The Continuous Hochschild Cochain Complex of a Scheme

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AbstractLet X be a separated finite type scheme over a noetherian base ring . There is a complex of topological -modules, called the complete Hochschild chain complex of X. To any -module —not necessarily quasi-coherent—we assign the complex of continuous Hochschild cochains with values in . Our first main result is that when X is smooth over there is a functorial isomorphismin the derived category , where .The second main result is that if X is smooth of relative dimension n and n! is invertible in K, then the standard maps induce a quasi-isomorphismWhen this is the quasi-isomorphism underlying the Kontsevich Formality Theorem.Combining the two results above we deduce a decomposition of the globalHochschild cohomologywhere is the relative tangent sheaf.