Spectral sequences for the cyclic cohomology of differential graded algebras

2025/08/23 by Phimister, Andrew
#FOS: Mathematics #K-Theory and Homology (math.KT)

paper · doi:10.48550/arxiv.2508.16869

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.

Citations

• The spectral sequence of a polycomplex
• An introduction to relative Calabi-Yau structures
• On differential graded categories
• Secondary Characteristic Classes and Cyclic Cohomology of Hopf Algebras

Related

• Spectral sequences for cyclic homology
• A spectral sequence for the Hochschild cohomology of a coconnective dga
• Cyclic theory for commutative differential graded algebras and s-cohomology
• Applications of the change-of-rings spectral sequence to the computation of Hochschild cohomology
• Cyclic Cohomology of Hopf algebras and Hopf Algebroids