Equivariant aspects of de-completing cyclic homology

TL;DR

This paper provides an equivariant homotopy-theoretic framework for polynomial periodic cyclic homology, connecting it to derived de Rham cohomology, and extends these ideas to topological analogues and noncommutative settings.

Contribution

It introduces an equivariant description of polynomial periodic cyclic homology, enabling Morita invariance and transparent comparison with derived de Rham cohomology, and extends to topological and noncommutative contexts.

Findings

Morita invariance of polynomial periodic cyclic homology without Noetherianness.

Transparent comparison between polynomial periodic cyclic homology and derived de Rham cohomology.

Extension of results to topological periodic cyclic homology and prime p=2.

Abstract

Derived de Rham cohomology turns out to be important in $p$-adic geometry, following Bhatt's discovery [Bha12] of conjugate filtration in char $p$, de-Hodge-completing results in [Bei12]. In [Kal18], Kaledin introduced an analogous de-completion of the periodic cyclic homology, called the polynomial periodic cyclic homology, equipped with a conjugate filtration in char $p$, and expected to be related to derived de Rham cohomology. In this article, using genuine equivariant homotopy structure on Hochschild homology as in [ABG+18, BHM22], we give an equivariant description of Kaledin's polynomial periodic cyclic homology. This leads to Morita invariance without any Noetherianness assumption as in [Kal18], and the comparison to derived de Rham cohomology becomes transparent. Moreover, this description adapts directly to "topological" analogues, which gives rise to a de-Nygaard-completion of the topological periodic cyclic homology, which admits an extension to linear categories over truncated Brown--Peterson spectra. As an application, we establish a noncommutative crystalline--de Rham comparison, which decompletes the result in [PV19], and extends it to prime $p=2$. We also compare polynomial periodic cyclic homology to topological Hochschild homology over $\mathbb F_p$, and produce a conjugate filtration in char p from our description.