  We classify (up to quasi-isomorphism) the free differential modules whose
homology is equal to a given module $M$ by developing a theory for deforming an
arbitrary free complex into a differential module. We use an iterative approach
to parameterize the deformations and obstructions in terms of certain Ext
groups, giving an algorithmic realization of a result of Brown-Erman. We apply
this theory to study certain rigidity properties of free resolutions and
related rank conjectures.
