Title: The Continuous Hochschild Cochain Complex of a Scheme

Publication status:
Canadian J. Math Vol. 54 (6), 2002, pp. 1319-1337.

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. 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_{X^2}(O_X,M), in the derived category

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. 

[Abstract as graphic]

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Warning: "Due to a processing error at the Publications Office of the Canadian Mathematical Society, the withdrawn paper Decomposition of the Hochschild Complex of a Scheme in Arbitrary Characteristic was printedin Volume 54, No. 4. This paper replaces the withdrawn one. The Editors of the Canadian Journal of Mathematics deeply regret this error."

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