Brown–Peterson cohomology

In mathematics, Brown–Peterson cohomology is a generalized cohomology theory introduced by, depending on a choice of prime p. It is described in detail by. 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 $$\Z_{(p)}$$ on generators $$v_n$$ in degrees $$2(p^n-1) $$ for $$n\ge 1$$.

$$\text{BP}_*(\text{BP})$$ 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.