On a theorem of Keller over a base ring

Let $X$ be a quasi-compact separated scheme over a base field. Keller proved a theorem stating that the cyclic homology of $X$ is canonically isomorphic to the cyclic homology of the dg category $\mathsf{P}\mathsf{e}\mathsf{r}\mathsf{f}(X)$ consisting of perfect complexes over $X$. This theorem shows the categorical nature of the cyclic homology. In this note, we generalize Keller’s theorem to allow $X$ be defined over a base commutative ring.