Cocycle for trivial group action

Definition
Suppose $$G$$ is a group and $$A$$ is an abelian group.

Definition in terms of cocycle for a group action
A $$n$$-cocycle for trivial group action is a $$n$$-defining ingredient::cocycle for a group action of $$G$$ on $$A$$, where the action is trivial.

Explicit definition
A $$n$$-cocycle for trivial group action of $$G$$ on $$A$$ is a function $$f:G^n \to A$$ satisfying the following for all $$(g_1,g_2,\dots,g_{n+1}) \in G^{n+1}$$:

$$\! f(g_2,g_3,\dots,g_{n+1}) + \left[ \sum_{i=1}^{n-1} (-1)^i f(g_1,g_2, \dots,g_ig_{i+1},\dots,g_n)\right] + (-1)^{n+1} f(g_1,g_2,\dots,g_n) = 0$$