Decomposition of the Hochschild complex of a scheme in arbitrary characteristic

Let X be a smooth separated scheme over a noetherian base ring K. The decomposition we are interested in is an isomorphism R ℋom_oX×KX(O_X, O_X) ≅ ⊕_q (Λ_ox^q τ_X/K)[-q] in the derived category D(Mod O_{x × K} x). Here τ_X/K is the relative tangent sheaf. Upon passing to cohomology sheaves such an isomorphism recovers the Hochschild-Kostant-Rosenberg Theorem. If K has characteristic 0 there is a decomposition that relies on a particular homomorphism of complexes from poly-tangents to continuous Hochschild cochains. We discuss sheaves of continuous Hochschild cochains on schemes and show why this approach to decomposition fails in positive characteristics. Our main result is a proof of the decomposition valid for any Gorenstein noetherian ring K of finite Krull dimension, regardless of characteristic. The proof is based on properties of minimal injective resolutions.