nLab Brown-Peterson spectrum

Contents

Context

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.

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 $\pi_{2n+1}(X_n)$.

References

The original article is

An alternate construction was noted by Priddy

A textbook account is in section 4 (pdf) of

The truncated version is discussed in

Last revised on March 22, 2017 at 01:15:52. See the history of this page for a list of all contributions to it.