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 also means that $A$ is a compact object in $D(A\otimes_R^L A^{op})$ (for simplicity, we assume that $A$ is cofibrant; in this case, the derived tensor product becomes the usual tensor product).

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

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

In fact, one is able to prove that $HH_\bullet(A)$ is a finitely generated $R$-module if $R$ is noetherian.

Thanks!