Hochschild cohomology commutes with adic completion

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 ${HH}^{n} (Â ∕ k, \hat{M})$ . To show this, we make a detailed study of derived completion as a functor $D (Mod A) \to D (Mod Â)$ 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 $k [[t_{1}, \dots, t_{n}]]$ over any noetherian ring $k$ , and open the door for a theory of Hochschild cohomology over formal schemes.

Keywords

Hochschild cohomology, adic completion

Mathematical Subject Classification 2010

Primary: 13D03

Secondary: 13J10, 14B15, 16E45, 13B35

Milestones

Received: 23 May 2015

Revised: 4 March 2016

Accepted: 16 May 2016

Published: 28 July 2016

Authors

Liran Shaul
Departement Wiskunde-Informatica Universiteit Antwerpen Middelheim Campus Middelheimlaan 1 2020 Antwerp Belgium

Vol. 10, No. 5, 2016

Liran Shaul

Abstract

Keywords

Mathematical Subject Classification 2010

Milestones

Authors