How to compute the periodic cyclic homology of this algebra

Question

Let $k=\mathbb{C}$ be the field of complex numbers. I consider the (DG) algebra $A:=k[x]/(x^2)$ such that $\deg(x)=-1$. My question is how to compute the periodic cyclic homology, Hochschild homology and Hochschild cohomology of this (DG) algebra? There are some references on computing those (co)homology of the ring of dual numbers $k[x]/(x^2)$. But in my case, $\deg(x)=-1$. Is there any reference on such computations?

Answer 1

You can use a derived version of the HKR theorem, i.e. $HH(A) = \text{Sym}_A^\bullet(\mathbb{L}_A[1])$ where $\mathbb{L}_A$ is the cotangent complex (over $k$). I'm not sure about a reference though.

Since $A$ is quasi-smooth, its cotangent complex will be in degrees $[-1, 0]$, and one can check using the fact that $\text{Spec}(A) = \{0\} \times_{\mathbb{A}^1} \{0\}$ that $\mathbb{L}_A \simeq \mathcal{N}^\vee_{\{0\}/\mathbb{A}^1}|_A \simeq A[1]$. Then, $HH(A) \simeq \text{Sym}_A^\bullet(A[2])$ (note that this is in even degree, so as a chain complex it is $\bigoplus_{n \geq 0} A[2n]$). You can do something similar for Hochschild cohomology using the tangent complex.

Now, the Connes $B$-operator is given by the de Rham differential. The module $\mathbb{L}_A$ is generated by $dx$, and the de Rham differential takes $x$ to $dx$. You can now calculate the various cyclic homologies, and in particular $HP(A) \simeq k(u)$ where $u \in H^2(BS^1; k)$ is the Chern class (note that since $dx$ is in degree $-2$, it behaves like a symmetric variable not an exterior one).

Answer 2

There's a spectral sequence starting with, for example, Hochschild cohomology of the cohomology of a DGA algebra $A$ and converging to the Hochschild cohomology of $A$, $E_2^{p,q}=H\!H^*H^*(A)\Rightarrow H\!H^*(A)$. In your case, it's just a case of keeping track of internal and cohomological degrees and then adding them, as there are no non-trivial differentials. If you want to see some more complicated examples, and how the spectral sequence relates to $A_\infty$ structure on $H^*(A)$, one place to look is my (long) preprint arXiv:2208.07913, "Classifying spaces of finite groups of tame representation type".