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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.

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