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.
|Number of pages||29|
|Journal||Algebra and Number Theory|
|State||Published - 1 Jan 2016|
- Adic completion
- Hochschild cohomology
ASJC Scopus subject areas
- Algebra and Number Theory