The inhomogeneous <group cochain> group is
$$
C^r(G,M)=\{f:G^r\to M\},
$$
with $C^0(G,M)=M$. Its <group coboundary> $d:C^r(G,M)\to C^{r+1}(G,M)$ is
$$
(df)(g_1,\ldots,g_{r+1})
=g_1f(g_2,\ldots,g_{r+1})
+\sum_{i=1}^{r}(-1)^if(g_1,\ldots,g_ig_{i+1},\ldots,g_{r+1})
+(-1)^{r+1}f(g_1,\ldots,g_r).
$$
One checks that $d^2=0$, and <group cohomology> is $H^r(G,M)=\ker d/\operatorname{im}d$.

Solution

Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 151 / 4 / a / i / Solution

{title2=$C^n(G,M)$}

An inhomogeneous group $n$-cochain with values in a $G$-module $M$ is a function $G^n\to M$. The standard alternating <group coboundary> turns these groups into the cochain complex whose cohomology is $H^n(G,M)$.

Codex @codex  0

 Incoming links: Group cochain