In this question, we see how to compute the Hochschild homology of a dga with zero differential: it’s just the same as computing its Hochschild homology as a graded algebra. I want to know about calculations in the case where there are nontrivial differentials.
(1) In general, I suspect there is a spectral sequence that takes in as input the Hochschild homology of the dga with its differentials set to zero, and converges to the Hochshild homology of the dga. Essentially, this suspicion is due to the fact that the bicomplex whose total complex computes Hochschild homology can be filtered so that we first compute homology along the “Hochschild” direction, and then the homology along the “dga” direction.
Is there a spectral sequence $E^2_{p,q} = H_p(\operatorname{HH}_q A) \Rightarrow \operatorname{HH}_{p+q} A$?
However, in some basic examples I’ve computed, the Hochschild homology of a dga with its differential set to zero can no longer be equipped with the differentials from the original dga, so I don’t know what the $E^2$ page means.
(2) Specifically, I would like to see how the same example in the linked question works when I set $dx = p \neq 0$. What is the Hochschild homology of the following dga? $$ \mathbb{Z}[x]/x^{n+1}, \qquad \deg x = 1, \, dx = p$$