When are two HKR isomorphisms equal?

Abstract

Let X↪S be a closed embedding of smooth schemes which splits to first order. An HKR isomorphism is an isomorphism between the shifted normal bundle NX/S[−1] and the derived self-intersection X×SRX. Given two different first order splittings of a closed embedding, one can obtain two HKR isomorphisms using a construction of Arinkin and Căldăraru. 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×X respectively. When i is the diagonal embedding, there are two natural projections from X×X to X. We show that the HKR isomorphisms defined by the two projections are equal over X, but not equal over X×X in general.