Trace map

There is a trace map defined as

$$\Tr\: M_r(A)\to A,\quad [a_{ij}]_{i,j=1}^r\mapsto \sum_{i=1}^r a_{ii}.$$

We can extend it to a map

$$\Tr\: M_r(A)^{\ox(n+1)} \to A^{\ox(n+1)},$$

$$[a_{i_0j_0}]\ox\hdots\ox[a_{i_nj_n}] \mapsto \sum_{k_0,k_1, \hdots, k_n} a_{k_0k_1}\ox a_{k_1k_2}\ox\hdots\ox a_{k_nk_0}$$

for any $r\geq 1$, $n\geq 0$. It induces a maps on Hochschild, cyclic, periodic cyclic and negative cyclic homology.

$$\HH_n(M_r(A))\to \HH_n(A), \quad \HC_n(M_r(A))\to \HC_n(A), \mbox{ etc.}$$

Let us take an idempotent $e^2=e$ in $M_r(A)$. Under the map $b$ in Hochschild complex for $M_r(A)$ we have

$$e^{\ox(n+1)}\mapsto \begin{cases}0 & n \text{ even}\\ e^{\ox n} & n \text{ odd}\end{cases}$$

In $C_n^{\la}(M_r(A))$ we have $e^{\ox(n+1)}=(-1)^n e^{\ox(n+1)}$. If $n$ is odd, then $[e^{\ox(n+1)}]=0$. If $n=2m$ is even, then $b[e^{\ox(n+1})]=0$, so $[e^{\ox(n+1)}]$ is a cycle, and we can define a map $[e]\mapsto [\Tr(e^{\ox(n+1)})]$,

$$\rK_0(A)\to \rH^{\la}_{2m}(M(A))\xrightarrow{\Tr} \rH^{\la}_{2m}(A),$$

$$M(A)=\bigcup_{r}M_r(A),\quad M_r(A)\hookrightarrow M_{r+1}(A), \quad \al\mapsto \twobytwo{\al}{0}{0}{0}.$$

We have to show that the element $[\Tr(e^{\ox(n+1)})]\in\HC_{2m}(A)$ depends only on the isomorphism class.

We have constructed a functorial map $\rK_0(A)\to \rH_{2m}^{\la}(A)$. Now we ask if we can construct a map $\rK_0(A)\to \HC_{2m}(A)$?

Recall the cyclic bicomplex $C_{\bullet\bullet}(A)$

$$\xymatrix{ \vdots \ar[d] & \vdots \ar[d] & \vdots \ar[d] & \... ...} & C_0 \ar[l]_{N} & C_0 \ar[l]_{1-t} & \hdots \ar[l]_{N}\\ }$$

Define
$$\begin{align*} y_i &:= (-1)^i\frac{(2i)!}{i!}e^{\ox(2i+1)}, \\ z_i &:= (-1)^{i-1}\frac{(2i)!}{2(i!)}e^{\ox(2i)}. \end{align*}$$


For the $B_{\bullet}C_{\bullet}$ we have to use $\ch([e]):=(y_n, y_{n-1}, \hdots, y_0)\in (\Tot(B_{\bullet}C_{\bullet}(A)))_n$.

We can define a map

$$\ch\:\rK_0(A)\to \drH^{ev}(A),\quad \ch([e]):= \Tr(e de de\hdots de).$$