MICROLOCAL EULER CLASSES AND HOCHSCHILD HOMOLOGY

We define the notion of a trace kernel on a manifold M. Roughly speaking, it is a sheaf on M x M for which the formalism of Hochschild homology applies. We associate a microlocal Euler class with such a kernel, a cohomology class with values in the relative dualizing complex of the cotangent bundle T*M over M, and we prove that this class is functorial with respect to the composition of kernels. This generalizes, unifies and simplifies various results from (relative) index theorems for constructible sheaves, D-modules and elliptic pairs.