Volume 26, issue 2 (2022)

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A universal Hochschild–Kostant–Rosenberg theorem

Tasos Moulinos, Marco Robalo and Bertrand Toën

Geometry & Topology 26 (2022) 777–874
Abstract

In this work we study the failure of the HKR theorem over rings of positive and mixed characteristic. For this we construct a filtered circle interpolating between the usual topological circle and a formal version of it. By mapping to schemes we produce this way an interpolation, realized in practice by the existence of a natural filtration, from Hochschild and (a filtered version of) cyclic homology to derived de Rham cohomology. In particular, we show that this recovers the filtration of Antieau (Ann. K–Theory 4 (2019) 505–519) and Bhatt, Morrow and Scholze (Publ. Math. Inst. Hautes Études Sci. 129 (2019) 199–310). The construction of our filtered circle is based on the theory of affine stacks and affinization introduced by the third author, together with some facts about schemes of Witt vectors.

Keywords
Hochschild homology, derived algebraic geometry, loop space, filtered circle, HKR theorem, de Rham cohomology
Mathematical Subject Classification
Primary: 14A30, 19D55
References
Publication
Received: 17 May 2020
Revised: 3 February 2021
Accepted: 4 March 2021
Published: 15 June 2022
Proposed: Stefan Schwede
Seconded: Richard P Thomas, Frances Kirwan
Authors
Tasos Moulinos
Université Paul Sabatier
Institut de Mathématiques de Toulouse (UMR 5219)
CNRS
Toulouse
France
Marco Robalo
Sorbonne Université
Faculté de sciences et ingénierie Pierre et Marie Curie
Institut de Mathématiques de Jussieu Paris Rive Gauche
CNRS
Paris
France
https://webusers.imj-prg.fr/~marco.robalo/
Bertrand Toën
Université Paul Sabatier
Institut de Mathématiques de Toulouse (UMR 5219)
CNRS
Toulouse
France
https://perso.math.univ-toulouse.fr/btoen/