From algebraic cobordism to 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.

Corrected a mistake in Prop. 8.2. Numbering matches published version

References in corpus (10)

• Higher Topos Theory
• A comparison of motivic and classical homotopy theories
• Comparison of cobordism theories
• Chern classes, K-theory and Landweber exactness over nonregular base schemes
• A universality theorem for Voevodsky's algebraic cobordism spectrum
• Motivic twisted K-theory
• Slices of motivic Landweber spectra
• Simplicial radditive functors
• Relations between slices and quotients of the algebraic cobordism spectrum
• The motivic Adams spectral sequence

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• Two-complete stable motivic stems over finite fields
• Localizations and completions in motivic homotopy theory
• Stable and Unstable operations in Algebraic Cobordism
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• Strong convergence in the motivic Adams spectral sequence
• Towards the dual motivic Steenrod algebra in positive characteristic
• On equivariant and motivic slices
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• Connective algebraic K-theory
• Chern classes from Morava K-theories to $p^n$-typical oriented theories
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• The slice spectral sequence for singular schemes and applications
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• Algebraic vector bundles and $p$-local A^1-homotopy theory
• Goerss--Hopkins obstruction theory for $\infty$-categories
• Cellular ${\mathbb{A}}^1$-homology and the motivic version of Matsumoto's theorem
• Algebraic elliptic cohomology and flops II: $SL$-cobordism
• Motivic stable cohomotopy and unimodular rows
• Transchromatic extensions in motivic and Real bordism
• ${\mathbb{E}}_{\infty }$ automorphisms of motivic Morava $E$-theories
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• Variations in ${\mathbb{A}}^1$ on a theme of Mohan Kumar
• Recursive formulas for the motivic Milnor basis
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• Galois equivariance and stable motivic homotopy theory
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• Squaring operations in the $RO(C_2)$-graded and real motivic Adams spectral sequences
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• Moduli stack of oriented formal groups and cellular motivic spectra over $\mathbf{C}$
• Algebraic Cobordism as a module over the Lazard ring