Periodic and negative cyclic homology

Recall the cyclic bicomplex

$$\xymatrix{ \vdots \ar[d] & \vdots \ar[d] &\vdots \ar[d] & \v... ...[l]_B & C_0 \ar[l]_B \\ C_1 \ar[d]_b & C_0 \ar[l]_B \\ C_0 }$$

which after passing to total complex gives a complex computing cyclic homology of an algebra. There is an obvious way to extend this bicomplex to the right using the same differentials

$$\xymatrix{ & \vdots \ar[d] & \vdots \ar[d] & \vdots \ar[d] &... ...ar[l]_B& C_0 \ar[l]_B \\ \hdots & C_1 \ar[l] & C_0 \ar[l]_B }$$

Furthermore we can repeat each row going down continuing the same pattern.

$$\xymatrix{ & \vdots \ar[d] & \vdots \ar[d] & \vdots \ar[d] &... ...[l] \ar[d] & C_0 \ar[l]_B \\ \hdots & C_0 \ar[l] \\ \hdots }$$ (3)

This is called a periodic bicomplex. If the columns of the cyclic bicomplex we started with were indexed by natural numbers starting from 0, then in the periodic bicomplex () we have columns indexed by integers.

To work with the total complex of the periodic bicomplex one should use the product instead of a sum. Otherwise one would get zero in the homology.