Groupes de Chow orientés

Abstract

In this work we study the oriented Chow groups. These groups were defined by J. Barge et F. Morel in order to understand when a projective module P of top rank over a ring A has a free factor of rank one, i.e is isomorphic to Q ⊕ A. We show first that these groups satisfy the same functorial properties as the classical Chow groups. Then we define for each locally free OX-module E (of constant rank n) over a regular scheme X an oriented top Chern class c n(E) which is a refinement of the usual top Chern class cn(E). The oriented class satisfies also good fonctorial properties. In particular, we get c n(P) = 0 if P is a projective module of rank n over a regular ring A of dimension n such that P ≃ Q ⊕ A. In further work we compute the top oriented Chow group of a regular ring A of dimension 2 and the top oriented Chow group of a regular R-algebra A of finite dimension. For such A, we get that if P is a projective module of rank equal to the dimension of the ring then c n(P) = 0 if and only if P ≃ Q ⊕ A. Finally, we examine the links between the oriented Chow groups and the Euler class groups defined by S. Bhatwadekar and R. Sridharan ([BS1]).