Hochschild cohomology commutes with adic completion

Liran Shaul

Research output: Contribution to journalArticlepeer-review

4 Scopus citations


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 HHn(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[[t1,….,tn]] over any noetherian ring k, and open the door for a theory of Hochschild cohomology over formal schemes.

Original languageEnglish
Pages (from-to)1001-1029
Number of pages29
JournalAlgebra and Number Theory
Issue number5
StatePublished - 1 Jan 2016
Externally publishedYes


  • Adic completion
  • Hochschild cohomology

ASJC Scopus subject areas

  • Algebra and Number Theory


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