{1}h(g_{2}).

We define CrossHom(G, M) to be the set of all (continuous) crossed homomorphisms from G to M. CrossHom(G, M) is made into an abelian group through the abelian group structure of M. We say that a crossed homomorphism h : G → M is principal if it has the form

h(g) = gm − m

for some fixed m ∈ M and all g ∈ G. (Such a map is indeed continuous, thanks to our assumption that M is discrete.) It is easy to check that every such map is a crossed homomorphism. We let PrincCrossHom(G, M) denote the subgroup of CrossHom(G, M) of principal crossed homomorphisms. With these definitions, it can be shown that

H^{1}(G, M) ≅ CrossHom(G, M)/ PrincCrossHom(G, M);

see [Se-LF, Chapter 7, Section 3]. (In fact, if one is working with general G and M (not necessarily profinite and discrete, respectively), one usually takes this as the definition of H^{1}(G, M); it is a theorem that in our case this agrees with the derived functor and direct limit definitions.) An element of CrossHom(G, M) is sometimes also called a 1-cocycle; an element of PrincCrossHom(G, M) is a 1-coboundary.

As an important special case of this, note that if G acts trivially on M, then

CrossHom(G, M) = Hom_{ℤ}(G, M)

and

PrincCrossHom(G, M) = 0,

so that

H^{1}(G, M) ≅ Hom_{ℤ}(G, M).

Using this description, we can see directly that inflation is injective. In this setting, for N closed and normal in G, inflation is induced by the natural map

CrossHom(G/N, M^{N}) → CrossHom(G, M)

defined by composing with the map G ↠ G/N. (It is clear that PrincCrossHom(G/N, M^{N}) maps to PrincCrossHom(G, M), so that we get a well-defined map on H^{1}’s.) To show injectivity we must check that if h : G/N → M^{N} is a crossed homomorphism, and if it becomes principal as a map G → M, say by h(g) = gm − m for some m ∈ M, then we can actually write h(g) = gm_{0} − m_{0} for some m_{0} ∈ M^{N}. But this is easy; since h vanishes on N, we must have nm = m for all n ∈ N. Thus we can take m = m_{0}, so inflation is injective on H^{1}’s.

1.3. The Basic Exact Sequence. We can actually extend our above injection

\xymatrix{H^{1}(G/N, M^{N}) \ar@{^{(}->}[r]^{inf} & H^{1}(G, M)}

to a longer exact sequence. We first recall how G/N acts on the cohomology group H^{1}(N, M). Most concretely, if h : N → M is a crossed homomorphism, we define γh : N → M for γ ∈ G/N as follows : first choose a lifting γ̃ ∈ G of γ. Now define

γh(x) = γ̃^{−1}h(γ̃xγ̃^{−1})

for x ∈ N. This makes sense since N is normal, and one can check easily that it is still a crossed homomorphism. Further, it is not hard to show that a different choice of γ̃ gives a crossed homomorphism which differs from h^{γ} by a principal crossed homomorphism, so that we really do get a well-defined action of G/N on H^{1}(N, M).

There is another way to define this action which is more abstract but somewhat cleaner. Recall that H^{j}(G, M) is contravariant in G and covariant in M. For any γ ∈ G, we can define maps

N ← N, M → M

by sending n ∈ N to γnγ^{−1} ∈ N and m ∈ M to mγ^{−1} ∈ M. These maps are compatible in the sense of [Se-LF, Chapter 7, Section 5] (this just means that the map M → M is a map of N-modules with the normal G-action on the second factor and the twisted action on the first factor) so that we get induced homomorphisms

H^{j}(N, M) →^{γ} H^{j}(N, M)

for all j. However, one can check easily that the action of any n ∈ N is in fact trivial; this is done by looking at the 0^{th} level and then either using the universal property of derived functors or else a dimension shifting argument. (See [Se-LF, Chapter 7, Section 5, Proposition 3].) Thus we get an action of G/N on all of the H^{j}(N, M).

We are now in a position to state the extension of our above injection. In it’s simplest form it is

0 ⟶ H^{1}(G/N, M^{N}) ⟶^{inf} H^{1}(G, M) ⟶^{res} H^{1}(N, M)^{G/N}.

