Low-dimensional cohomology of groups. Let G be a profinite topological group, and M a discrete G-module. We have

H^{j}(G; M) = lim_{→} H^{j}(G/H; M^{H}) = lim_{→} Ext_{Z[G/H]}^{j}(Z, M^{H}),

where H ⊂ G runs through all open subgroups (of finite index) in G. Dimension 0 : We have H^{0}(G; M) = M^{G} this equality being, by most treatments, the very definition of H^{0}, and the higher H^{j}’s are the right-derived functors attached to the left-exact functor M ↦ M^{G}. Perhaps the thing to notice here, if anything, is that the topology of G doesn’t affect H^{0}(G; M). Dimension 1 : Here

H^{1}(G; M) = lim_{→} H^{1}(G/H; M^{H}) = ∪H^{1}(G/H; M^{H}),

the limit/and/union being again taken over all open subgroups H (of finite index) in G. The point here is that the groups H^{1}(G/H; M^{H}) are subgroups of H^{1}(G; M). More about this in a moment. A basic interpretation of H^{1} is as follows: H^{1}(G; M) ≅ the quotient group of (continuous) crossed homomorphisms from G to M modulo the subgroup of “principal crossed homomorphisms”. If N ⊂ G is a closed normal subgroup, consider the basic exact sequence:

0 → H^{1}(G/N, M^{N}) → H^{1}(G, M) → H^{1}(N, M)^{G/N}.

Here the mappings are the natural mappings, and Exercise. Prove the exactness by directly appealing to the crossed-homomorphisms interpretation. The basic exact sequence is really the beginning of the Spectral Sequence

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

In particular, we can extend the basic exact sequence above to an exact sequence

0 → H^{1}(G/N, M^{N}) → H^{1}(G, M) → H^{1}(N, M)^{G/N} → H^{2}(G/N, M^{N}).

Cup products for “low dimensions” : For A a commutative ring, and A-modules M_{1}, M_{2} endowed with A-linear discrete G-module structures, we have functorial cup products

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

In particular, when i = 0 and j = 1, this computes to be the evident homomorphism if you interpret H^{1} as crossed homomorphisms (Exercise: What is that evident homomorphism? And when i = 1 and j = 1, ... Interpretations of Galois cohomology in dimension 1 : torsors. Let M be now a finite Galois module over a field K. The cohomology group H^{1}(G_{K}, M) can be interpreted as the group of isomorphism classes of “M-torsors”. To define the notion of M-torsor, we will be considering finite G_{K}-sets T: these are, by definition, (finite) sets equipped with continuous G_{K}-action, i.e., equipped with an action which factors through a finite quotient group of G_{K}.

Definition. Let M be a finite G_{K}-module. An M-torsor is a G_{K}-set T, together with a G_{K}-equivariant action of the abelian group M on the set T, i.e. , a mapping α : M × T → T which satisfies the standard rules for an action. Moreover, we require that the action α makes M act principally and transitively on T and and the action satisfies the rule: α(gm, gt) = gα(m · t) for all g ∈ G_{K}, m ∈ M, and t ∈ T.

Exercise: For finite G_{K}-modules M, establish a canonical isomorphism between H^{1}(G, M) and the set of isomorphism classes of M-torsors.

Remark: Torsors over algebraic groups. Lang’s Theorem.

Interpretations of Galois cohomology in dimension 1 : descent theory. Explain the notion of twisting by a 1-cocycle, and connect H^{1}(G_{K}, \underline{Aut}(E)) with twists of E, for various kinds of E. Brauer-Severi varieties. Explain Hilbert’s Theorem 90 this way.

Tate Local Duality. Local Class Field theory gives the canonical identification

H^{2}(G_{K}, μ(K̄)) ≡ Q/Z.

For fun, a quick check that the tame version of this is true. That is. for ℓ ≠ p, we have:

H^{1}(I_{ℓ}, μ_{ℓ}(K̄)) ≡ Q_{ℓ}/Z_{ℓ},

and use the basic Spectral Sequence to deduce that

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

Define the Cartier dual of the module M to be M^{∗} := Hom_{Z}(M, μ). Suppose that M is finite. Then

Theorem. (Tate Local Duality):

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

is a perfect pairing in the sense that (a) the cohomology groups H^{∗}(G_{K}, M) and H^{∗}(G_{K}, M^{∗}) are all finite and the pairing is perfect. In particular, the only nonzero dimsensions of cohomology are for i = 0, 1, 2.

The most important pairing occurs in the “middle dimension”:

H^{1}(G_{K}, M) × H^{1}(G_{K}, M^{∗}) → Q/Z.

Cheating Example: Put M = Z/NZ with trivial G_{K}-action. Then

H^{1}(G_{K}, M) = Hom_{cont.} (G_{K}, Z/NZ),

and

H^{1}(G_{K}, M^{∗}) = H^{1}(G_{K}, μ_{N}) = K^{∗}/K^{∗N},

(by Kummer theory, so: the pairing gives an isomorphism

Hom_{cont.} (G_{K}, Z/NZ) ≡ Hom (K^{∗}/K^{∗N}, Q/Z).

From this we deduce a natural isomorphism

G_{K}^{ab}/G_{K}^{abN} ≡ K^{∗}/K^{∗N},

and these isomorphisms are compatible for N increasing “multiplicatively” and they compile to give (in the projective limit) the local class field theory isomorphism G_{K}^{ab} → K̂^{∗}/K^{∗N} (where the hat refers to profinite completion).

Applying the basic exact sequence.

0 → H^{1}(G_{k}, M^{I_{k}}) → H^{1}(G_{K}, M) → H^{1}(I_{K}, M)^{G_{k}} → 0.

If M is unramified, then this reads:

0 → H^{1}(G_{k}, M) → H^{1}(G_{K}, M) → Hom_{G_{k}} (I_{K})^{ab}, M) → 0.

If M is unramified and of cardinality a power of a prime number ℓ ≠ p, then

Hom_{G_{k}} ((I_{K})^{ab}, M) = Hom_{G_{k}} Z_{ℓ}(1), M) = M(−1)^{G_{k}},

giving the following evaluation of the basic exact sequence

0 → H^{1}(G_{k}, M) → H^{1}(G_{K}, M) → M(−1)^{G_{k}} → 0,

and, of course

0 → H^{1}(G_{k}, M^{∗}) → H^{1}(G_{K}, M^{∗}) → M^{∗}(−1)^{G_{k}} → 0,

but since M^{∗}(−1) = M^{∧} (Pontrjagin duality) we have

0 → H^{1}(G_{k}, M^{∗}) → H^{1}(G_{K}, M^{∗}) → (M^{∧})^{G_{k}} → 0.

Remark. There are two ways of doing this (see [Milne] Ch. 1).

The finite and singular parts of 1-dimensional cohomology. Axiomatized version