## Abstract

For a flat commutative k-algebra A such that the enveloping algebra A ⊗_{k} A is noetherian, given a finitely generated bimodule M, we show that the adic completion of the Hochschild cohomology module HH^{n}(Â/k, M) is naturally isomorphic to HHn.AO=k; MO /. To show this, we make a detailed study of derived completion as a functor D(Mod A) → D(Mod Â) over a nonnoetherian ring A, prove a flat base change result for weakly proregular ideals, and prove that Hochschild cohomology and analytic Hochschild cohomology of complete noetherian local rings are isomorphic, answering a question of Buchweitz and Flenner. Our results make it possible for the first time to compute the Hochschild cohomology of k[[t_{1},….,t_{n}]] over any noetherian ring k, and open the door for a theory of Hochschild cohomology over formal schemes.

Original language | English |
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Pages (from-to) | 1001-1029 |

Number of pages | 29 |

Journal | Algebra and Number Theory |

Volume | 10 |

Issue number | 5 |

DOIs | |

State | Published - 1 Jan 2016 |

Externally published | Yes |

## Keywords

- Adic completion
- Hochschild cohomology

## ASJC Scopus subject areas

- Algebra and Number Theory