Periodic cyclic homology and derived de Rham cohomology

TL;DR

This paper constructs filtrations on cyclic homology theories of schemes using advanced tools like the Beilinson t-structure and the Hochschild-Kostant-Rosenberg theorem, linking them to derived de Rham cohomology.

Contribution

It introduces a new approach to filtrations on cyclic homology using the Beilinson t-structure, extending previous constructions to broader contexts.

Findings

Filtrations on negative cyclic and periodic cyclic homologies are constructed.

Graded pieces of these filtrations correspond to the Hodge-completion of derived de Rham cohomology.

Connections to prior work by Loday and Bhatt-Morrow-Scholze are established.

Abstract

We use the Beilinson $t$-structure on filtered complexes and the Hochschild-Kostant-Rosenberg theorem to construct filtrations on the negative cyclic and periodic cyclic homologies of a scheme $X$ with graded pieces given by the Hodge-completion of the derived de Rham cohomology of $X$. Such filtrations have previously been constructed by Loday in characteristic zero and by Bhatt-Morrow-Scholze for $p$-complete negative cyclic and periodic cyclic homology in the quasisyntomic case.