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]$ ?