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?
2 Answers
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 quasismooth, 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).

2$\begingroup$ It could be remarked that it's not a coincidence that $HP(A)$ turns out to be equal to $HP(k)$, the periodic cyclic homology of the base field: it's a theorem of Goodwillie's that periodic cyclic homology in characteristic zero is invariant under nilpotent thickenings, and our $k[x]/x^2$ is a nilpotent thickening of $k$. As far as references go, this particular example is considered in Section 5.4 (especially, 5.4.14) of Loday's book 'Cyclic homology'. $\endgroup$– SashaPCommented Aug 20, 2023 at 16:01

$\begingroup$ @SashaP Thanks! I might also add a paper by Bhatt: arxiv.org/abs/1207.6193 $\endgroup$ Commented Aug 20, 2023 at 17:46

$\begingroup$ @SashaP Thanks for the reference, but it seems that the examples in Loday's book is about $k[x]/x^2$ with $deg(x)=0$, but in my question, the $deg(x)=1$, does this matter? $\endgroup$ Commented Aug 22, 2023 at 17:44

$\begingroup$ @user41650 The reference sciencedirect.com/science/article/pii/0040938385900552 Theorem III.5.1 might do it. $\endgroup$ Commented Aug 22, 2023 at 18:01

$\begingroup$ @user41650 My mistake, I should have rather referenced Theorem 5.4.12 in the same section. $\endgroup$– SashaPCommented Aug 22, 2023 at 18:13
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 nontrivial 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".