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motivic Thom spectrum

Contents

Idea

The motivic Thom spectrum MGLMGL 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 MUMU, Voevodsky defined the algebraic cobordism spectrum MGL SMGL_S in the stable motivic homotopy category by the formula

MGL S=colim nΩ P 1 nTh(V n), MGL_S = colim_{n\to\infty} \Omega^n_{\mathbf{P}^1} Th(V_n),

where SS is the base scheme, Th(V n)Th(V_n) is the (infinite suspension of) the Tho space? of the tautological vector bundle V nV_n over the infinite Grassmannian of nn-planes Gr(n)Gr(n). More precisely, Gr(n)Gr(n) is defined as the colimit of the smooth SS-schemes Gr(n,k)Gr(n,k) as kk\to\infty, and V nV_n is similarly the colimit of the tautological vector bundles over Gr(n,k)Gr(n,k). The notation Ω P 1 n\Omega^n_{\mathbf{P}^1} denotes the nnth P 1\mathbf{P}^1-loop space.

Properties

Like the motivic Eilenberg-Mac Lane spectrum? HZH\mathbf{Z} and the algebraic K-theory spectrum KGLKGL, MGLMGL is an E E_\infty-motivic ring spectrum.

The spectrum MGLMGL 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.