Suppose we have the following data:


1) A group ring $\mathbb{Z}[G]$, where $G$ is a torsion free group.


2) $M\_{\bullet}$ a bounded (above and below) chain complex of $\mathbb{Z}[G]$-modules such that each $M\_{i}$ is a finitely generated free $\mathbb{Z}[G]$-module. 


My question is the following:
If the homology of $M\_{\bullet}\otimes\_{\mathbb{Z}[G]}\mathbb{Z}$ is trivial ie 
$H\_{n}(M\_{\bullet}\otimes\_{\mathbb{Z}[G]}\mathbb{Z})=0$ for all $n\in \mathbb{Z}$ does it imply that $$H\_{n}(M\_{\bullet})=0$$ for all $n\in \mathbb{Z}$



Faithfully flat modules over a group algebra
No way. Let $G={\mathbb Z}$ so that ${\mathbb Z}[G]={\mathbb Z}[x,x^{-1}]$. Then use the complex
$$\ldots \rightarrow 0 \rightarrow 0 \rightarrow {\mathbb Z}[x,x^{-1}] \xrightarrow{1-x+x^2} {\mathbb Z}[x,x^{-1}] \rightarrow 0 \rightarrow 0 \rightarrow \ldots$$


