Caractères de Chern, traces équivariantes et géométrie algébrique dérivée

Abstract

The purpose of this work is to provide details about the construction of the Chern character for categorical sheaves mentioned in our previous work "Chern character, loop spaces and derived algebraic geometry". For this, we introduce and study the notion of rigid symmetric monoidal \infty-category. We show how trace maps can be constructed in this higher categorical setting, and using a recent work of Hopkins-Lurie we prove the existence of a "cyclic trace", which is the main ingredient in the construction of the Chern character. Our Chern character is then constructed for any pair (T,A), consisting of a \infty-topos T and a stack of rigid symmetric monoidal \infty-categories A on T. We propose two main applications of this construction. First of all, we show how to recover the Chern character of perfect complexes on schemes, with values in cyclic homology, and show how it can be extended to an interesting new Chern character for Artin stacks. Our second application provides invariants of famillies of dg-categories. A consequence of the existence of these invariants is the construction of a Gauss-Manin connexion on the cyclic homology complex of such a familly, generalizing previous constructions by Getzler and of Dolgushev-Tamarkin-Tsygan. Another consequence is the construction of the "character sheaf" associated to a representation of an algebraic group into a dg-category, which is a categorification of the character function of a linear representation. Finally, for a familly of saturated dg-categories we construct a "secondary Chern character", taking values in a new cohomology theory called "secondary cyclic homology".