Group cohomology is a useful language for expressing the results of class field theory, among (many) other things. There are a few ways I could introduce this. I could define them as derived functors (i.e. as a special case of ) or satellites, which would be the most general, but I try to keep my posts somewhat self-contained. I could define them additionally as cochains or coboundaries. I’ve decided to give an axiomatic definition, which will include the previous ones.

**Axioms for Cohomology **

First of all, here is a useful way of constructing a right -module from an abelian group : consider and let act on the left factor. Such modules are said to be **co-induced**. In the case of , this yields a left-module too by the anti-involution sending . Note that each -module can be imbedded inside a co-induced one.

Given -modules (i.e. -modules) with an exact sequence

the sequence of fixed points

is exact except at the end. The cohomology functors extend this into a long exact sequence.

Here are the axioms:

- For each , is a covariant functor
- for
- Given an exact sequence , there is a long exact sequence
which is functorial in the short exact sequence .

- For co-induced, if .

I think there’s a definite analogy between this approach and the Eilenberg-Steenrod axioms for homology theories, except that in this case on already has a more general theory (derived functors), so an axiomatic approach is less useful.

Anyway, the third axiom shows that the form a -functor, a notion introduced by Grothendieck which basically means a sequence of functors from one abelian category to another with a long exact sequence

associated functorially to a short exact sequence . The last condition means that the are “effaceable.” Grothendieck proved a theorem that if given two -functors with effaceable in this way, a natural transformation extends uniquely to natural transformations which make the diagram

commutative. I won’t prove this theorem, since it’s a messy sequence of diagram chases, but you can find it in books on homological algebra. For us, this will be important.

Anyway, the disadvantage of using an axiomatic approach is that we have to give an existence and uniqueness proof.

Theorem 1Up to natural isomorphism, there is precisely one -functor satisfying the above axioms. If is a projective resolution in (with the trivial -module), then

where on the left the refers to the cohomology of a cochain complex.

Ok, first of all, the last part of the theorem gives us a clean way to prove existence: just fix the projective resolution , and check the axioms. The first is evident. Next, let be the end of the resolution; then

is exact, which proves 2. For 3, given an exact sequence , there is a natural short exact sequence **of complexes**

by projectivity, so taking the associated long exact sequence gives a natural long exact sequence of of the right form. Finally, if for an abelian group, we have

and for abelian the latter cochain complex has trivial cohomology after the first index, because splits as -complexes.

Uniqueness follows because given two such families of functors , we have transformations which is the identity, and which by Grothendieck’s theorem yields natural transformations ; the composition of these extend the identity transformations and must be the identity.

Alternatively, one often defines as the derived functors of the left-exact functor ; the uniqueness in the above theorem says that deriving the functor in either of the two variables gives the same result. This fact can be extended to or over a ring.

**Homology **

Group homology is dual to cohomology, except that one starts with the right exact functor , where is the **augmentation ideal** of generated by . The axioms are as follows.

- For each , is a covariant functor
- for
- Given an exact sequence , there is a long exact sequence
which is functorial in the short exact sequence .

- For induced (i.e. ), if .

The form a -functor in the opposite direction.

Theorem 2Up to natural isomorphism, there is precisely one -functor satisfying the above axioms. If is a projective resolution in (with the trivial -module), then

where on the left the refers to the cohomology of a cochain complex.

In the tensor product, is considered as a right -module via its left-module structure and the anti-involution on .The rest of the proof is similar, since

Next, I’ll discuss the canonical resolution of and how one can compute cohomology and homology.

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