---
tag: S8S72AA9C8
kind: concept
name: Brown–Peterson cohomology
msc: [55N22, 55P42]
specializes: [H28PSZR7XG]
sources:
  - url: https://en.wikipedia.org/wiki/Brown–Peterson_cohomology
same_as:
  wikidata: Q17097796
  wikipedia: Brown–Peterson cohomology
---

## Summary

Brown–Peterson cohomology is a [[generalized cohomology theory|H28PSZR7XG]] attached to a prime $p$, represented by a [[ring spectrum|BAW0T68GWK]] $BP$.
After [[complex cobordism|5CZHFNY9F3]] $MU$ is localized at $p$, it splits as a wedge of suspensions of copies of $BP$.
So $BP$ keeps all the $p$-local information of $MU$ in a much smaller object, with coefficient ring $\pi_* BP = \Z_{(p)}[v_1, v_2, \ldots]$ and $\abs{v_i} = 2(p^i - 1)$.
It is the basic tool of the Adams–Novikov spectral sequence and of chromatic homotopy theory.

## Explanations

### L1 human

Complex cobordism is a very powerful cohomology theory, but it is large and hard to compute with.
When one only cares about one prime $p$, most of it repeats the same pattern many times.
Brown–Peterson cohomology is one copy of this pattern.
Its coefficients are polynomials in variables $v_1, v_2, \ldots$, and for $p = 2$ these sit in degrees $2, 6, 14, 30, \ldots$, that is $2(2^i - 1)$.
The variable $v_n$ is the source of the "height $n$" layer studied in chromatic homotopy theory.