Derived Algebraic Cobordism

TL;DR

This paper develops a new theory of bivariant cobordism for derived schemes, generalizing previous algebraic bordism and connecting it to motivic cohomology, with applications to vector bundles and Chow cohomology.

Contribution

It introduces a comprehensive bivariant cobordism theory for derived schemes, extending algebraic bordism and relating it to motivic cohomology and K-theory.

Findings

Bivariant cobordism satisfies the projective bundle formula.

Constructs cobordism Chern classes for vector bundles.

Proposes a candidate for Chow cohomology of schemes.

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.