The motivic cobordism for group actions

TL;DR

This paper develops an equivariant version of motivic cobordism for algebraic group actions, establishing its properties and applications to classifying spaces and quotient stacks in algebraic geometry.

Contribution

It introduces an equivariant motivic cobordism theory for group actions, extending motivic cobordism to smooth quotient stacks and analyzing its properties and applications.

Findings

Defined equivariant motivic cobordism as an oriented cohomology theory.

Established a natural transformation to equivariant motivic cohomology.

Applied the theory to study cobordism rings of classifying spaces and cycle class maps.

Abstract

For a linear algebraic group $G$ over a field $k$, we define an equivariant version of the Voevodsky's motivic cobordism $MGL$. We show that this is an oriented cohomology theory with localization sequence on the category of smooth $G$-schemes and there is a natural transformation from this functor to the functor of equivariant motivic cohomology. We give several applications. In particular, we use this equivariant motivic cobordism to study the cobordism ring of the classifying spaces and the cycle class maps from the algebraic to the singular cohomology of such spaces. This theory of motivic cobordism allows us to define the theory of motivic cobordism on the category of all smooth quotient stacks.