This can be proved directly, working with crossed homomorphisms; see [Se-LF, Chapter 7, Section 6]. We can get a more general result by using the full Hochschild-Serre spectral sequence: for N a closed normal subgroup of G and M a discrete G-module this is a second stage, first quadrant, cohomological spectral sequence

E_{2}^{pq} = H^{p}(G/N, H^{q}(N, M)) ⇒ H^{p+q}(G, M).

The exact sequence of low degree terms takes the form

0 ⟶ H^{1}(G/N; M^{N}) ⟶^{inf} H^{1}(G; M) ⟶^{res} H^{1}(N; M)^{G/N}
⟶ H^{2}(G/N; M^{N}) ⟶^{inf} H^{2}(G; M).

For the construction of the Hochschild-Serre spectral sequence, see [Wei, Chapter 5]; for a proof of the above exact sequence and some generalizations, see [Wes-IR].

1.4. Cup Products. We briefly recall the notion of cup products; for a statement of the main properties, see [Se-LF, Chapter 8, Section 3]. For proofs of these properties, see [AW, Section 7].

Let M_{1} and M_{2} be A-modules, where A is some commutative ring, which are also discrete G-modules for some profinite group G. We further suppose that the action of G is A-linear; this is equivalent to saying that M_{1} and M_{2} are A[G]- modules rather than just ℤ[G]-modules. We endow the tensor product M_{1} ⊗_{A} M_{2} with a structure of G-module by setting g(m_{1} ⊗ m_{2}) = gm_{1} ⊗ gm_{2}. It is also, of course, still an A-module. The cup product is then a pairing

H^{i}(G, M_{1}) ⊗_{A} H^{j}(G, M_{2}) → H^{i+j}(G, M_{1} ⊗_{A} M_{2}).

We will usually write m_{1} ∪ m_{2} for the cup product of m_{1} ∈ H^{i}(G, M_{1}) and m_{2} ∈ H^{j}(G, M_{2}).

Often one takes cup products in a situation where there is also a natural map

M_{1} ⊗_{A} M_{2} → M_{3}

for some A[G]-module M_{3}, in which case we can consider the natural map

H^{i}(G, M_{1}) ⊗_{A} H^{j}(G, M_{2}) → H^{i+j}(G, M_{3})

induced by cup product and the above map.

2. Galois Cohomology

2.1. Torsors for Finite G_{K}-modules. For this section we fix a field K and let G_{K} = Gal(K_{s}/K) be its absolute Galois group. We further assume that M is a finite G_{K}-module. (M is also assumed to be discrete and the action continuous, of course.) We will refer to finite sets on which G_{K} acts as G_{K}-sets. What we say below is actually true for general groups, but we will confine ourself to the case of G_{K}.

Definition 2. An M-torsor is a G_{K}-set T together with a G_{K}-equivariant principal homogeneous action of M,

α : M × T → T.

We will write α(m, t) as m + t, and by G_{K}-equivariance we mean that g(m + t) = gm + gt. Recall that a principal homogeneous action is one which is simply transitive; that is, for any fixed t_{0} ∈ T, the map

M → T

given by sending m to m + t_{0} is a bijection.

Thus an M-torsor is a G_{K}-set which looks exactly like M, except that it isn’t itself an abelian group; by choosing a t_{0} to act as the identity it can be made into one. So somehow a M-torsor is a copy of M without a distinguished identity element.

There are other ways to describe M-torsors which can be somewhat more illuminating. For each t_{0} ∈ T we get a commutative diagram

\xymatrix{M × T \ar[r]^{α} & T \\ M × M \ar[r] \ar[u]^{≅} & M \ar[u]_{≅}}

where the bottom map is just the group law on M, the first vertical isomorphism sends (m_{1}, m_{2}) to (m_{1}, m_{2} + t_{0}) and the second vertical isomorphism sends m to m + t_{0}. This diagram expresses both the fact that α is a group action and that it is principal homogeneous.

