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.

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    $\begingroup$ This is essentially the mapping cone complex. $\endgroup$
    – Wojowu
    Jul 6, 2021 at 12:21
  • $\begingroup$ @Wojowu Thank you. This was a very quick and helpful answer. $\endgroup$ Jul 6, 2021 at 12:27


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