Hochschild cohomology for Lie algebroids

TL;DR

This paper extends Hochschild (co)homology to ringed spaces relative to Lie algebroids, providing computational methods and linking jetbundles to formal groupoids, thus enriching the algebraic geometry toolkit.

Contribution

It introduces a new definition of Hochschild (co)homology for ringed spaces with Lie algebroids and relates jetbundles to formal groupoids, offering computational techniques.

Findings

Hochschild (co)homology can be computed via standard complexes.

Jetbundles over Lie algebroids are formal groupoids.

The framework generalizes existing algebraic geometry concepts.

Abstract

We define the Hochschild (co)homology of a ringed space relative to a locally free Lie algebroid. Our definitions mimic those of Swan and Caldararu for an algebraic variety. We show that our (co)homology groups can be computed using suitable standard complexes. Our formulae depend on certain natural structures on jetbundles over Lie algebroids. In an appendix we explain this by showing that such jetbundles are formal groupoids which serve as the formal exponentiation of the Lie algebroid.