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We classify (up to quasi-isomorphism) the free differential modules whosehomology is equal to a given module $M$ by developing a theory for deforming anarbitrary free complex into a differential module. We use an iterative approachto parameterize the deformations and obstructions in terms of certain Extgroups, giving an algorithmic realization of a result of Brown-Erman. We applythis theory to study certain rigidity properties of free resolutions andrelated rank conjectures.

Differential modules and Deformations of Free Complexes