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Brown-Peterson spectrum

Contents

Context

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

Stable Homotopy theory

Higher algebra

Contents

Idea

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

Definition

Theorem

For each prime pp there is an unique commutative ring spectrum BPB P which is a retract of MU (p)M U_{(p)} such that the map MU (p)BPMU_{(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 pp-local complex oriented cohomology theories in that 𝔾 BP\mathbb{G}_{B P} is universal among pp-local, p-typical formal group laws.

(…)

Relation to pp-typical formal groups

BPB 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 BPBP-Steenrod algebra BP (BP)BP_\bullet(BP) is described by the Adams-Quillen theorem.

Relation to Adams-Novikov spectral sequence

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

Ext BP *(BP)(BP *,BP *). Ext_{BP_\ast(BP)}(BP_\ast, BP_\ast) \,.

recalled e.g. as Ravenel, theorem 1.4.2

As a CW spectrum

The spectrum BPB P can be constructed as a CW spectrum (cf. Priddy 1980) starting from the pp-local sphere spectrum S 0=X 0S^0 = X_0 by minimally attaching cells to X nX_n to kill π 2n+1(X n)\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.