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

The inhomogeneous group cochain group iswith $C^0(G,M)=M$. Its group coboundary $d:C^r(G,M)\to C^{r+1}(G,M)$ isOne checks that $d^2=0$, and group cohomology is $H^r(G,M)=\ker d/\mathrm{im}d$.