I read the article "Defomrations of algebras in noncommutative geometry" by Schedler. In Definition 3.7.9. he gives the definition of Calabi-Yau algebra of dimensi on d as algebras that are homological smooth with $HH^{*}(A,A \otimes_k A)=A[-d]$ as graded algebras.

My question: For finite dimensional algebras, is that the same as algebras with $A$ having finite global dimension and $HH^{d}(A,A \otimes_k A)$ (which should be the same as $ Ext^{d}(D(A),A)$) isomorphic to $A$ as bimodules while $HH^{i}(A,A \otimes_k A)=0$ for $i \neq d$ (including $i=0$?) Im confused by the graded algebras thing when one only has one nonzero term.