Let $A$ be a differential graded algebra over a commutative ring $R$. Suppose that $H_*(A)=0$, i.e. $A$ is acyclic.

**Question:** Does this imply that the Hochschild homology $HH_*(A)$ also vanishes identically? If not, what hypotheses does one need to impose on $R$ and $A$? (In the case I am mostly interested in, $R$ is a polynomial ring in one variable.)

**Remark:** if we assume that $R$ is a field, then it's a general fact that one can cook up an $A_{\infty}$-algebra structure on $H_*(A)$ such that $A \to H_*(A)$ is an $A_{\infty}$ quasi-isomorphism. Moreover, $A_{\infty}$-algebra quasi-isomorphisms are invertible and $HH_*(-)$ is functorial. So it follows that that $HH_*(A)=0$ is $H_*(A)=0$. So the result should be true in this case.

Homological algebra. $\endgroup$ – Denis Nardin Apr 19 at 18:58