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From algebraic cobordism to motivic cohomology. (English) Zbl 1382.14006
Summary: Let $$S$$ be an essentially smooth scheme over a field of characteristic exponent $$c$$. We prove that there is a canonical equivalence of motivic spectra over $$S$$ $\mathrm{MGL}/(a_1,a_2,\dots)[1/c] \simeq H\mathbb{Z}[1/c]$ where $$H\mathbb{Z}$$ is the motivic cohomology spectrum, MGL is the algebraic cobordism spectrum, and the elements $$a_n$$ are generators of the Lazard ring. We discuss several applications including the computation of the slices of $$\mathbb{Z}[1/c]$$-local Landweber (see the work of M. Spitzweck [“Relations between slices and quotients of the algebraic cobordism spectrum”, Homology Homotopy Appl. 12, No. 2, 335–351 (2010; doi:10.4310/HHA.2010.v12.n2.a11); J. $$K$$-Theory 9, No. 1, 103–117 (2012; Zbl 1249.14008)] exact motivic spectra and the convergence of the associated slice spectral sequences.

##### MSC:
 14F43 Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies) 14F42 Motivic cohomology; motivic homotopy theory 55N22 Bordism and cobordism theories and formal group laws in algebraic topology 55U35 Abstract and axiomatic homotopy theory in algebraic topology
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