group cohomology

Suppose a group G acts on an abelian group A, and you only care about the elements of A that G leaves completely alone — the fixed points. Picking out fixed points is a clean operation, but it is lossy: two different situations can have the same fixed points yet differ in how G twists everything else. Group cohomology is the systematic record of that lost information. The zeroth piece is just the fixed points; the higher pieces measure, in increasingly subtle ways, the obstructions and twistings that fixed points alone cannot see.

Precisely, let G be a group and A a G-module — an abelian group on which G acts by automorphisms, equivalently a module over the group ring Z[G]. The fixed-point functor A |-> A^G = { a in A : g·a = a for all g } is left exact but not exact, so it has right derived functors. Its n-th right derived functor evaluated at A is the n-th cohomology group H^n(G, A), with H^0(G, A) = A^G. Concretely H^n(G, A) is computed as the cohomology of the cochain complex of functions G^n -> A built from a free resolution of Z over Z[G], most often the bar resolution.

The low-degree groups have famous meanings: H^1(G, A) classifies crossed homomorphisms up to principal ones (and, for nonabelian A, classifies torsors or twisted forms), while H^2(G, A) classifies extensions of G by A with the given action. Higher H^n encode obstructions to lifting and gluing problems. Equivalently and invariantly, H^n(G, A) = Ext^n over Z[G] of the trivial module Z with A, which is why all the standard homological machinery — long exact sequences, dimension shifting, derived functors — applies verbatim.