# A Universal HKR Theorem

## TL;DR

This paper investigates the limitations of the HKR theorem in positive and mixed characteristic rings by constructing a filtered circle that interpolates between Hochschild, cyclic homology, and derived de Rham cohomology, recovering known filtrations.

## Contribution

It introduces a novel filtered circle construction based on affine stacks and Witt vectors, providing a new perspective on the HKR theorem's failure and its relation to existing filtrations.

## Key findings

- Constructed a filtered circle interpolating between topological and formal circles.
- Revealed a natural filtration connecting Hochschild, cyclic homology, and derived de Rham cohomology.
- Recovers the filtrations of Antieau and Bhatt-Morrow-Scholze.

## 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 and Bhatt-Morrow-Scholze. 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.