# Brown–Peterson cohomology

In mathematics, Brown–Peterson cohomology is a generalized cohomology theory introduced by Edgar H. Brown and Franklin P. Peterson (1966), depending on a choice of prime p. It is described in detail by Douglas Ravenel (2003, Chapter 4). Its representing spectrum is denoted by BP.

## Complex cobordism and Quillen's idempotent

Brown–Peterson cohomology BP is a summand of MU(p), which is complex cobordism MU localized at a prime p. In fact MU(p) is a wedge product of suspensions of BP.

For each prime p, Daniel Quillen showed there is a unique idempotent map of ring spectra ε from MUQ(p) to itself, with the property that ε([CPn]) is [CPn] if n+1 is a power of p, and 0 otherwise. The spectrum BP is the image of this idempotent ε.

## Structure of BP

The coefficient ring $\pi _{*}({\text{BP}})$ is a polynomial algebra over $\mathbb {Z} _{(p)}$ on generators $v_{n}$ in degrees $2(p^{n}-1)$ for $n\geq 1$ .

${\text{BP}}_{*}({\text{BP}})$ is isomorphic to the polynomial ring $\pi _{*}({\text{BP}})[t_{1},t_{2},\ldots ]$ over $\pi _{*}({\text{BP}})$ with generators $t_{i}$ in ${\text{BP}}_{2(p^{i}-1)}({\text{BP}})$ of degrees $2(p^{i}-1)$ .

The cohomology of the Hopf algebroid $(\pi _{*}({\text{BP}}),{\text{BP}}_{*}({\text{BP}}))$ is the initial term of the Adams–Novikov spectral sequence for calculating p-local homotopy groups of spheres.

BP is the universal example of a complex oriented cohomology theory whose associated formal group law is p-typical.