Equivariant Algebraic Cobordism and Equivariant Formal Group Laws

Original · EN

We introduce an equivariant algebraic cobordism theory Ωᵍ 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 Ωby 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 LG(F) in place of the Lazard ring L. We show that localization property and homotopy invariance property hold in Ωᵍ. We also prove the surjectivity of the canonical map from LG(F) to Ωᵍ(Spec k). Moreover, we give some comparison results with Ω, 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 Ωᵍ is independent of the choice of F.