Hochschild cohomology commutes with adic completion

Abstract

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 $H{H}^n(A/k,M)$ is naturally isomorphic to $H{H}^n(\hat{A}/k,\hat{M})$. To show this, we (1) make a detailed study of derived completion as a functor $D(A)\to D(\hat{A})$ 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.

23 pages. Final version, to appear in Algebra and Number Theory

References in corpus (1)

• Tensor product of dualizing complexes over a field

Cited by in corpus (3)

• Derived Categories
• Injective DG-modules over non-positive DG-rings
• Extended Local Cohomology and Local Homology