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!

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!