Equivariant Cobordism of Schemes

Abstract

Let k be a field of characteristic zero. For a linear alge- braic group G over k acting on a scheme X, we define the equivariant algebraic cobordism of X and establish its basic properties. We ex- plicitly describe the relation of equivariant cobordism with equivariant Chow groups, K-groups and complex cobordism. We show that the rational equivariant cobordism of a G-scheme can be expressed as the Weyl group invariants of the equivariant cobordism for the action of a maximal torus of G. As applications, we show that the rational algebraic cobordism of the classifying space of a complex linear algebraic group is isomorphic to its complex cobordism.