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Hochschild cohomology commutes with adic completion
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 (1)
make a detailed study of derived completion as a functor over a non-noetherian ring ; (2) prove a flat base change
result for weakly proregular ideals; and (3) Prove that Hochschild cohomology
and analytic Hochschild cohomology of complete noetherian local rings are
isomorphic, answering a question of Buchweitz and Flenner. Our results makes 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.Comment: 23 pages. Final version, to appear in Algebra and Number Theor
Adic reduction to the diagonal and a relation between cofiniteness and derived completion
We prove two results about the derived functor of -adic completion: (1)
Let be a commutative noetherian ring, let be a flat noetherian
-algebra which is -adically complete with respect to some ideal
, such that is essentially of finite type over , and let
be finitely generated -modules. Then adic reduction to the diagonal
holds: . A similar result is given in the case where are
not necessarily finitely generated. (2) Let be a commutative ring, let
be a weakly proregular ideal, let be an -module, and
assume that the -adic completion of is noetherian (if is noetherian,
all these conditions are always satisfied). Then \mbox{Ext}^i_A(A/a,M) is
finitely generated for all if and only if the derived -adic
completion \L\hat{\Lambda}_{a}(M) has finitely generated cohomologies over
. The first result is a far reaching generalization of a result of
Serre, who proved this in case is a field or a discrete valuation ring and
.Comment: 12 pages. Final version, to appear in Proceedings of the AM
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