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The HKR theorem for cohomology in characteristic zero says that if $R$ is a regular, commutative $k$ algebra ($char(k) = 0$) then a certain map $\bigwedge^* Der(R) \to CH^*(R,R)$ (where $\wedge^* Der(R)$ has zero differential) is a quasi-isomorphism of dg vector spaces, that is, it induces an isomorphism of graded vector spaces on cohomology.

Can the HKR morphism be extended to an $A_\infty$ morphism? Is there a refinement in this spirit to make up for the fact that it is not, on the nose, a morphism of dg-algebras?

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Yes there is. It was noted by Kontsevich long time ago that the HKR quasi-isomorphism on cochains can be corrected to give a quasi-isomorphism of dg-algebras and thus induce an $A_\infty$ quasi-isomorphism of minimal models. The correction is very natural - one needs to compose the HKR map it the contraction by the square root of the Todd class, where the latter is understood as a polynomial of the Atiyah class. This story has been studied in great detail in the past few years and has been generalized further to give Tsygan formality which is a quasi-isomorphism of $\infty$-calculi. This was proven by Dolgushev-Tamarkin-Tsygan and also by Calaque-Rossi-van den Bergh.

The literature on the subject is huge but you should get a good sense of the results if you look at this survey by Dolgushev-Tamarkin-Tsygan and at this paper of Calaque-Rossi-van den Bergh. There are also many interesting references listed in these papers, for instance the works of Caldararu on the Mukai pairing.

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