Group cohomology

In abstract algebra, homological algebra, algebraic topology and algebraic number theory, as well as in applications to group theory proper, group cohomology arises as a sequence of functors Hⁿ. The H⁰ functor can be described directly as the subgroup of G-invariant elements, for any abelian group on which a group acts (by endomorphisms). Then the other functors, for n = 1, 2, 3 ... arise because taking invariants doesn't respect exact sequences.

In more concrete terms, if B is a subgroup of A mapped to itself by the action of G, it isn't in general true that the invariants in A/B are found as the quotient of the invariants in A by the invariants in B: being invariant 'up to something in B' is broader. The difference then lies in H¹(B). That is, by general procedures (cf. snake lemma) a long exact sequence is constructed.

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 (Schur's multiplicator) and in simple algebras (Brauer), both of these latter being connected with H². 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 to class field theory provided theorems 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 analoguous theory for Lie algebras is formally similar, starting with the corresponding definition of invariant. It is much applied in representation theory.