Group cohomology

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}.
Contents 
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 Gmodules: a Gmodule 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 Gmodule M, it is natural to consider the subgroup of Ginvariant 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 H^{1}(G,N) precisely measures the difference. The group cohomology functors H^{n} 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 Gmodules 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 Gmodules 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 nth cohomology group of G with coefficients in M". H^{0}(G,M) is identified with M^{G}.
In practice, one often computes the cohomology groups using the following fact: if
 <math>0\to L \to M \to N \to 0<math>
is a short exact sequence of Gmodules, then a long exact sequence
 <math>0\to L^G\to M^G\to N^G\to H^1(G,L) \to H^1(G,M) \to H^1(G,N)\to H^2(G,L)\to \cdots<math>
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 C^{n}(G,M) be the set of all functions from G^{n} to M:
 C^{n}(G,M) = { φ : G^{n} → M }
This is an abelian group; its elements are called the ncochains. We further define group homomorphisms
 d^{ n} : C^{n}(G,M) → C^{n+1}(G,M)
by
 <math>
d^n(\phi)(g_1,...,g_{n+1}) = g_1\cdot \phi(g_2,...,g_{n+1}) <math>
 <math>
+ \sum_{i=1}^n (1)^i \phi(g_1,...,g_{i1},g_ig_{i+1},g_{i+2}, ...,g_{n+1}) <math>
 <math>
+ (1)^{n+1} \phi(g_1,...,g_n)\; <math> 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 ncocycles as
 Z^{n}(G,M) = ker(d^{n}) for n≥ 0
and the group of ncoboundaries as
 B_{0}(G,M) = {0} and B^{n}(G,M)= image(d^{ n1}) for n≥ 1
and
 H^{ n}(G,M) = Z^{n}(G,M) / B^{n}(G,M).
Yet another approach is to treat Gmodules as modules over the group ring ZG and use Ext functors:
 H^{n}(G,M) = Ext^{n}_{ZG}(Z,M).
Here Z is treated as the trivial Gmodule: 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 EilenbergMacLane space K(G, 1) (whose fundamental group is G and whose higher homotopy groups vanish); the nth 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 Hmodule, then we have a naturally induced morphism H^{n}(H,M) → H^{n}(G,M) (where in the latter case, M is treated as a Gmodule via f).
If M is a trivial Gmodule (i.e. the action of G on M is trivial), the second cohomology group <math>H^2(G;M)<math> is in onetoone 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 H^{1}), 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 H^{2}. 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 moduletheoretic point of view this was integrated into the CartanEilenberg theory, and topologically into an aspect of the construction of the classifying space BG for Gbundles.
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 post1960 have been made (continuous cocycles, Tate's redefinition) but the basic outlines remain the same.
The analogous theory for Lie algebras, introduced by JeanLouis 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 (http://www.math.uu.nl/people/crainic/chapter2.pdf), PDF file.