nLab motivic Thom spectrum

Idea

The motivic Thom spectrum $MGL$ is the motivic spectrum in the stable motivic homotopy category representing algebraic cobordism. It is the algebraic or motivic analogue of the Thom spectrum in the stable homotopy category which represents complex cobordism.

Definition

In analogy with the complex cobordism spectrum $MU$, Voevodsky defined the algebraic cobordism spectrum $MGL_S$ in the stable motivic homotopy category by the formula

$$MGL_S = colim_{n\to\infty} \Omega^n_{\mathbf{P}^1} Th(V_n),$$

where $S$ is the base scheme, $Th(V_n)$ is the (infinite suspension of) the Thom space of the tautological vector bundle $V_n$ over the infinite Grassmannian of $n$-planes $Gr(n)$. More precisely, $Gr(n)$ is defined as the colimit of the smooth $S$-schemes $Gr(n,k)$ as $k\to\infty$, and $V_n$ is similarly the colimit of the tautological vector bundles over $Gr(n,k)$. The notation $\Omega^n_{\mathbf{P}^1}$ denotes the $n$th $\mathbf{P}^1$-loop space.

Properties

Like the motivic Eilenberg-Mac Lane spectrum? $H\mathbf{Z}$ and the algebraic K-theory spectrum $KGL$, $MGL$ is an $E_\infty$-motivic ring spectrum.

The spectrum $MGL$ was used by Voevodsky in his proof of the Bloch-Kato conjecture. It is also the starting point of chromatic motivic homotopy theory.

Related concepts

• algebraic cobordism
• complex cobordism
• Thom spectrum
• motivic homotopy theory
• Snaith's theorem
• motivic Eilenberg-Mac Lane spectrum?
• algebraic K-theory

References

• Vladimir Voevodsky, $\mathbf{A}^1$-Homotopy Theory, Doc. Math., Extra Vol. ICM 1998(I), 417-442, web.

• Ivan Panin, K. Pimenov, Oliver Röndigs, A universality theorem for Voevodsky’s algebraic cobordism spectrum, Homology, Homotopy and Applications, 2008, 10(2), 211-226, arXiv.

• Ivan Panin, K. Pimenov, Oliver Röndigs, On the relation of Voevodsky’s algebraic cobordism to Quillen’s K-theory, Inventiones mathematicae, 2009, 175(2), 435-451, DOI, arXiv.

• Markus Spitzweck, Algebraic cobordism in mixed characteristic, arXiv.

• Marc Hoyois, From algebraic cobordism to motivic cohomology, pdf, arXiv.

• David Gepner, Victor Snaith, On the motivic spectra representing algebraic cobordism and algebraic K-theory, Documenta Math. 2008 (arXiv:0712.2817).

A discussion on MathOverflow:

• Interdependence between A^1-homotopy theory and algebraic cobordism, MO/36659.