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 $HH^n(\widehat{A}/k,\widehat{M})$. To show this, we (1) make a detailed study of derived completion as a functor $D(A) \to D(\widehat{A})$ over a non-noetherian ring $A$; (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 $k[[t_1,\dots,t_n]]$ over any noetherian ring $k$, 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 $a$-adic completion: (1) Let $K$ be a commutative noetherian ring, let $A$ be a flat noetherian $K$-algebra which is $a$-adically complete with respect to some ideal $a\subseteq A$, such that $A/a$ is essentially of finite type over $K$, and let $M,N$ be finitely generated $A$-modules. Then adic reduction to the diagonal holds: $A\otimes^{L}_{ A\hat{\otimes}_{K} A } ( M\hat{\otimes}^{L}_{K} N ) \cong M \otimes^{L}_A N$. A similar result is given in the case where $M,N$ are not necessarily finitely generated. (2) Let $A$ be a commutative ring, let $a\subseteq A$ be a weakly proregular ideal, let $M$ be an $A$-module, and assume that the $a$-adic completion of $A$ is noetherian (if $A$ is noetherian, all these conditions are always satisfied). Then \mbox{Ext}^i_A(A/a,M) is finitely generated for all $i\ge 0$ if and only if the derived $a$-adic completion \L\hat{\Lambda}_{a}(M) has finitely generated cohomologies over $\hat{A}$. The first result is a far reaching generalization of a result of Serre, who proved this in case $K$ is a field or a discrete valuation ring and $A = K[[x_1,\dots,x_n]]$.Comment: 12 pages. Final version, to appear in Proceedings of the AM