# Contents

## 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 Tho 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.

## References

A discussion on MathOverflow:

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

Last revised on July 20, 2017 at 04:33:22. See the history of this page for a list of all contributions to it.