Spectral sequences for the cyclic cohomology of differential graded algebras

TL;DR

This paper introduces new spectral sequences for computing the cyclic cohomology of differential graded algebras, explores their properties, and extends these concepts to dg-categories, providing new tools for algebraic topology and noncommutative geometry.

Contribution

It constructs novel spectral sequences for cyclic cohomology of dgas and extends these methods to dg-categories, advancing computational techniques in homological algebra.

Findings

New spectral sequences for cyclic cohomology of dgas

Results on low-dimensional cyclic cohomology

Extension of spectral sequences to dg-categories

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.