Motivic Brown–Peterson invariants of the rationals

Ormsby, Kyle M¹ ; Østvær, Paul²

¹ Department of Mathematics, MIT, Cambridge, MA 02139, USA
² Department of Mathematics, University of Oslo, 0316 Oslo, Norway, Department of Mathematics, MIT, Cambridge, MA 02139, USA

Geometry & topology, Tome 17 (2013) no. 3, pp. 1671-1706.

Résumé

Let $\mathsf{BP}\langle n\rangle$, $0\le n\le \infty$, denote the family of motivic truncated Brown–Peterson spectra over $\mathbb{Q}$. We employ a “local-to-global” philosophy in order to compute the bigraded homotopy groups of $\mathsf{BP}\langle n\rangle$. Along the way, we produce a computation of the homotopy groups of $\mathsf{BP}\langle n\rangle$ over ${\mathbb{Q}}_2$, prove a motivic Hasse principle for the spectra $\mathsf{BP}\langle n\rangle$, 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 $\mathbb{Q}$.

Détail

DOI : 10.2140/gt.2013.17.1671

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

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