## Homology, Homotopy and Applications

- Homology Homotopy Appl.
- Volume 10, Number 2 (2008), 211-226.

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

Ivan Panin, Konstantin Pimenov, and Oliver Röndigs

#### 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].

#### Article information

**Source**

Homology Homotopy Appl., Volume 10, Number 2 (2008), 211-226.

**Dates**

First available in Project Euclid: 1 September 2009

**Permanent link to this document**

https://projecteuclid.org/euclid.hha/1251811074

**Mathematical Reviews number (MathSciNet)**

MR2475610

**Zentralblatt MATH identifier**

1162.14013

**Subjects**

Primary: 14F05: Sheaves, derived categories of sheaves and related constructions [See also 14H60, 14J60, 18F20, 32Lxx, 46M20] 55N22: Bordism and cobordism theories, formal group laws [See also 14L05, 19L41, 57R75, 57R77, 57R85, 57R90] 55P43: Spectra with additional structure ($E_\infty$, $A_\infty$, ring spectra, etc.)

**Keywords**

Algebraic cobordism motivic ring spectra

#### Citation

Panin, Ivan; Pimenov, Konstantin; Röndigs, Oliver. A universality theorem for Voevodsky's algebraic cobordism spectrum. Homology Homotopy Appl. 10 (2008), no. 2, 211--226. https://projecteuclid.org/euclid.hha/1251811074