Given a group extension $0\to K\to G\to Q\to 1$, we define $B_n$ as the free $\Bbb Z[Q]$-module on $Q^n$. And then we want to make a exact sequence $\cdots\to B_3\to B_2\to B_1\to B_0$, where the homomorphism between these $B_n$'s is $d_n$'s. And then the author define $d_3:B_3\to B_2$ as $d_3[x|y|z]=x[y|z]-[xy|z]+[x|yz]-[x|y]$. However, I feel weird that is such definition make sense? The domain of $d_3$ is $B_3$, which is isomorphic to $\oplus_{i\in Q^3}\Bbb Z[Q]$. And any element in $\oplus_{i\in Q^3}\Bbb Z[Q]$ is of the form $(\underbrace{\cdots\cdots}_{\text{has}~|Q^n|~\text{many component}})$. So how can $d_3$ be defined as $d_3[x|y|z]=x[y|z]-[xy|z]+[x|yz]-[x|y]$? It doesn't seem match.
