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 is a polynomial algebra over \Z_{(p)} on generators v_n in degrees 2(p^n-1) for n\ge 1. is isomorphic to the polynomial ring over with generators t_i in of degrees 2 (p^i-1). The cohomology of the Hopf algebroid 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.