Let $A$ be a unital associative algebra over a commutative noetherian ring $R$.Assume that $A$ is homologically smooth which means that $A\in D_{perf}(A\otimes_R^L A^{op})$ which means that $A$ is compact object in $D_{perf}(A\otimes_R^L A)$(for simplicity,we assume that $A$ is cofibrant,in this case,derived tensor product becomes usual tensor product)

The question is if $A\in D_{perf}(R)$($A$ is proper over $R$),is hochschild homology of $A$,i.e:$HH_.(A)$ projective $R$-module? If it is not true,any counter example?

Notice that $R$ is not field $k$ here,it is just a commutative ring.

In fact,one is able to prove that $HH_.(A)$ is finite generate $R$-module if $R$ is noetherian.

Thanks!