Hochschild cohomology commutes with adic completion

Abstract

For a flat commutative $k$-algebra $A$ such that the enveloping algebra $A\otimes_k A$ is noetherian, given a finitely generated bimodule $M$, we show that the adic completion of the Hochschild cohomology module $HH^n(A/k,M)$ is naturally isomorphic to $HH^n(\widehat{A}/k,\widehat{M})$. To show this, we (1) make a detailed study of derived completion as a functor $D(A) \to D(\widehat{A})$ over a non-noetherian ring $A$; (2) prove a flat base change result for weakly proregular ideals; and (3) Prove that Hochschild cohomology and analytic Hochschild cohomology of complete noetherian local rings are isomorphic, answering a question of Buchweitz and Flenner. Our results makes it possible for the first time to compute the Hochschild cohomology of $k[[t_1,\dots,t_n]]$ over any noetherian ring $k$, and open the door for a theory of Hochschild cohomology over formal schemes.