# Modules and Group Cohomology

Let  be a  module. This means that is an abelian group equipped with a left action of , i.e., a group homomorphism , where denotes the group of bijections that preserve the group structure on . Alternatively,  is a module over the ring in the usual sense of module. For example, with the trivial action is a module over any group , as is for any positive integer . Another example is , which acts via multiplication on .

For each integer there is an abelian group called the th cohomology group of  acting on . The general definition is somewhat complicated, but the definition for is fairly concrete. For example, the 0th cohomology group

for all

is the subgroup of elements of that are fixed by every element of .

The first cohomology group

is the group of -cocycles modulo -coboundaries, where

such that

and if we let denote the set-theoretic map , then

There are also explicit, and increasingly complicated, definitions of for each in terms of certain maps modulo a subgroup, but we will not need this.

For example, if has the trivial action, then , since for any . Also, . If , then since is finite there are no nonzero homomorphisms , so .

If is any abelian group, then

is a -module. We call a module constructed in this way co-induced.

The following theorem gives three properties of group cohomology, which uniquely determine group cohomology.

Theorem 11.2.1   Suppose is a finite group. Then
1. We have .
2. If is a co-induced -module, then for all .
3. If is any exact sequence of -modules, then there is a long exact sequence

Moreover, the functor is uniquely determined by these three properties.

We will not prove this theorem. For proofs see [Cp86, Atiyah-Wall] and [Ser79, Ch. 7]. The properties of the theorem uniquely determine group cohomology, so one should in theory be able to use them to deduce anything that can be deduced about cohomology groups. Indeed, in practice one frequently proves results about higher cohomology groups by writing down appropriate exact sequences, using explicit knowledge of , and chasing diagrams.

Remark 11.2.2   Alternatively, we could view the defining properties of the theorem as the definition of group cohomology, and could state a theorem that asserts that group cohomology exists.

Remark 11.2.3   For those familiar with commutative and homological algebra, we have

where is the trivial -module.

Remark 11.2.4   One can interpret as the group of equivalence classes of extensions of by , where an extension is an exact sequence

such that the induced conjugation action of on is the given action of  on . (Note that acts by conjugation, as is a normal subgroup since it is the kernel of a homomorphism.)

Subsections
William Stein 2012-09-24