DG-algebras and derived A∞-algebras

A differential graded algebra can be viewed as an A(infinity)-algebra. By a theorem of Kadeishvili, a dga over a field admits a quasi-isomorphism from a minimal A(infinity)-algebra. We introduce the notion of a derived A(infinity)-algebra and show that any dga A over an arbitrary commutative ground ring k is equivalent to a minimal derived A(infinity)-algebra. Such a minimal derived A(infinity)-algebra model for A is a k-projective resolution of the homology algebra of A together with a family of maps satisfying appropriate relations. As in the case of A(infinity)-algebras, it is possible to recover the dga up to quasi-isomorphism from a minimal derived A(infinity)-algebras model. Hence the structure we are describing provides a complete description of the quasi-isomorphism type of the dga.