  Let H be a differential graded Hopf algebra over a field k. This paper gives
an explicit construction of a triple cochain complex that defines the
Hochschild-Cartier cohomology of H. A certain truncation of this complex is the
appropriate setting for deforming H as an H(q)-structure. The direct limit of
all such truncations is the appropriate setting for deforming H as a strongly
homotopy associative structure. Sign complications are systematically
controlled. The connection between rational perturbation theory and the
deformation theory of certain free commutative differential graded algebras is
clarified.
