Cocycle for a group action

Definition

Suppose $G$ is a group and $A$ is an abelian group, with an action $\varphi$ of $G$ on $A$. In other words, $\varphi$ is a homomorphism of groups from $G$ to $\operatorname {Aut} (A)$, the automorphism group of $A$.

Definition in terms of bar resolution

A $n$-cocycle is an element in the $n^{th}$ cocycle group for the Hom complex from the bar resolution of $G$ to $A$, in the sense of $\mathbb {Z} G$-modules.

Explicit definition

For $n$ a nonnegative integer, a $n$-cocycle for the action $\varphi$ of $G$ on $A$ is a function $f:G^{n}\to A$ such that, for all $g_{1},g_{2},\dots ,g_{n+1}\in G$:

$\!\varphi (g_{1})(f(g_{2},g_{3},\dots ,g_{n+1}))+\left[\sum _{i=1}^{n-1}(-1)^{i}f(g_{1},g_{2},\dots ,g_{i}g_{i+1},\dots ,g_{n})\right]+(-1)^{n+1}f(g_{1},g_{2},\dots ,g_{n})=0$

If we suppress the symbol $\varphi$ and denote the action by $\cdot$, this becomes:

$\!g_{1}\cdot f(g_{2},g_{3},\dots ,g_{n+1}))+\left[\sum _{i=1}^{n-1}(-1)^{i}f(g_{1},g_{2},\dots ,g_{i}g_{i+1},\dots ,g_{n})\right]+(-1)^{n+1}f(g_{1},g_{2},\dots ,g_{n})=0$

In particular, when the action is trivial, this is equivalent to saying that:

$\!f(g_{2},g_{3},\dots ,g_{n+1})+\left[\sum _{i=1}^{n-1}(-1)^{i}f(g_{1},g_{2},\dots ,g_{i}g_{i+1},\dots ,g_{n})\right]+(-1)^{n+1}f(g_{1},g_{2},\dots ,g_{n})=0$

Particular cases

$n$	Condition for being a $n$-cocycle	Further information
1	For all $g_{1},g_{2}\in G$, we have $\!g_{1}\cdot f(g_{2})-f(g_{1}g_{2})+f(g_{1})=0$, equivalently $\!f(g_{1}g_{2})=f(g_{1})+g_{1}\cdot f(g_{2})$	1-cocycle for a group action
2	For all $g_{1},g_{2},g_{3}\in G$, we have $\!g_{1}\cdot f(g_{2},g_{3})-f(g_{1}g_{2},g_{3})+f(g_{1},g_{2}g_{3})-f(g_{1},g_{2})=0$, equivalently $g_{1}\cdot f(g_{2},g_{3})+f(g_{1},g_{2}g_{3})=f(g_{1}g_{2},g_{3})+f(g_{2},g_{3})$	2-cocycle for a group action
3	For all $g_{1},g_{2},g_{3},g_{4}\in G$, we have $\!g_{1}\cdot f(g_{2},g_{3},g_{4})-f(g_{1}g_{2},g_{3},g_{4})+f(g_{1},g_{2}g_{3},g_{4})-f(g_{1},g_{2},g_{3}g_{4})+f(g_{1},g_{2},g_{3})=0$, or equivalently, $\!g_{1}\cdot f(g_{2},g_{3},g_{4})+f(g_{1},g_{2}g_{3},g_{4})+f(g_{1},g_{2},g_{3})=f(g_{1}g_{2},g_{3},g_{4})+f(g_{1},g_{2},g_{3}g_{4})$	3-cocycle for a group action