Spectral sequences for the cyclic cohomology of differential graded algebras

Abstract

We construct a number of new spectral sequences for calculating the cyclic cohomology $HC^*_{dg}(A)$ of a differential graded algebra (dga). With these spectral sequences we prove some results about the low dimensional cyclic cohomology and demonstrate the existence of various maps between $HC^*_{dg}(A)$ and $HH^*_{dg}(A)$. We also briefly introduce variations on Hochschild and cyclic cohomology of a dga, namely the $n$-th partial Hochschild cohomology and $n$-partial cyclic cohomology. Finally, we show how these results can be extended naturally to the dg-category setting. In particular we define the Hochschild and cyclic cohomolgy of dg-categories and show that the spectral sequences we have constructed can be used in this setting as well.