I am reading some materials about derived category. For one step in the proof of one theorem, it seems using the following consequence:
Let $C_{\bullet}, D_{\bullet}$ be two chains of abelian grouos ($C_{\bullet}:... C_{n}\to C_{n-1}\to...C_{0}\to 0$), $f=(f_{n})$ be a chain map from $C_{\bullet}$ to $D_{\bullet}$. If every $C_{n}$ and $\mathrm{H}_{n}(C_{\bullet})$ are free abelian groups and $f$ induced $0$ homomorphism, then $f$ is homotopic to $0$.
I try to construct chain homotopy $l_{n}: C_{n} \to D_{n+1}$. Since $f_{0}$ induced $0$ homomorphism, $f_{0}(C)\subseteq d_{1}(D_{1})$. Thus we could lift $f_{0}$ to $l_{0}: C_{0} \to D_{1}$. But I do not know how to construct $l_{1}$. Also since $\mathrm{H}_{n}$ are free, $\mathrm{Ker}d_{n}=\mathrm{Im} d_{n+1}\oplus \mathrm{H}_{n}$. I guess in order to construct $l_{n}$, we need to use this.
Could you tell me how to construct $l_{n}$? Thanks a lot.