Let $k$ be a field, $R$ is a commutative algebra over $k$ and $A$ is an associative algebra over $R$. There is a morphism of commutative algebras $R \to T$. Is it possible to reduce calculation of Hochschild homology $HH_*(A\otimes_R T)$ (over basic field $k$) to $HH_*(A)$? I'm mostly interested in a situation when $T$ is separable over $R$, or even more specific $T=R/I$.
According to the following paper:
Geller, S.; Weibel, C.: Étale descent for Hochschild and cyclic homology, Comment. Math. Helv. 66 (1991), no. 3, 368–388;
—theorem (0.1)— you need that $T$ is étale over $R$. This holds in your case only if the ideal $I$ is generated by an idempotent. Notice that "étale" = "unramified + flat". And the notion of unramifiedness is related to the condition of separable ring extension, but it depends on which of the several used definitions of "separable" you consider.