One-cocycle

For a $G$-module $M$, a one-cocycle is a map $f:G\to M$ satisfying $f(gh)=f(g)+g\cdot f(h)$.

A continuous one-cocycle for a Galois group acting on an abelian module $M$ is a map satisfying the displayed identity. Cocycles modulo maps $\sigma \mapsto \sigma u-u$ form $H^1(K,M)$ in Galois cohomology. For an elliptic curve, choosing $mQ=P$ gives the cocycle $\sigma \mapsto \sigma Q-Q$ in $E[m]$, which represents the Kummer map of an elliptic curve.

A crossed homomorphism from $G$ to a $G$-module $M$ is a map $f:G\to M$ satisfyingThus crossed homomorphisms are precisely one-cocycles.

For $m\in M$, the map $g\mapsto g\cdot m-m$ is a principal crossed homomorphism. It is the group coboundary of the zero-cochain $m$, so first group cohomology is the quotient of crossed homomorphisms by principal crossed homomorphisms.