Combining all of these diagrams over T, we get a diagram

\xymatrix{(M × T) × T \ar[r]^{α×1} & (T) × T \\ (M × M) × T \ar[r] \ar[u]^{≅} & (M) × T \ar[u]_{≅}}

where all of the maps on the last T -factor are the identity, and the vertical maps now send (m_{1}, m_{2}, t) to (m_{1}, m_{2} + t, t) and (m, t) to (m + t, t).

Considering the last T factor as a sort of parameter space for the other commutative diagrams, it might make somewhat more sense to write the previous diagram as

\xymatrix{(M × T)_{T} \ar[r]^{α} & T_{T} \\ (M × M)_{T} \ar[r] \ar[u]^{≅} & M_{T} \ar[u]_{≅}}

2.2. Connetions between Torsors and H^{1}(G, M). We first recall that it is possible to put a group structure on the set of M-torsors. Let T_{1} and T_{2} be two M-torsors. Consider the finite set T_{1} × T_{2} with its natural product G_{K}-action. M × M acts naturally on T_{1} × T_{2}, and we define an equivalence relation on T_{1} × T_{2} by decreeing that

(t_{1}, t_{2}) ∼ (t_{1} + m, t_{2} − m)

for all m ∈ M. We define the Baer sum T_{1} ⊙ T_{2} of T_{1} and T_{2} to be the quotient of T_{1} ⊗ T_{2} by this equivalence relation. One can easily check that T_{1} ⊙ T_{2} is actually an M-torsor, with its M-action defined by only having M act on the first factor.

We are now in a position to state our main theorem on torsors. We say that two M-torsors T_{1} and T_{2} are isomorphic as M-torsors if there is a isomorphism of G_{K}-sets between them commuting with the action of M.

Theorem 2.1. H^{1}(G_{K}, M) is naturally isomorphic to the set of M-torsors up to isomorphism of M-torsors. It is in fact an isomorphism of groups where the M-torsors are made into a group by the Baer sum.

Proof. This is actually fairly easy. One uses the identification

H^{1}(G_{K}, M) = \underrightarrow{lim}_{H◁G} Ext_{ℤ[G_{K}/H]}^{1}(ℤ, M^{H}),

together with the description of this group as isomorphism classes of extensions

0 → M → ε → ℤ → 0.

Given such an extension, the corresponding torsor is simply the inverse image of 1 ∈ ℤ in ε. We leave the details to the reader.

Recall that the category of G_{K}-sets is actually equivalent to the category of étale algebras over K. (An étale algebra is one which is flat and unramified; see [Mi-EC, Chapter 1, Section 3, p. 21] for the definition of an unramified algebra.) In the case that G_{K} acts trivially on M, we have that

H^{1}(G_{K}, M) ≅ Hom_{cts}(G_{K}, M)

classifies étale algebras over K with M-action. If the homomorphism G_{K} → M is actually surjective, then it corresponds to a field extension L/K with Galois group M.

2.3. Torsors for Algebraic Groups. There is a similar theory of torsors for algebraic groups. We replace M by an algebraic group Γ defined over K (recall that an algebraic group is an algebraic variety with a group structure such that the multiplication and inversion maps are actually morphisms of algebraic varieties), and for simplicity we assume that Γ is smooth as an algebraic variety over K. With appropriate definitions everything we say can be generalized to the case where Γ is non-commutative, but we will concentrate on the commutative case. (Note that if Γ is non-commutative we haven’t even defined the cohomology set (it isn’t even a group) H^{1}(G_{K}, Γ(K_{s})).)

In this setting, a Γ-torsor T is an algebraic variety over K together with a morphism

Γ × T ⟶^{α} T

satisfying the same conditions as our map α above. Specifically, we have a diagram

\xymatrix{(Γ × T) × T \ar[r]^{α×1} & (T) × T \\ (Γ × Γ) × T \ar[r] \ar[u]^{u}_{≅} & (Γ) × T \ar[u]_{v}^{≅}}

where the bottom map is the multiplication morphism on Γ and the identity on T,

u = id ×α × id

(where α operates on the last two factors Γ × T) and

v = α × id .

On points, then,

u(γ_{1}, γ_{2}, t) = (γ_{1}, γ_{2} + t, t)

and

v(γ, t) = (γ + t, t).

