The continuous Hochschild cochain complex of a scheme

Let X be a separated finite type scheme over a noetherian base ring double-struck K sign. There is a complex ê^.(X) of topological script O sign_X-modules, called the complete Hochschild chain complex of X. To any script O sign_X-module script M sign - not necessarily quasi-coherent - we assign the complex ℋom_{script O sign X}^cont (ê^.(X), script M sign) of continuous Hochschild cochains with values in script M sign. Our first main result is that when X is smooth over double-struck K sign there is a functorial isomorphism ℋom_{script O sign X}^cont (ê^.(X), script M sign) ≅ R ℋom_{script O sign X2} (script O sign_X, script M sign) in the derived category D(Mod script O sign_X2), where X²: = X ×_{double-struck K sign} X. The second main result is that if X is smooth of relative dimension n and n! is invertible in double-struk K sign, then the standard maps π: ê^-q(X) → Ω_{X/double-struck K sign}^q induce a quasi-isomorphism ℋom_{script O sign X} (⊕_q Ω_{X/double-struck K sign}^q[q], script M sign) → ℋom_{script O sign X}^cont (ê^.(X), M). When M = script O sign_X this is the quasi-isomorphism underlying the Kontsevich Formality Theorem. Combining the two results above we deduce a decomposition of the global Hochschild cohomology Ext_{script O sign X2}ⁱ(script O sign_X, script M sign) ≅ ⊕ H^i-q (X· (∧ script T sign_{X/double-struck K sign}) ⊗_{script O sign X} script M sign). where script T sign_{X/double-struck K sign} is the relative tangent sheaf.