Global Homotopies for HKR Theorems in Differential Geometry

The classical Hochschild-Kostant-Rosenberg (HKR) Theorem in Differential Geometry gives a quasi-isomorphism between the differentiable Hochschild complex of the algebra of smooth functions on a manifold and the multivector fields on it. The HKR morphism plays an important role in deformation theory, as it allows us to study existence and classification of deformations of associative algebras. Even though the HKR Theorem has been known for a long time, available proofs are often of a local nature and are hard to generalize to more structured situations. We will present a novel proof for the HKR Theorem using a symbol calculus and a van Est-double complex. This strategy will allow for an explicit global homotopy and can easily be adapted to various situations.