Hochschild cohomology commutes with adic completion

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,….,t_n]] over any noetherian ring k, and open the door for a theory of Hochschild cohomology over formal schemes.