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 group (mathematics) using a sequence of functors H n .

= Motivation =

A general paradigm in group theory is that a group G should be studied via its 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: : MG = { 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 G -modules is a category theory (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 module (mathematics) 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 functor. 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 . H 0( G , M ) is identified with M G .

In practice, one often computes the cohomology groups using the following fact: if :0 o L o M o N o 0 is a short exact sequence of G -modules, then a long exact sequence :0 o L^G o M^G o N^G o H^1(G,L) o H^1(G,M) o H^1(G,N) o H^2(G,L) o cdots 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 function (mathematics)s from G n to M : : C n ( G , M ) = { φ : Gn → M } This is an abelian group; its elements are called the n-cochains . We further define group homomorphisms : d n : C n ( G , M ) → C n +1( G , M ) by : d^n(varphi)(g_{1},dots,g_{n+1}) = g_{1}cdot varphi(g_{2},dots,g_{n+1})

:: + sum_{i=1}^{n} (-1)^{i} varphi(g_{1},dots,g_{i-1},g_{i} g_{i+1},g_{i+2},dots,g_{n+1})

:: + (-1)^{n+1} varphi(g_{1},dots,g_{n})

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 : B 0( 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 ) = Ext n 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 H^2(G;M) is in one-to-one correspondence with the set of central extension of G by M (up to a natural equivalence relation).

= History and relation to other fields =

Early recognition of group cohomology came in the Emmy 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 Saunders Mac Lane and Lyndon; from a module-theoretic point of view this was integrated into the Henri Cartan-Samuel Eilenberg theory, and topologically into an aspect of the construction of the classifying space BG for G -fiber bundle.

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 =