$A_\infty$-modules over $A_\infty$-algebras and Hochschild cohomology for modules over algebras

For each left graded module $M$ over a graded algebra $A$, a Hochschild cochain complex $S^*(A,M)$ whose homology is responsible for the existence of nontrivial structures of $A_\infty$-modules over $A_\infty$-algebras on the given module is constructed.