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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2021/iii/paper-151/4/a/i/solution