nLab Brown-Peterson spectrum





Special and general types

Special notions


Extra structure



Stable Homotopy theory

Higher algebra



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.



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).


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).


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 *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)

Last revised on November 30, 2020 at 17:55:30. See the history of this page for a list of all contributions to it.