$D_\infty$-differential $E_\infty$-algebras and spectral sequences of $D_\infty$-differential modules

In the present paper, we introduce the concept of a filtered $E_\infty$-algebra, construct spectral sequences for such algebras, and apply them to multiplicative cohomological spectral sequences of bundles. The existence of the structure of $D_\infty$-differential $A_\infty$-algebra in cohomological spectral sequences of bundles over fields is proved and the initial multiplicative component of this structure at the second term of the spectral sequence is calculated.