#### Vol. 10, No. 5, 2016

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Hochschild cohomology commutes with adic completion

### Liran Shaul

Vol. 10 (2016), No. 5, 1001–1029
##### 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 ${HH}^{n}\left(Â∕\mathbb{k},\stackrel{̂}{M}\right)$. 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.