**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
Preprint:**

postscipt file

Journal
format pdf

**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Canadian Mathematical Society, the withdrawn paper Decomposition of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 This paper replaces the withdrawn one.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)