Title:When are two HKR isomorphisms 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 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 $X\times X$ 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.