Derived Algebraic Cobordism

Abstract

We construct and study a theory of bivariant cobordism of derived schemes. Our theory provides a vast generalization of the algebraic bordism theory of characteristic 0 algebraic schemes, constructed earlier by Levine and Morel, and a (partial) non-$\Ab^1$-invariant refinement of the motivic cohomology theory $MGL$ in Morel--Voevodsky's stable motivic homotopy theory. Our main result is that bivariant cobordism satisfies the projective bundle formula. As applications of this, we construct cobordism Chern classes of vector bundles, and establish a strong connection between the cobordism cohomology rings and the Grothendieck ring of vector bundles. We also provide several universal properties for our theory. Additionally, our algebraic cobordism is also used to construct a candidate for the elusive theory of Chow cohomology of schemes.