Vector Bundles and Chern Classes

Abstract

We will construct, for any vector bundle E on a scheme X, Chern class operations $$c_{i}\left ( E \right)\cap \_:A_{k}\,X\rightarrow A_{k-i}\,X,$$ satisfying properties expected from topology. From the special case of line bundles done in § 2.5, we first construct inverse Chern classes, or Segre classes, which are then inverted to produce Chern classes. The first Chern class operations are also used to describe A * E and A * P (E) in terms of A * X.