The Continuous Hochschild Cochain Complex of a Scheme

Abstract

Let X be a separated finite type scheme over a noetherian base ring K. There is a complex C(X) of topological O_X-modules on X, called the complete Hochschild chain complex of X. To any O_X-module M - not necessarily quasi-coherent - we assign the complex Hom^{cont}_X(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 RHom_{X2}(O_X,M), in the derived category D(Mod(O_{X2})). 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 Hom^{cont}_X(C(X),M) in derived category D(Mod(O_X)). When M = O_X 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.