# nLab Brown-Peterson spectrum

Contents

cohomology

### Theorems

#### Stable Homotopy theory

stable homotopy theory

Introduction

# Contents

#### Higher algebra

higher algebra

universal algebra

# Contents

## Idea

The localization of complex cobordism cohomology theory $MU$ at a prime $p$, hence the p-localization $MU_{(p)}$, decomposes as a direct sum. The direct summands are the Brown-Peterson spectra.

## Definition

###### Theorem

For each prime $p$ there is an unique commutative ring spectrum $B P$ which is a retract of $M U_{(p)}$ such that the map $MU_{(p)} \to B P$ is multiplicative and such that

1. (…)

2. (…)

3. (…)

Due to (Brown-Peterson 66), recalled as (Ravenel, theorem, 4.1.12).

## Properties

### Universal characterization

(…)

The formal group law of Brown-Peterson cohomology theory is universal for $p$-local complex oriented cohomology theories in that $\mathbb{G}_{B P}$ is universal among $p$-local, p-typical formal group laws.

(…)

### Relation to $p$-typical formal groups

$B P$ is related to p-typical formal groups as MU is to formal groups.

### Hopf algebroid structure on dual BP-Steenrod algebra

The structure of Hopf algebroid over a commutative base on the dual $BP$-Steenrod algebra $BP_\bullet(BP)$ is described by the Adams-Quillen theorem.

### Relation to Adams-Novikov spectral sequence

The $p$-component of the $E^2$-term of the Adams-Novikov spectral sequence for the sphere spectrum, hence the one converging to the stable homotopy groups of spheres $\pi_\ast(\mathbb{S})$ is

$Ext_{BP_\ast(BP)}(BP_\ast, BP_\ast) \,.$

recalled e.g. as Ravenel, theorem 1.4.2

### As a CW spectrum

The spectrum $B P$ can be constructed as a CW spectrum (cf. Priddy 1980) starting from the $p$-local sphere spectrum $S^0 = X_0$ by minimally attaching cells to $X_n$ to kill $\pi_{2n+1}(X_n)$.

The original article is

An alternate construction was noted by Priddy

A textbook accounts:

The truncated version is discussed in

On the Hopf invariant and e-invariant in BP-theory:

• Yasumasa Hirashima, On the $BP_\ast$-Hopf invariant, Osaka J. Math., Volume 12, Number 1 (1975), 187-196 (euclid:ojm/1200757733)

• Martin Bendersky, The BP Hopf Invariant, American Journal of Mathematics, Vol. 108, No. 5 (Oct., 1986) (jstor:2374595)