## Abstract

Let X be a separated finite type scheme over a noetherian base ring double-struck K sign. There is a complex ê^{.}(X) of topological script O sign_{X}-modules, called the complete Hochschild chain complex of X. To any script O sign_{X}-module script M sign - not necessarily quasi-coherent - we assign the complex ℋom_{script O sign X}^{cont} (ê^{.}(X), script M sign) of continuous Hochschild cochains with values in script M sign. Our first main result is that when X is smooth over double-struck K sign there is a functorial isomorphism ℋom_{script O sign X}^{cont} (ê^{.}(X), script M sign) ≅ R ℋom_{script O sign X2} (script O sign_{X}, script M sign) in the derived category D(Mod script O sign_{X2}), where X^{2}: = X ×_{double-struck K sign} X. The second main result is that if X is smooth of relative dimension n and n! is invertible in double-struk K sign, then the standard maps π: ê^{-q}(X) → Ω_{X/double-struck K sign}^{q} induce a quasi-isomorphism ℋom_{script O sign X} (⊕_{q} Ω_{X/double-struck K sign}^{q}[q], script M sign) → ℋom_{script O sign X}^{cont} (ê^{.}(X), M). When M = script O sign_{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_{script O sign X2}^{i}(script O sign_{X}, script M sign) ≅ ⊕ H^{i-q} (X· (∧ script T sign_{X/double-struck K sign}) ⊗_{script O sign X} script M sign). where script T sign_{X/double-struck K sign} is the relative tangent sheaf.

Original language | English |
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Pages (from-to) | 1319-1337 |

Number of pages | 19 |

Journal | Canadian Journal of Mathematics |

Volume | 54 |

Issue number | 6 |

DOIs | |

State | Published - 1 Jan 2002 |

## Keywords

- Derived categories
- Hochschild cohomology
- Schemes

## ASJC Scopus subject areas

- General Mathematics