Equivariant Algebraic Cobordism and Equivariant Formal Group Laws

TL;DR

This paper develops an equivariant algebraic cobordism theory ^G for algebraic varieties with G-action, extending previous theories by integrating formal group laws and proving key properties and isomorphisms.

Contribution

It introduces a new equivariant algebraic cobordism theory ^G that incorporates (G, F)-formal group laws and demonstrates its fundamental properties and relations to existing theories.

Findings

^G satisfies localization and homotopy invariance.

The canonical map from L_G(F) to ^G(Spec k) is surjective.

The theory ^G is independent of the choice of F.

Abstract

We introduce an equivariant algebraic cobordism theory \Omega^G for algebraic varieties with G-action, where G is a split diagonalizable group scheme over a field k. It is done by combining the construction of the algebraic cobordism theory \Omega by F. Morel and M. Levine, with the notion of (G, F)-formal group law with respect to a complete G-universe and complete G-flag F as introduced by M. Cole, J. P. C. Greenlees and I. Kriz. In particular, we use their corresponding representing ring L_G(F) in place of the Lazard ring L. We show that localization property and homotopy invariance property hold in \Omega^G. We also prove the surjectivity of the canonical map from L_G(F) to \Omega^G(Spec k). Moreover, we give some comparison results with \Omega, the equivariant algebraic cobordism theory introduced by J. Heller and J. Malagon-Lopez, the equivariant K-theory and Tom Dieck equivariant cobordism theory (when k = C). In particular, we proved the equivariant Conner-Floyd isomorphism when char k = 0. Finally, we show that our definition of \Omega^G is independent of the choice of F.