In abstract algebra, homological algebra, algebraic topology and algebraic number theory, as well as in applications to group theory proper, group cohomology is a way to study groups using a sequence of functors H ⁿ.

Motivation

A general paradigm in group theory is that a group G should be studied via its group representations. A slight generalization of those representations are the G-modules: a G-module is an abelian group M together with a group action of G on M, with every element of G acting as an automorphisms of M. In the sequel we will write G multiplicatively and M additively.

Given such a G-module M, it is natural to consider the subgroup of G-invariant elements:

M^G = { x in M : gx = x for all g in G }

Now, if N is a submodule of M (i.e. a subgroup of M mapped to itself by the action of G), it isn't in general true that the invariants in M/N are found as the quotient of the invariants in M by the invariants in N: being invariant 'up to something in N' is broader. The first group cohomology H¹(G,N) precisely measures the difference. The group cohomology functors Hⁿ in general measure the extent to which taking invariants doesn't respect exact sequences. This is expressed by a "long exact sequence".

Formal constructions

The collection of all G-modules is a category (the morphisms are group homomorphisms f with the property f(gx) = g(f(x)) for all g in G and x in M). This category of G-modules is an abelian category with enough injectives (since it is isomorphic to the category of all modules over the group ring ZG).

Sending each module M to the group of invariants M^G yields a functor from this category to the category Ab of abelian groups. This functor is left exact. We may therefore form its derived functors; their values are abelian groups and they are denoted by H ⁿ(G,M), "the n-th cohomology group of G with coefficients in M". H⁰(G,M) is identified with M^G.

In practice, one often computes the cohomology groups using the following fact: if

is a short exact sequence of G-modules, then a long exact sequence

is induced.

Rather than using the machinery of derived functors, we can also define the cohomology groups more concretely, as follows. For n≥0, we let Cⁿ(G,M) be the set of all functions from Gⁿ to M:

Cⁿ(G,M) = { φ : Gⁿ → M }

This is an abelian group; its elements are called the n-cochains. We further define group homomorphisms

d ⁿ : Cⁿ(G,M) → Cⁿ⁺¹(G,M)

by

These are known as the coboundary homomorphisms. The crucial thing to check here is

d ⁿ⁺¹ o d ⁿ = 0

thus we have a chain complex and we can compute cohomology: define the group of n-cocycles as

Zⁿ(G,M) = ker(dⁿ)    for n≥ 0

and the group of n-coboundaries as

B₀(G,M) = {0}  and   Bⁿ(G,M)= image(d ^n-1)    for n≥ 1

and

H ⁿ(G,M) = Zⁿ(G,M) / Bⁿ(G,M).

Yet another approach is to treat G-modules as modules over the group ring ZG and use Ext functors:

Hⁿ(G,M) = Extⁿ_ZG(Z,M).

Here Z is treated as the trivial G-module: every element of G acts as the identity. These Ext groups can also be computed via a projective resolution of Z, the advantage being that such a resolution only depends on G and not on M.

Finally, group cohomology can be related to topological cohomology theories: to the group G we construct the Eilenberg-MacLane space K(G, 1) (whose fundamental group is G and whose higher homotopy groups vanish); the n-th cohomology of this space with coefficients in M (in the topological sense) is the same as the group cohomology of G with coefficients in M.

Properties

Group cohomology depends contravariantly on the group G, in the following sense: if f : G → H is a group homomorphism and M is an H-module, then we have a naturally induced morphism Hⁿ(H,M) → Hⁿ(G,M) (where in the latter case, M is treated as a G-module via f).

If M is a trivial G-module (i.e. the action of G on M is trivial), the second cohomology group H²(G;M) is in one-to-one correspondence with the set of central extensions of G by M (up to a natural equivalence relation).

History and relation to other fields

Early recognition of group cohomology came in the Noether's equations of Galois theory (an appearance of cocycles for H¹), and the factor sets of the extension problem for groups (Issai Schur's multiplicator ) and in simple algebras (Richard Brauer, the Brauer group), both of these latter being connected with H². The first theorem of the subject can be identified as Hilbert's Theorem 90.

Some general theory was supplied by Mac Lane and Lyndon; from a module-theoretic point of view this was integrated into the Cartan-Eilenberg theory, and topologically into an aspect of the construction of the classifying space BG for G-bundles.

The application in algebraic number theory to class field theory provided theorems valid for general Galois extensions (not just abelian extensions).

Some refinements in the theory post-1960 have been made (continuous cocycles, Tate's redefinition) but the basic outlines remain the same.

The analogous theory for Lie algebras, introduced by Jean-Louis Koszul , is formally similar, starting with the corresponding definition of invariant. It is much applied in representation theory, and is closely connected with the BRST quantization of theoretical physics.

External links

• Turkelli, Szilágyi, Lukács: Cohomology of Groups, PDF file.