Calabi-Yau algebra for finite dimensional algebras

Question

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.

Answer 1

It is not particularly productive to think about finite-dimensional Calabi–Yau algebras, at least if you are grading everything in degree $0$ (and possibly even if you have a more interesting grading—I have never seriously thought about this case), since they are all semi-simple.

Let $A$ be a non-zero finite-dimensional Calabi–Yau algebra. Then one can find a non-zero map $S\to P$ for some projective $P$ and finite-dimensional module $S$ (e.g. take the socle of an indecomposable projective). But then the formula

$$\operatorname{Hom}_A(S,P)=\mathrm{D}\operatorname{Ext}^d_A(P,S),$$

which being $d$-Calabi–Yau implies, can't hold for $d>0$ since the right-hand side is always zero. So the only possibility is $d=0$, meaning in particular that $\operatorname{gldim}{A}=0$, so $A$ is semi-simple.