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# The Continuous Hochschild Cochain Complex of a Scheme

Published:2002-12-01
Printed: Dec 2002
• Amnon Yekutieli
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## Abstract

Let $X$ be a separated finite type scheme over a noetherian base ring $\mathbb{K}$. There is a complex $\widehat{\mathcal{C}}^{\cdot} (X)$ 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 $\mathcal{H}om^{\cont}_{\mathcal{O}_X} \bigl( \widehat{\mathcal{C}}^{\cdot} (X), \mathcal{M} \bigr)$ 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 $$\mathcal{H}om^{\cont}_{\mathcal{O}_X} \bigl( \widehat{\mathcal{C}}^{\cdot} (X), \mathcal{M} \bigr) \cong \R \mathcal{H}om_{\mathcal{O}_{X^2}} (\mathcal{O}_X, \mathcal{M})$$ in the derived category $\mathsf{D} (\Mod \mathcal{O}_{X^2})$, 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 $\pi \colon \widehat{\mathcal{C}}^{-q} (X) \to \Omega^q_{X/ \mathbb{K}}$ induce a quasi-isomorphism $$\mathcal{H}om_{\mathcal{O}_X} \Bigl( \bigoplus_q \Omega^q_{X/ \mathbb{K}} [q], \mathcal{M} \Bigr) \to \mathcal{H}om^{\cont}_{\mathcal{O}_X} \bigl( \widehat{\mathcal{C}}^{\cdot} (X), \mathcal{M} \bigr).$$ When $\mathcal{M} = \mathcal{O}_X$ this is the quasi-isomorphism underlying the Kontsevich Formality Theorem. Combining the two results above we deduce a decomposition of the global Hochschild cohomology $$\Ext^i_{\mathcal{O}_{X^2}} (\mathcal{O}_X, \mathcal{M}) \cong \bigoplus_q \H^{i-q} \Bigl( X, \bigl( \bigwedge^q_{\mathcal{O}_X} \mathcal{T}_{X/\mathbb{K}} \bigr) \otimes_{\mathcal{O}_X} \mathcal{M} \Bigr),$$ where $\mathcal{T}_{X/\mathbb{K}}$ is the relative tangent sheaf.
 Keywords: Hochschild cohomology, schemes, derived categories
 MSC Classifications: 16E40 - (Co)homology of rings and algebras (e.g. Hochschild, cyclic, dihedral, etc.) 14F10 - Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials [See also 13Nxx, 32C38] 18G10 - Resolutions; derived functors [See also 13D02, 16E05, 18E25] 13H10 - Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.) [See also 14M05]

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