Let $C$ be a chain complex over a principal ideal domain $R$. How can I construct a chain complex $F$ of free $R$-modules which is quasi-isomorphic to $C$?
Edit: It is well known how to do this with a chain complex concentrated in degree $0$, and we could take that as motivation. In that case, a projective resolution allows us to construct a chain complex with the same homological properties, but which is more tractible (e.g. because a quasi-isomorphism of projectives is a homotopy equivalence).