Cochain complex for a group action

Definition
Given a group $$G$$ acting as automorphisms on an Abelian group $$A$$, the cochain complex for this group action is defined as follows:


 * The $$n^{th}$$ cochain group is the group of all maps from $$G^n$$ to $$A$$ (by maps here is meant set-theoretic maps)
 * The boundary map, or derivation, from the $$n^{th}$$ cochain group to the $$(n+1)^{th}$$ cochain group, is defined as follows:

$$(d_nf)(g_1,g_2,\ldots,g_n,g_{n+1}) = g_1.f(g_2,g_3,\ldots,g_{n+1}) - f(g_1g_2,g_3,\ldots,g_{n+1}) + f(g_1,g_2g_3,\ldots,g_{n+1}) \ldots + (-1)^{n+1}f(g_1,g_2,\ldots,g_n)$$

The lack of a left-right interchange symmetry is because the group is acting from the left side. If the group were acting on the left as well as the right, we would have a $$g_{n+1}$$ acting on the right in the last term.

The fact that this is a cochain complex follows from the (not-too-hard) fact that:

$$d_n \circ d_{n-1} = 0$$ The cohomology of this cochain complex is defined as the cohomology of the group action.

Cocycle groups
The $$n^{th}$$ cocycle group, denoted as $$Z^n(G)$$, is defined as the kernel of the map $$d_n$$. In other words, it is the additive group of those functions $$f:G^n \to A$$ such that:

$$g_1.f(g_2,g_3,\ldots,g_{n+1}) - f(g_1g_2,g_3,\ldots,g_{n+1}) + f(g_1,g_2g_3,\ldots,g_{n+1}) \ldots + (-1)^{n+1}f(g_1,g_2,\ldots,g_n) = 0$$

for all tuples $$(g_1,g_2,\ldots,g_{n+1}) \in G^{n+1}$$

See also: 1-cocycle for a group action, 2-cocycle for a group action, 3-cocycle for a group action

Coboundary groups
The $$n^{th}$$ coboundary group, $$B^n(G)$$, is defined as the image of the map $$d_{n-1}$$.

Note that $$B^n(G)$$ is a subgroup of $$Z^n(G)$$, by the fact that this is a complex.

Cohomology groups
The $$n^{th}$$ cohomology group, $$H^n(G)$$ is defined as the quotient $$Z^n(G)/B^n(G)$$. It is in other words the homology of the cochain complex.

Particular cohomology groups ,specially the first ,second and third ones, are of direct significance even in finite group theory. These are discussed in separate articles.

See also:


 * First cohomology group for a group action
 * Second cohomology group for a group action
 * Third cohomology group for a group action