Volume 17, issue 3 (2013)

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Motivic Brown–Peterson invariants of the rationals

Kyle M Ormsby and Paul Arne Østvær

Geometry & Topology 17 (2013) 1671–1706
Abstract

Let $\mathsf{BP}〈n〉$, $0\le n\le \infty$, denote the family of motivic truncated Brown–Peterson spectra over $ℚ$. We employ a “local-to-global” philosophy in order to compute the bigraded homotopy groups of $\mathsf{BP}〈n〉$. Along the way, we produce a computation of the homotopy groups of $\mathsf{BP}〈n〉$ over ${ℚ}_{2}$, prove a motivic Hasse principle for the spectra $\mathsf{BP}〈n〉$, and reprove several classical and recent theorems about the $K$–theory of particular fields in a streamlined fashion. We also compute the bigraded homotopy groups of the 2–complete algebraic cobordism spectrum $\mathsf{MGL}$ over $ℚ$.

Keywords
motivic Adams spectral sequence, algebraic cobordism, algebraic $K$–theory, Hasse principle
Mathematical Subject Classification 2010
Primary: 55T15
Secondary: 19D50, 19E15