Cyclic Modules with $\infty$-Simplicial Faces and the Cyclic Homology of $A_\infty$-Algebras

A chain bicomplex for $A_\infty$-algebras, which generalizes the Tsygan chain bicomplex in the theory of cyclic homology of associative algebras, is constructed by using the techniques of differential modules with $\infty$-simplicial faces and $D_\infty$-differential modules. For homotopy unital $A_\infty$-algebras, an exact sequence generalizing the Connes–Tsygan exact sequence for unital associative algebras is obtained.