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 appears in deformation quantization as first degree of Kontsevich’s formality map. 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. As examples, we will present HKR Theorems for submanifolds and surjective submersions, as well as equivariant versions for both Lie group and Lie algebra actions.