Algebraic cobordism and Grothendieck groups over singular schemes

Abstract

A theorem of Levine-Morel states that algebraic cobordism groups are isomorphic to (multiplicative) Grothendieck groups over smooth schemes. We extend this theorem to singular schemes. As a consequence, we provide a new proof of the singular Riemann-Roch theorem of Baum-Fulton-MacPherson and a new type of Riemann-Roch theorem with respect to pullbacks of locally complete morphisms.