Equivariant Intersection Theory
نویسندگان
چکیده
The purpose of this paper is to develop an equivariant intersection theory for actions of linear algebraic groups on algebraic schemes. The theory is based on our construction of equivariant Chow groups. They are algebraic analogues of equivariant cohomology groups which have all the functorial properties of ordinary Chow groups. In addition, they enjoy many of the properties of equivariant cohomology. Previous work ([Br], [Gi], [Vi]) defined equivariant Chow groups using only invariant cycles on X. However, there are not enough invariant cycles on X to define equivariant Chow groups with nice properties, such as being a homotopy invariant, or having an intersection product whenX is smooth (see Section 3.5). In the definition of this paper, which is modeled after Borel’s definition of equivariant cohomology, an equivariant class is represented by an invariant cycle on X ×V , where V is a representation of G. By enlarging the definition of equivariant cycle, we obtain a rich theory, which is closely related to other aspects of group actions on schemes. After establishing the basic properties of equivariant Chow groups, this paper is mainly devoted to the relationship between equivariant Chow groups and Chow groups of quotient schemes and stacks. If G is a linear algebraic group acting on a scheme X, denote by Ai (X) the i-th equivariant Chow group of X. Suppose that G acts properly (hence with finite stabilizers) on
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