Vol. 10, No. 5, 2016

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Hochschild cohomology commutes with adic completion

Liran Shaul

Vol. 10 (2016), No. 5, 1001–1029

For a flat commutative k-algebra A such that the enveloping algebra A kA is noetherian, given a finitely generated bimodule M, we show that the adic completion of the Hochschild cohomology module HHn(Ak,M) is naturally isomorphic to HHn(Âk,M̂). To show this, we make a detailed study of derived completion as a functor D(ModA) D(ModÂ) 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,,tn]] over any noetherian ring k, and open the door for a theory of Hochschild cohomology over formal schemes.

Hochschild cohomology, adic completion
Mathematical Subject Classification 2010
Primary: 13D03
Secondary: 13J10, 14B15, 16E45, 13B35
Received: 23 May 2015
Revised: 4 March 2016
Accepted: 16 May 2016
Published: 28 July 2016
Liran Shaul
Departement Wiskunde-Informatica
Universiteit Antwerpen
Middelheim Campus
Middelheimlaan 1
2020 Antwerp