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

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

The cohomology of the Hopf algebroid ${\displaystyle (\pi _{*}({\text{BP}}),{\text{BP}}_{*}({\text{BP}}))}$ 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.