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Publication# 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]).

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Related concepts (42)

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Chern class

In mathematics, in particular in algebraic topology, differential geometry and algebraic geometry, the Chern classes are characteristic classes associated with complex vector bundles. They have since become fundamental concepts in many branches of mathematics and physics, such as string theory, Chern–Simons theory, knot theory, Gromov-Witten invariants. Chern classes were introduced by . Chern classes are characteristic classes. They are topological invariants associated with vector bundles on a smooth manifold.

Intersection theory

In mathematics, intersection theory is one of the main branches of algebraic geometry, where it gives information about the intersection of two subvarieties of a given variety. The theory for varieties is older, with roots in Bézout's theorem on curves and elimination theory. On the other hand, the topological theory more quickly reached a definitive form. There is yet an ongoing development of intersection theory. Currently the main focus is on: virtual fundamental cycles, quantum intersection rings, Gromov-Witten theory and the extension of intersection theory from schemes to stacks.

Regular sequence

In commutative algebra, a regular sequence is a sequence of elements of a commutative ring which are as independent as possible, in a precise sense. This is the algebraic analogue of the geometric notion of a complete intersection. For a commutative ring R and an R-module M, an element r in R is called a non-zero-divisor on M if r m = 0 implies m = 0 for m in M. An M-regular sequence is a sequence r1, ..., rd in R such that ri is a not a zero-divisor on M/(r1, ..., ri-1)M for i = 1, ..., d.

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