Stefan Waldmann: The HKR Theorem: a new look at a classical result

Abstract

In this talk I will give an overview on a new approach to this classical theorem in differential geometry: the HKR theorem computes the differential Hochschild cohomology of the algebra of smooth functions to be the multivector fields on the manifold. The proofs known in the literature make use of local computations with rather unclear geometric nature. The situtation clarifies if one uses a global symbol calculus to encode the multidifferential operators by means of certain tensor fields. Then the Hochschild complex becomes the complex of group cohomology of an abelian group whose cohomology can most easily be compute by means of the van Est maps. This point if view has the advantage to split the problem into a geometric one (global symbol calculus) and a purely algebraic and universal one.

I will indicate several applications to HKR theorems for invariant Hochschild cohomologies, Hochschild cohomologies with nontrivial coefficients etc. leading to explicit computations of cohomologies not known before. The results are based on a recent work with Marvin Dippell, Chiara Esposito and Jonas Schnitzer.