Motivic cohomology, localized Chern classes, and local terms

Abstract

Let \({c : C \rightarrow X \times X}\) be a correspondence with C and X quasi-projective schemes over an algebraically closed field k. We show that if \({u_\ell : c_1^*\mathbb{Q}_\ell \rightarrow c_2^!\mathbb{Q}_\ell}\) is an action defined by the localized Chern classes of a c2-perfect complex of vector bundles on C, where l is a prime invertible in k, then the local terms of ul are given by the class of an algebraic cycle independent of l. We also prove some related results for quasi-finite correspondences. The proofs are based on the work of Cisinski and Deglise on triangulated categories of motives.