I am reading a book that uses a lot of homological algebra and I am not familiar with the tools of it. To prove a part of a theorem I suspect the following must be true.
Let $X$ be an infinite $G$-set. Consider a chain complex $(C_{*}(X), \partial)$ where $C_n(X)$ is the free abelian group generated by $X^{n+1}$ and $\partial((x_0,\ldots,x_n))=\sum_i (-1)^{i} (x_0, \ldots, x_{i-1}, x_{i+1}, \ldots, x_n)$.
If $C_n^{\neq}(X)$ is free abelian group generated by the set $\{(x_0, \ldots,x_n)~|~x_i \neq x_j \forall i \neq j\}$ and $C_n^D(X)$ is free abelian group generated by the set $\{(x_0, \ldots, x_n)~|~x_i =x_{i+1} ~\textrm{for some}~0 \leq i \leq n-1\}$, then is it true that for any $G$ module $M$, $H_n(C_n ^{\neq} (X) \otimes_{\mathbb{Z}[G]}M) \cong H_n(C_n^{nor}(X) \otimes_{\mathbb{Z}[G]} M)?$
Here $C_n^{nor}(X)= C_n(X)/C_n^{D}(X)$ and is called normalized complex.