Periodic Cyclic Homology over Q

TL;DR

This paper provides a comprehensive description of periodic cyclic homology for derived schemes over characteristic zero rings, linking it explicitly to the Hodge completed derived de Rham complex, and extends prior results to non-smooth cases.

Contribution

It offers a complete characterization of periodic cyclic homology in terms of the Hodge completed derived de Rham complex, extending earlier work to non-smooth algebras.

Findings

Explicit description of periodic cyclic homology via derived de Rham complex

Extension of Loday-Quillen computations to non-smooth algebras

Conditions for the exhaustiveness of the HKR-filtration on periodic cyclic homology

Abstract

Let $X$ be a derived scheme over an animated commutative ring of characteristic 0. We give a complete description of the periodic cyclic homology of $X$ in terms of the Hodge completed derived de Rham complex of $X$. In particular this extends earlier computations of Loday-Quillen to non-smooth algebras. Moreover, we get an explicit condition on the Hodge completed derived de Rham complex, that makes the HKR-filtration on periodic cyclic homology constructed by Antieau and Bhatt-Lurie exhaustive.