Motivic Brown–Peterson invariants of the rationals

Ormsby, Kyle M; Østvær, Paul

doi:10.2140/gt.2013.17.1671

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Motivic Brown–Peterson invariants of the rationals

Kyle M Ormsby and Paul Arne Østvær

Geometry & Topology 17 (2013) 1671–1706

DOI: 10.2140/gt.2013.17.1671

Abstract

Let $BP 〈 n 〉$ , $0 \leq n \leq \infty$ , 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 $BP 〈 n 〉$ . Along the way, we produce a computation of the homotopy groups of $BP 〈 n 〉$ over $ℚ_{2}$ , prove a motivic Hasse principle for the spectra $BP 〈 n 〉$ , 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 $MGL$ over $ℚ$ .

Keywords

motivic Adams spectral sequence, algebraic cobordism, algebraic $K$–theory, Hasse principle

Mathematical Subject Classification 2010

Primary: 55T15

Secondary: 19D50, 19E15

References

Publication

Received: 27 August 2012

Revised: 25 February 2013

Accepted: 8 March 2013

Published: 21 June 2013

Proposed: Haynes Miller

Seconded: Paul Goerss, Ralph Cohen

Authors

Kyle M Ormsby
Department of Mathematics MIT Cambridge, MA 02139 USA

Paul Arne Østvær
Department of Mathematics University of Oslo 0316 Oslo Norway	Department of Mathematics MIT Cambridge, MA 02139 USA

Volume 17, issue 3 (2013)