From algebraic cobordism to motivic cohomology

TL;DR

This paper establishes a key equivalence between algebraic cobordism and motivic cohomology spectra after inverting the characteristic exponent, and applies this to prove spectral sequence convergence for certain motivic spectra.

Contribution

It proves the equivalence of MGL modulo the ideal generated by the a_i after inverting the characteristic exponent c.

Findings

The canonical map becomes an equivalence after inverting c.

Convergence of the Atiyah-Hirzebruch spectral sequence for Z[1/c]-local Landweber spectra.

Provides a bridge between algebraic cobordism and motivic cohomology.

Abstract

Let S be an essentially smooth scheme over a field of characteristic exponent c. Let MGL and HZ denote the algebraic cobordism spectrum and the motivic cohomology spectrum over S, respectively. We show that the canonical map MGL/(a1, a2, ...) -> HZ induced by the additive orientation of motivic cohomology becomes an equivalence after inverting c. As an application, we prove the convergence of the Atiyah-Hirzebruch spectral sequence for all Z[1/c]-local Landweber exact motivic spectra.