TITLE: Does homology detect chain homotopy equivalence?
QUESTION [22 upvotes]: Is the following true: If two chain complexes of free abelian groups have isomorphic homology modules then they are chain homotopy equivalent.

REPLY [14 votes]: Yes, this is true, and it does not matter whether the complexes are bounded from any side (nor of course does it matter whether the homology is finitely generated).  This is so because:

The homotopy category of free abelian groups is equivalent to the derived category of abelian groups.  This holds even for unbounded complexes, since the category of abelian groups has a finite homological dimension.
Any complex of abelian groups is quasi-isomorphic to its homology, since the category of abelian groups has homological dimension 1.