#### 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