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Abstract
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For a flat commutative
-algebra
such that the
enveloping algebra
is noetherian, given a finitely generated bimodule
,
we show that the adic completion of the Hochschild cohomology module
is naturally
isomorphic to
.
To show this, we make a detailed study of derived completion as a functor
over a
nonnoetherian ring
,
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
over any
noetherian ring
,
and open the door for a theory of Hochschild cohomology over formal schemes.
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Keywords
Hochschild cohomology, adic completion
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Mathematical Subject Classification 2010
Primary: 13D03
Secondary: 13J10, 14B15, 16E45, 13B35
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Milestones
Received: 23 May 2015
Revised: 4 March 2016
Accepted: 16 May 2016
Published: 28 July 2016
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