Derived algebraic cobordism

TL;DR

This paper develops a new cohomology theory based on derived schemes, establishing its equivalence with algebraic cobordism in characteristic zero and providing new generators and relations.

Contribution

It introduces a derived algebraic cobordism theory with characteristic-independent pull-backs, aligning with Levine and Morel's algebraic cobordism in characteristic zero.

Findings

The new theory agrees with classical algebraic cobordism in characteristic zero.

It provides a characteristic-independent framework for pull-backs.

New generators and relations for algebraic cobordism are established.

Abstract

We construct a cohomology theory using quasi-smooth derived schemes as generators and an analogue of the bordism relation using derived fibre products as relations. This theory has pull-backs along all morphisms between smooth schemes independent of any characteristic assumptions. We prove that in characteristic zero, the resulting theory agrees with algebraic cobordism as defined by Levine and Morel. We thus obtain a new set of generators and relations for algebraic cobordism.