Hochschild cohomology commutes with adic completion

Abstract

For a flat commutative $\mathbb{k}$-algebra $A$ such that the enveloping algebra $A{\otimes }_{\mathbb{k}}A$ is noetherian, given a finitely generated bimodule $M$, we show that the adic completion of the Hochschild cohomology module ${HH}^n\left(A∕\mathbb{k},M\right)$ is naturally isomorphic to ${HH}^n\left(Â∕\mathbb{k},\widehat{M}\right)$. To show this, we make a detailed study of derived completion as a functor $D\left(ModA\right)\to D\left(ModÂ\right)$ 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 $\mathbb{k}\left[\left[t_1,\dots ,t_n\right]\right]$ over any noetherian ring $\mathbb{k}$, and open the door for a theory of Hochschild cohomology over formal schemes.