It follows from this definition that T is isomorphic to K over K̄ (it might not be isomorphic over K since it might not even have any points defined over K), so T is necessarily smooth. Also, we can define a Baer sum in an analagous way to that above.

We have in this setting the following analogue of Theorem 2.1.

Theorem 2.2 (Grothendieck). If Γ is a smooth algebraic group over K, then H^{1}(G_{K}, Γ(K_{s})) is naturally isomorphic to the group of Γ-torsors (defined over K) up to isomorphism.

Proof. The proof of this theorem is quite difficult, in contrast to the fairly immediate proof of Theorem 2.1. The fact that Γ is smooth is absolutely essential; if Γ isn’t smooth, then we must allow flat coverings rather than merely étale coverings.

For a proof in the much easier case that Γ is an elliptic curve, see [Si, Chapter 10, Theorem 3.6].

2.4. The Local Invariant. We now consider the algebraic group 𝔾_{m} over a local field K (in our usual sense). More accurately, we will fix an algebraic closure K̄ of K and consider the attached G_{K}-module K̄^{∗}. We will denote its torsion subgroup by μ(K̄^{∗}); it is just the group of roots of unity in K̄^{∗}. (Since K is perfect it will actually contain all of the roots of unity, but in general it may not contain the p^{th} roots of unity, where p is the characteristic of K.)

In this setting we have the following fundamental theorem of local class field theory.

Theorem 2.3. Let K be a local field. Then there is a canonical isomorphism inv, called the invariant map,

H^{2}(G_{K}, μ(K̄^{∗})) ⟶^{≅} ℚ/ℤ.

Proof. See [Se-LF, Chapter 13, Section 3]. This theorem is actually often stated as

H^{2}(G_{K}, K̄^{∗}) ⟶^{≅} ℚ/ℤ,

but either form can easily be obtained from the other.

In this section we will use our knowledge of the structure of G_{K} to compute the l-adic component of H^{2}(G_{K}, μ(K̄^{∗})); that is, we will show that H^{2}(G_{K}, μ_{l^{∞}}(K̄^{∗})) is canonically isomorphic to ℚ_{l}/ℤ_{l}, at least in the case l ≠ p, where p is the characteristic of the residue field of K. Here μ_{l^{∞}}(K̄^{∗}) is the group of l-power roots of unity in K̄^{∗}.

Our main tools will be the Hochschild-Serre spectral sequence and the following lemma.

Lemma 2.4. Let G be a pro-p group and let M be a discrete G-module in which each element is killed by a power of l. If p ≠ l, then for all j ≥ 1,

H^{j}(G, M) = 0.

Proof. Recall that

H^{j}(G, M) = \underrightarrow{lim}_{H◁G}H^{j}(G/H, M^{H}),

the direct limit being over the open, normal subgroups of G. Choose x ∈ H^{j}(G/H, M^{H}) and let c : (G/H)^{j} → M^{H} be a j-cocyle representing x. (See [Se-LF, Chapter 7, Section 3].) Since G/H is finite, c takes on only finitely many values in M^{H}. Thus there must be some n ≥ 0 such that l^{n}c = 0, and therefore l^{n}x = 0. But G/H has order p^{m} for some m, so p^{m}x = 0 by [Se-LF, Chapter 8, Section 2, Corollary 1]. So, since l ≠ p, we must have x = 0. Thus every group in the direct limit is 0, so H^{j}(G, M) = 0 as well.

We will use various exact sequences from Lecture 4, Sections 1.3 and 2.3. First, we have an exact sequence

1 → P_{K} → G_{K} → T_{K} → 1

where P_{K} is a p-group. The Hochschild-Serre spectral sequence for this exact sequence collapses to the bottom row by Lemma 2.4, since P_{K} is pro-p. Thus we get an isomorphism

H^{2}(G_{K}, μ_{l^{∞}}(K̄^{∗})) ≅ H^{2}(T_{K}, μ_{l^{∞}}(K̄^{∗})).

Next we recall that we have an exact sequence

1 → ∏_{r≠p,l} ℤ_{r}(1) → T_{K} → T_{l,K} → 1.

