Let $G$ be a group and let $M$ be a $G$-module. We denote by $(C(G,M),d)$ the corresponding cochain complex obtained via the standard resolution.
Suppose that $f:M\to M$ is a morphism of $G$-modules. We define the cochain complex $(C(G,M,f),d)$ by $C^n(G,M,f)=C^n(G,M)\times C^{n+1}(G,M)$, with $d^n:C^n(G,M,f)\to C^{n+1}(G,M,f)$ given by $d^n(a,b)=(f(b)-da,db)$.
It is an easy exercise to see that $d^{n+1}d^n=0$.
My question is whether this is already known. (Maybe with some other notation.) My knowledge of cohomology of groups is quite limited. Maybe some expert of the field saw this cochain complex somewhere.
