Group cohomology

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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 Hⁿ.

1 Motivation
2 Formal constructions
3 Connections with Topological Cohomology Theories
4 Properties
5 History and relation to other fields
6 See also

[edit] 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 endomorphism 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:

$M^{G} = \lbrace x \in M : \forall g \in G \ gx=x \rbrace$

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.

[edit] 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 Z[G]).

Sending each module M to the group of invariants M^G yields a functor from this category to the category $\mathfrak{Ab}$ of abelian groups. This functor is left exact but not necessarily right exact. We may therefore form its right derived functors; their values are abelian groups and they are denoted by Hⁿ (G,M), "the n-th cohomology group of G with coefficients in M". H⁰(G,M) is identified with M^G.

[edit] Long Exact Sequences

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

$0 \to L \to M \to N \to 0$

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

$0\to L^G\to M^G\to N^G\to H^1(G,L) \to H^1(G,M) \to H^1(G,N)\to H^2(G,L)\to \cdots$

is induced.

[edit] Cochain Complexes

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

$C^{n} (G,M) = \lbrace \varphi : G^{n} \rightarrow M \rbrace$

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

$d^{n}: C^{n} (G,M) \rightarrow C^{n+1}(G,M)$

$d^n(\varphi)(g_{0},\dots,g_{n}) = g_{0}\cdot \varphi(g_{1},\dots,g_{n})$

$+ \sum_{i=0}^{n-1} (-1)^{i+1} \varphi(g_{0},\dots,g_{i-1},g_{i} g_{i+1},g_{i+2},\dots,g_{n})$

$+ (-1)^{n+1} \varphi(g_{0},\dots,g_{n-1})$

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

$d^{n+1} \circ d^{n} = 0$

thus we have a chain complex and we can compute cohomology. For $n \geq 0$ define the group of n-cocycles as:

$Z^{n}(G,M) = \operatorname{ker}(d^{n})$

and the group of $n$ -coboundaries as

$\begin{cases} B^{0}(G,M) = {0} \\ B^{n}(G,M)= \operatorname{im}(d^{n-1}) \ n \geq 1 \end{cases}$

and

H n (G, M) = Z n (G, M) / B n (G, M)

[edit] The functors $\operatorname{Ext}^n$

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

$H^{n}(G,M) = \operatorname{Ext}^{n}_{\mathbb{Z}[G]}(\mathbb{Z},M)$ .

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.

[edit] Connections with Topological Cohomology Theories

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.

[edit] 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).

[edit] 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). The cohomological part of class field theory was axiomatized as the theory of class formations. Galois cohomology is a large field, and now basic in the theories of algebraic groups and étale cohomology (which builds on it).

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, called Lie algebra cohomology and largely developed after early papers in the late 1940s, 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.