About the Hochschild-Kostant-Rosenberg theorem for differentiable manifolds

TL;DR

This paper provides techniques and a proof for the Hochschild-Kostant-Rosenberg theorem on differentiable manifolds, including definitions of multidifferential operators, polyderivations, and a cup product structure.

Contribution

It introduces a coordinate-free approach to the theorem, defining key concepts and constructing algebraic structures to facilitate understanding and proof.

Findings

Constructed a cup product on polyderivations corresponding to wedge product.

Provided a coordinate-free proof of the Hochschild-Kostant-Rosenberg theorem.

Clarified the algebraic structures underlying differentiable manifolds.

Abstract

In this notes it will be provided a set of techniques which can help one to understand the proof of the Hochschild-Kostant-Rosenberg theorem for differentiable manifolds. Precise definitions of multidiferential operators and polyderivations on an algebra are given, allowing to work on these concepts, when the algebra is an algebra of functions on a differentiable manifold, in a coordinate free description. Also, it will be constructed a cup product on polyderivations which corresponds on (Hochschild) cohomology to wedge product on multivector fields. At the end, a proof of the above mentioned theorem will be given.