On the structure of algebraic cobordism

In this paper we investigate the structure of algebraic cobordism of Levine-Morel as a module over the Lazard ring with the action of Landweber-Novikov and symmetric operations on it. We show that the associated graded groups of algebraic cobordism with respect to the topological filtration Ω(r)⁎(X) are unions of finitely presented L-modules of very specific structure. Namely, these submodules possess a filtration such that the corresponding factors are either free or isomorphic to cyclic modules L/I(p,n)x where deg⁡x≥pn−1p−1. As a corollary we prove the Syzygies Conjecture of Vishik on the existence of certain free L-resolutions of Ω⁎(X), and show that algebraic cobordism of a smooth surface can be described in terms of K0 together with a topological filtration.