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 $n\in \mathbb{Z}$