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Let
$X$
be a separated finite type scheme over a noetherian base ring
$\mathbb{K}$
. There is a complex
${{\hat{C}}^{\cdot }}\left( X \right)$
of topological
${{\mathcal{O}}_{X}}$
-modules, called the complete Hochschild chain complex of
$X$
. To any
${{\mathcal{O}}_{X}}$
-module
$\mathcal{M}$
—not necessarily quasi-coherent—we assign the complex
$Hom_{{{\mathcal{O}}_{X}}}^{\text{cont}}\left( {{{\hat{C}}}^{\cdot }}\left( X \right),\,\mathcal{M} \right)$
of continuous Hochschild cochains with values in
$\mathcal{M}$
. Our first main result is that when
$X$
is smooth over
$\mathbb{K}$
there is a functorial isomorphism
$$Hom_{{{\mathcal{O}}_{X}}}^{\text{cont}}\left( {{C}^{\cdot }}\left( X \right),\,M \right)\,\cong \,\text{R}\,Hom_{{{\mathcal{O}}_{X}}^{2}}^{{}}\,\left( {{\mathcal{O}}_{X}},\,M \right)$$
in the derived category
$\text{D}\left( \text{Mod}\,{{\mathcal{O}}_{{{X}^{2}}}} \right)$
, where
${{X}^{2}}\,:=\,X\,{{\times }_{\mathbb{K}}}\,X$
.
The second main result is that if
$X$
is smooth of relative dimension
$n$
and
$n!$
is invertible in
$\mathbb{K}$
, then the standard maps
$\text{ }\!\!\pi\!\!\text{ }\,\text{:}\,{{\hat{C}}^{-q}}\left( X \right)\,\to \,\Omega _{X/\mathbb{K}}^{q}$
induce a quasi-isomorphism
$$Hom_{{{\mathcal{O}}_{X}}}^{{}}\,\left( \underset{q}{\mathop \oplus }\,\,\Omega _{X/\mathbb{K}}^{q}\,\left[ q \right],\,M \right)\,\,\to \,Hom_{{{\mathcal{O}}_{X}}}^{\text{cont}}\left( {{{\hat{C}}}^{\cdot }}\left( X \right),\,M \right).$$
When
$M\,=\,{{\mathcal{O}}_{X}}$
this is the quasi-isomorphism underlying the Kontsevich Formality Theorem.
Combining the two results above we deduce a decomposition of the global Hochschild cohomology
$$\text{Ext}_{{{\mathcal{O}}_{{{X}^{2}}}}}^{i}\,\left( {{\mathcal{O}}_{X}}\,,\,M \right)\,\cong \,\underset{q}{\mathop \oplus }\,\,\,{{\text{H}}^{i-q}}\,\left( X,\,\left( \underset{{{\mathcal{O}}_{X}}}{\overset{q}{\mathop \Lambda }}\,\,{{T}_{X/\mathbb{K}}} \right)\,{{\otimes }_{{{\mathcal{O}}_{X}}}}\,M \right),$$
where
${{T}_{X/\mathbb{K}}}$
is the relative tangent sheaf.
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