The Continuous Hochschild Cochain Complex of a Scheme

Abstract

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.