$(DA)_\infty$-modules over $(DA)_\infty$-algebras and spectral sequences

We introduce the construction of a $(DA)_\infty$-(co)module over a $(DA)_\infty$-(co) algebra and study its main homotopy properties. We establish a connection between $(DA)_\infty$-(co)modules over $(DA)_\infty$-(co)algebras and spectral sequences, and thus obtain the structure of an $A_\infty$-comodule over the Milnor $A_\infty$-coalgebra on the homology of any spectrum directly from the differentials of the Adams spectral sequence of this spectrum.