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# Homology, Homotopy and Applications

## Volume 10 (2008)

### Number 2

### A universality theorem for Voevodsky’s algebraic cobordism spectrum

Pages: 212 – 226

DOI: https://dx.doi.org/10.4310/HHA.2008.v10.n2.a11

#### Authors

#### 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 $\mathrm{SH}(S)$ of $\mathbf{P}^1$-spectra, equipped with the symmetric monoidal structure described in [7]. The algebraic cobordism $\mathbf{P}^1$-spectrum $\mathrm{MGL}$ is considered as a commutative monoid equipped with a canonical orientation $th^{\mathrm{MGL}} \in \mathrm{MGL}^{2,1}(\mathrm{Th}(\mathcal O(-1)))$. For a commutative monoid $E$ in the category $\mathrm{SH}(S)$, it is proved that the assignment $\varphi \mapsto \varphi(th^{\mathrm{MGL}})$ identifies the set of monoid homomorphisms $\varphi\colon \mathrm{MGL} \to E$ in the motivic stable homotopy category $\mathrm{SH}(S)$ with the set of all orientations of $E$. This result generalizes a result of G. Vezzosi in [12].

#### Keywords

algebraic cobordism; motivic ring spectra

#### 2010 Mathematics Subject Classification

14F05, 55N22, 55P43

Published 1 January 2008