Since ∏_{r≠p,l} ℤ_{r}(1) has no pro-l component, the same argument as above shows that

H^{2}(T_{K}, μ_{l^{∞}}(K̄^{∗})) ≅ H^{2}(T_{l,K}, μ_{l^{∞}}(K̄^{∗})),

and thus that

H^{2}(G_{K}, μ_{l^{∞}}(K̄^{∗})) ≅ H^{2}(T_{l,K}, μ_{l^{∞}}(K̄^{∗})).

Next we use the exact sequence

1 → ℤ_{l}(1) → T_{l,K} → ℤ̂ → 1.

This time the Hochschild-Serre spectral sequence is entirely zero outside of the bottom left 2 × 2 square, since ℤ̂ and ℤ_{l}(1) cohomology vanish on torsion modules in all dimensions greater than or equal to 2. (See [Se-LF, Chapter 13, Section 1, Proposition 2].) From this we get an isomorphism Thus

H^{2}(G_{K}, μ_{l^{∞}}(K̄^{∗})) ≅ H^{1}(ℤ̂, H^{1}(ℤ_{l}(1), μ_{l^{∞}}(K̄^{∗}))).
Now, μ_{l^{∞}}(K̄^{∗}) ⊆ K^{ur}, so ℤ_{l}(1) acts trivially on it. Thus,
H^{1}(ℤ_{l}(1), μ_{l^{∞}}(K̄^{∗})) = Hom(ℤ_{l}(1), μ_{l^{∞}}(K̄^{∗})) = Hom(ℤ_{l}, ℚ_{l}/ℤ_{l}) ≅ ℚ_{l}/ℤ_{l},

since as abelian groups ℤ_{l}(1) is just ℤ_{l} and μ_{l^{∞}}(K̄^{∗}) is just ℚ_{l}/ℤ_{l}. (All Hom-groups are continuous homomorphisms, of course.) In fact, H^{1}(ℤ_{l}(1), μ_{l^{∞}}(K̄^{∗})) even has trivial ℤ̂ = Gal(K^{ur}/K)-action, since ℤ̂ acts via the l-cyclotomic character on each and the actions cancel. Thus,

H^{1}(ℤ̂, H^{1}(ℤ_{l}(1), μ_{l^{∞}}(K̄^{∗}))) = Hom(ℤ̂, ℚ_{l}/ℤ_{l}) ≅ ℚ_{l}/ℤ_{l}.

Thus, as promised, we have shown that

H^{2}(G_{K}, μ_{l^{∞}}(K̄^{∗})) ≅ ℚ_{l}/ℤ_{l}.

Of course, this is not yet the full determination of the invariant map. Most importantly, we have not computed the p-part; our methods above would be completely ineffective in that case.

3. Tate Local Duality

Let K be a local field and let M be a discrete G_{K}-module. We further assume that M is finite. We define the Pontrjagin dual M^{∨} of M by

M^{∨} = Hom(M, ℚ/ℤ),

with the usual adjoint G_{K}-action. We further define the Cartier dual M^{∗} by

M^{∗} = Hom(M, μ(K̄^{∗})),

again with the adjoint G_{K}-action. M^{∨} and M^{∗} are isomorphic as abelian groups, but not as G_{K}-modules, and in both cases the double dual is simply M.

Pontrjagin duality is simply the obvious assertion that there is a natural, G_{K}- equivariant, perfect pairing

M ⊗ M^{∨} → ℚ/ℤ.

Similarly, Cartier duality is the assertion that there is a natural G_{K}-equivariant perfect pairing

M ⊗ M^{∗} → μ(K̄^{∗}).

Tate local duality is essentially the extension of Cartier duality to cohomology groups.

Theorem 3.1 (Tate Local Duality). Let K and M be as above. Then, for all i, H^{i}(G_{K}, M) is finite, and there are perfect pairings

H^{i}(G_{K}, M) ⊗ H^{2−i}(G_{K}, M^{∗}) → ℚ/ℤ

induced by cup product, Cartier duality and the invariant map.

Proof. See [Mi-ADT, Chapter 1, Section 2].