In his paper Homological algebra of homotopy algebras, Hinich shows that if two dg algebras are quasi-isomorphic, they have equivalent derived categories. Is it true that if X is a perfect complex of (A,B)-bimodules giving an equivalence D(A) ---> D(B) one can produce a zig-zag of quasi-isomorphisms connecting A and B? That is, does derived equivalence imply that A and B are isomorphic in Ho(dg-Alg)?
There is a quasi-iso A --> RHom_B(X,X), for example. So I am guessing using this and an inverse to the equivalence one may produce something of the shape A --> M --> N <-- M' <-- B