Even and odd part of the Hochschild and cyclic homology of a super-algebra

Question

Let $A$ be a $\mathbb Z_2$-graded $k$-algebra, where $k$ is a field of characteristic $0$. Then we know that the tensor product of $A$ with itself is also $\mathbb Z_2$-graded by

$$(A\otimes_k A)_0:=A_0\otimes_kA_0\oplus A_1\otimes_k A_1\tag{1}\label{1}$$

and

$$(A\otimes_k A)_1:=A_0\otimes_kA_1\oplus A_1\otimes_k A_0.\tag{2}\label{2}$$

The above $\mathbb Z_2$-grading of the tensor product induces a $\mathbb Z_2$-grading on the $\mathbb Z_2$-graded Hochschild homology and on the $\mathbb Z_2$-graded cyclic homology of $A$.

Is there an explicit formula for the even and the odd part the $\mathbb Z_2$-graded Hochschild homology $HH_n(A)$ and the $\mathbb Z_2$-graded cyclic homology $HC_n(A)$ similar to Identities \eqref{1} and \eqref{2}?

In particular, I would like to understand the grading on the left- and the right-hand side of the baby example \begin{gather*} HH_n(\mathbb C[\epsilon])=HH_n(\mathbb C[\epsilon])_0\oplus HH_n(\mathbb C[\epsilon])_1=\mathbb C\omega_n^1\oplus\mathbb C\omega_n^2,\tag{3}\label{3} \\ HC_n(\mathbb C[\epsilon])=HC_n(\mathbb C[\epsilon])_0\oplus HC_n(\mathbb C[\epsilon])_1=HC_n(\mathbb C)\oplus\mathbb C,\tag{4}\label{4} \end{gather*}

where $\omega_n^1:=1\otimes\epsilon^{\otimes n}$, $\omega_n^2:=\epsilon^{\otimes n+1}$ and $\mathbb C[\epsilon]$ with $\epsilon^2=0$ and $\deg(\epsilon)=1$ is the super-algebra of dual numbers. I am not absolutely certain about the generators of $HC_n(\mathbb C)$ and $\mathbb C$, so I would appreciate it if someone could give the cycles which generate these spaces. But more importantly, I would like to know which part is the even part and which part is the odd part on the right-hand side of \eqref{3} and \eqref{4}?

Answer 1

I would like to share a partial answer to my own question. It should be taken with cautiousness because it might not be correct. I will improve it over time.

For every $n\in\mathbb Z_{\geq0}$, let us define the even part $HH_n(\mathbb C[\epsilon])_0$ and $HC_n(\mathbb C[\epsilon])_0$ of the $\mathbb Z_2$-graded Hochschild and cyclic homology $HH_n(\mathbb C[\epsilon])$ and $HC_n(\mathbb C[\epsilon])$, respectively, as the subspace generated by the homology classes of the $n$-cycles with an even number $k\leq n$ of epsilons, while the odd part $HH_n(\mathbb C[\epsilon])_1$ and $HC_n(\mathbb C[\epsilon])_1$ as the subspace, generated by those $n$-cycles consisting of an odd number $l\leq n$ of epsilons.

Therefore, for every even number $n$, we have

$$HH_n(\mathbb C[\epsilon])_0\cong\mathbb C\omega_n^1$$ $$HH_n(\mathbb C[\epsilon])_1\cong\mathbb C\omega_n^2.$$

On the contrary, for every odd number $n$, we have

$$HH_n(\mathbb C[\epsilon])_0\cong\mathbb C\omega_n^2$$ $$HH_n(\mathbb C[\epsilon])_1\cong\mathbb C\omega_n^1.$$

Similarly, for the $\mathbb Z_2$-graded cyclic homology of $\mathbb C[\epsilon]$, we have:

When $n$ is even, then

$$HC_n(\mathbb C[\epsilon])_0=\mathbb C\omega_n^0,$$ $$HC_n(\mathbb C[\epsilon])_1=\mathbb C\omega_n^2,$$

where $\omega_n^0:=1^{\otimes n+1}$. When $n$ is odd,

$$HC_n(\mathbb C[\epsilon])_0=\mathbb C\omega_n^2$$ $$HC_n(\mathbb C[\epsilon])_1=\{0\}.$$