A universality theorem for Voevodsky's algebraic cobordism spectrum

Abstract

An algebraic version of a theorem of Quillen is proved. More precisely, for a regular Noetherian scheme $S$ of finite Krull dimension, we consider the motivic stable homotopy category SH($S$) of $\mathbb{P}^1$-spectra, equipped with the symmetric monoidal structure described in [7]. The algebraic cobordism $\mathbb{P}^1$-spectrum MGL is considered as a commutative monoid equipped with a canonical orientation$th^{MGL} \in \rm{MGL}^{2,1}(\rm{Th}(\mathcal{O}(-1)))$. For a commutative monoid $E$ in the category SH($S$), it is proved that the assignment $\varphi \longmapsto \varphi(th^{\rm{MGL}})$ identifies the set of monoid homomorphisms $\varphi : \rm {MGL} \longmapsto E$ in the motivic stable homotopy category SH($S$) with the set of all orientations of $E$. This result generalizes a result of G. Vezzosi in [12].