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.



Why should noncommutative CYs be dgas?
For any reasonable smooth scheme the category $D^b(X)$ is equivalent to perfect DG-modules over a DG-algebra. So, the world of schemes is embedded into the world of DG-algebras. This is why we think about DG-algebras as about noncommutative spaces.


