Algebraic Cobordism as a module over the Lazard ring

TL;DR

This paper investigates the structure of the algebraic cobordism ring as a module over the Lazard ring, revealing relations in positive codimensions and computing the cobordism ring for curves using symmetric operations.

Contribution

It extends previous results by showing the presence of relations in positive codimensions and proves a graded version of the structure of algebraic cobordism as a Lazard module.

Findings

Algebraic cobordism ring has relations in positive codimensions.

Computed the cobordism ring of a curve.

Extended the generator results to include relations in positive codimensions.

Abstract

In this paper we study the structure of the Algebraic Cobordism ring of a variety as a module over the Lazard ring, and show that it has relations in positive codimensions. We actually prove the stronger graded version. This extends the result of M.Levine-F.Morel claiming that this module has generators in non-negative codimensions. As an application we compute the Algebraic Cobordism ring of a curve. The main tool is Symmetric Operations in Algebraic Cobordism.