The Continuous Hochschild Cochain Complex of a Scheme

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Let X be a separated finite type scheme over a noetherian base ring K. There is a complex C(X) of topological Oₓ-modules on X, called the complete Hochschild chain complex of X. To any Oₓ-module M - not necessarily quasi-coherent - we assign the complex Homcontₓ(C(X),M) of continuous Hochschild cochains with values in M. Our first main result is that when X is smooth over K there is a functorial isomorphism between the complex of continuous Hochschild cochains and RHomₓ₂(Oₓ,M), in the derived category D(Mod(Oₓ₂)). The second main result is that if X is smooth of relative dimension n and n! is invertible in K, then the standard map from Hochschild chains to differential forms induces a decomposition of Homcontₓ(C(X),M) in derived category D(Mod(Oₓ)). When M = Oₓ this is the precisely the quasi-isomorphism underlying the Kontsevich Formality Theorem. Combining the two results above we deduce a decomposition of the global Hochschild cohomology with values in M.