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doctoral thesis

Groupes de Chow orientés

Fasel, Jean  
2005

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]).

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