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2016 Hochschild cohomology commutes with adic completion
Liran Shaul
Algebra Number Theory 10(5): 1001-1029 (2016). DOI: 10.2140/ant.2016.10.1001


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.


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Liran Shaul. "Hochschild cohomology commutes with adic completion." Algebra Number Theory 10 (5) 1001 - 1029, 2016.


Received: 23 May 2015; Revised: 4 March 2016; Accepted: 16 May 2016; Published: 2016
First available in Project Euclid: 16 November 2017

zbMATH: 1362.13014
MathSciNet: MR3531360
Digital Object Identifier: 10.2140/ant.2016.10.1001

Primary: 13D03
Secondary: 13B35, 13J10, 14B15, 16E45

Rights: Copyright © 2016 Mathematical Sciences Publishers


Vol.10 • No. 5 • 2016
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