Brown-Peterson cohomology from Morava E-theory

TL;DR

This paper demonstrates that the p-completed Brown-Peterson spectrum can be embedded into a product of Morava E-theory spectra, extending known results from spaces to spectra and analyzing group cohomology structures.

Contribution

It proves the retract property of the p-completed Brown-Peterson spectrum within Morava E-theory spectra and extends cohomological results to a broader spectral context.

Findings

Brown-Peterson spectrum is a retract of Morava E-theory spectra

Good groups are characterized by Brown-Peterson cohomology

Rational factorizations lift to integral cohomology with bounded torsion

Abstract

We prove that the $p$-completed Brown-Peterson spectrum is a retract of a product of Morava $E$-theory spectra. As a consequence, we generalize results of Ravenel-Wilson-Yagita and Kashiwabara from spaces to spectra and deduce that the notion of good group is determined by Brown-Peterson cohomology. Furthermore, we show that rational factorizations of the Morava $E$-theory of certain finite groups hold integrally up to bounded torsion with height-independent exponent, thereby lifting these factorizations to the rationalized Brown-Peterson cohomology of such groups.