When are two HKR isomorphisms equal?

TL;DR

This paper investigates conditions under which two HKR isomorphisms, arising from different first order splittings of a closed embedding, are equal, using a generalized Atiyah class to characterize their equality.

Contribution

It introduces the generalized Atiyah class for vector bundles associated with embeddings and splittings, providing criteria for the equality of HKR isomorphisms.

Findings

HKR isomorphisms from different splittings are generally not equal over the product space

The two HKR isomorphisms are equal over the original scheme X

The generalized Atiyah class characterizes when HKR isomorphisms are equal

Abstract

Let $X\hookrightarrow S$ be a closed embedding of smooth schemes which splits to first order. An HKR isomorphism is an isomorphism between the shifted normal bundle $\mathbb{N}_{X/S}[-1]$ and the derived self-intersection $X\times^R_SX$. Given two different first order splittings of a closed embedding, one can obtain two HKR isomorphisms using a construction of Arinkin and C\u{a}ld\u{a}raru. A priori, it is not known if the two isomorphisms are equal or not. We define the generalized Atiyah class of a vector bundle on $X$ associated to a closed embedding and two first order splittings. We use the generalized Atiyah class to give sufficient and necessary conditions for when the two HKR isomorphisms are equal over $X$ and over $X\times X$ respectively. When $i$ is the diagonal embedding, there are two natural projections from $X\times X$ to $X$. We show that the HKR isomorphisms defined by the two projections are equal over $X$, but not equal over $X\times X$ in general.