Hochschild (Co)homology for Lie Algebroids

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 formulas depend on certain natural structures on jet bundles over Lie algebroids. In an appendix, we explain this by showing that such jet bundles are formal groupoids which serve as the formal exponentiation of the Lie algebroid.