Equivariant aspects of de-completing cyclic homology

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.