Brown-Peterson spectra in stable A^1-homotopy theory

TL;DR

This paper characterizes morphisms from the algebraic cobordism spectrum to oriented spectra in stable A^1-homotopy theory, and constructs a motivic BP-spectrum at each prime p, extending Quillen's classical topological results.

Contribution

It provides a classification of ring spectrum morphisms via formal group laws and constructs motivic BP-spectra analogous to classical topological spectra.

Findings

Characterization of morphisms from MGL to oriented spectra via formal group laws.

Construction of motivic Quillen idempotents at each prime p.

Definition of the motivic BP-spectrum in the stable A^1-homotopy setting.

Abstract

We characterize ring spectra morphisms from the algebraic cobordism spectrum $\QTR{Bbb}{MGL}$ (\QCITE{cite}{}{Vo1}) to an oriented spectrum $\QTR{Bbb}{E}$ (in the sense of Morel \QCITE{cite}{}{Mo}) via formal group laws on the ''topological'' subring $E^{*}=\oplus_iE^{2i,i}$ of $E^{**}$. This result is then used to construct for any prime $p$ a motivic Quillen idempotent on $\QTR{Bbb}{MGL}_{(p)}$. This defines the $BP$-spectrum associated to the prime $p$ as in Quillen's \QCITE{cite}{}{Q1} for the complex-oriented topological case.