Filtrations and cohomology III: cohomology of $E_\infty$ rings

TL;DR

This paper explores filtrations derived from de Rham-type cohomology theories for $E_ $ rings, including examples like Hochschild and Hodge filtrations, advancing understanding of their structures and relationships.

Contribution

It introduces and analyzes filtrations on $E_ $ rings' cohomology theories, providing new insights into their structures and interrelations.

Findings

HKR filtration on topological Hochschild homology examined

Hodge filtration on $E_ $ infinitesimal cohomology analyzed

Connections between different filtrations established

Abstract

We discuss filtrations arising from de Rham-type cohomology theories for $E_\infty$ rings and $E_n$ rings. Examples include the HKR filtration on relative topological Hochschild homology, the Hodge filtration on $E_\infty$ infinitesimal cohomology, and the Hodge filtration on $E_\infty$ de Rham cohomology.