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Periodic cyclic homology over $\mathbb{Q}$

Konrad Bals

Vol. 9 (2024), No. 1, 119–142
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 and Feigin–Tsygan to nonsmooth and non-lci 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.

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Keywords
periodic cyclic homology, Hochschild homology, Hodge completed derived de Rham complex, derived schemes, Tate filtration
Mathematical Subject Classification
Primary: 13D03, 19D55
Secondary: 19E08, 55P42
Milestones
Received: 5 April 2023
Revised: 3 January 2024
Accepted: 3 January 2024
Published: 25 May 2024
Authors
Konrad Bals
Mathematisches Institut
Universität Münster
Münster
Germany