Title:
The Continuous Hochschild Cochain Complex of a Scheme
Publication status: Canadian J. Math Vol. 54
(6), 2002, pp. 1319-1337.
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. 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
D(Mod(O_{X^2})).
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.
Electronic
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Warning: "Due to a processing error at the Publications Office of the Canadian Mathematical Society, the withdrawn paper Decomposition of the Hochschild Complex of a Scheme in Arbitrary Characteristic was printedin Volume 54, No. 4. This paper replaces the withdrawn one.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."
(updated 13/3/2011)