Choices of HKR isomorphisms

TL;DR

This paper characterizes the set of multiplicative HKR isomorphisms across all derived schemes, linking them to filtered formal exponential maps and showing that over a field of characteristic zero, this set is isomorphic to the multiplicative group.

Contribution

It establishes a natural bijection between HKR isomorphism choices and filtered formal exponential maps via Cartier duality.

Findings

The set of HKR isomorphisms corresponds to filtered formal exponential maps.

Over characteristic zero fields, the set of choices is isomorphic to the field's multiplicative group.

The paper clarifies the structure of HKR isomorphisms in derived algebraic geometry.

Abstract

In this short note we record the fact that the set of multiplicative HKR natural equivalences defined simultaneously for all derived schemes, functorialy splitting the HKR-filtration and rendering the circle action compatible with the de Rham differential, is, via Cartier duality, in a natural bijection with the set of filtered formal exponential maps $ \widehat{\mathbb{G}_a}\to \widehat{\mathbb{G}_m}$. In particular, when the base $k$ is a field of characteristic zero, the set of choices is $k^\ast$.