Crossed homomorphism

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.