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Microlocal Euler classes and Hochschild homology

Part of: Partial differential equations (Co)homology theory

Published online by Cambridge University Press:  18 July 2013

Masaki Kashiwara  and
Pierre Schapira
Masaki Kashiwara
Affiliation:
Research Institute for Mathematical Sciences, Kyoto University, Japan Department of Mathematical Sciences, Seoul National University, Republic of Korea ([email protected])
Pierre Schapira
Affiliation:
Institut de Mathématiques, Université Pierre et Marie Curie, France Mathematics Research Unit, University of Luxemburg, Luxemburg ([email protected])
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Abstract

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We define the notion of a trace kernel on a manifold $M$. Roughly speaking, it is a sheaf on $M\times M$ 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 ${T}^{\ast } M$ 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, $\mathscr{D}$-modules and elliptic pairs.

Keywords

sheavesD-modulesmicrolocal sheaf theoryEuler classes

MSC classification

Secondary: 14F05: Sheaves, derived categories of sheaves and related constructions 35A27: Microlocal methods; methods of sheaf theory and homological algebra in PDE
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 13 , Issue 3 , July 2014 , pp. 487 - 516
Copyright
©Cambridge University Press 2013 

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