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Let $(Q,W)$ be a quiver with potential, and $D$ be Ginzburg's DG algebra associated to it (as explained in and other places), so that it is a 3-Calabi-Yau algebra. I want to relate the Hochschild homology of $D$ and the cyclic homology of $D$.

Consider the long exact sequence relating them, $\cdots\to HC_{n+2} \to HC_{n}\to HH_n \to HC_{n-1}\to \cdots$. In our case, we know $HC_{\bullet\ge3}(D)=0$ and $HH_{\bullet\ge4}(D)=0$, and also $\overline{HC}_2(D)\simeq HH_3(D)$. We also have $HH_0(D)=HC_0(D)$.

The remaining interesting part is

$0\to HC_1(D)\to HH_2(D)\to HC_2(D)\to HC_0(D)\to HH_1(D)\to HC_1(D)\to 0$.

What is known about the periodicity map $S:HC_2(D)\to HC_0(D)$ in this particular case when $D$ is given as above? Is it a zero map? Or is it an isomorphism?

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