Algebraic Cobordism as a module over the Lazard ring

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 Levine and Morel (Algebraic Cobordism. In: Springer Monographs in Mathematics, 2007) 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 (Vishik, Symmetric operations for all primes and Steenrod operations in Algebraic Cobordism, 2013)