For a (commutative) CY manifold of dimension $n$, Serre duality implies that there are ifunctorial isomorphisms $$\operatorname{Hom}_{D^b(X)}(E,F)\cong\operatorname{Hom}_{D^b(X)}(F,E[n])^*$$ in the bounded derived category of coherent sheaves and $[n]$ denotes the $n$-fold shift.

Kontsevich and Soibelman define a differential graded algebra $A$ to be Calabi-Yau of dimension $n$ if its (perfect) derived category satisfies the formula analogous to those of $D^b(X)$ above. Ginzburg gives an alternative (but inequivalent) definition of the Calabi-Yau property by requiring an $A$-bimodule isomorphism $A\to A^![n]$ where $A^!=\mathbf{R}\operatorname{Hom}_{A^e}(A,A^e)$. Either way, in both of these definitions, a Calabi-Yau algebra is always required to be a *differential graded* algebra.

Question:Why should a noncommutative CY space be differential graded?

Any thoughts or explanations or references would be much appreciated.