Equivariant cobordism of schemes

Abstract

Let $k$ be a field of characteristic zero. For a linear algebraic group $G$ over $k$ acting on a scheme $X$, we define the equivariant algebraic cobordism of $X$ and establish its basic properties. We explicitly 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.