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

Complex cobordism

In mathematics, complex cobordism is a generalized cohomology theory related to cobordism of manifolds. Its spectrum is denoted by MU. It is an exceptionally powerful cohomology theory, but can be quite hard to compute, so often instead of using it directly one uses some slightly weaker theories...

 MU localized at a prime p. In fact MU_(p) is a wedge product of suspensions of BP.

For each prime p, 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 π_*(BP) is a polynomial algebra over Z_(p) on generators v_n of dimension 2(pⁿ − 1) for n ≥ 1.

BP_*(BP) is isomorphic to the polynomial ring π_*(BP)[t₁, t₂, ...] over π_*(BP) with generators t_i in BP_2(pⁱ−1)(BP) of degrees 2(pⁱ−1).

The cohomology of the Hopf algebroid (π_*(BP), BP_*(BP)) is the initial term of the Adams–Novikov spectral sequence for calculating p-local homotopy groups of spheres

Homotopy groups of spheres

In the mathematical field of algebraic topology, the homotopy groups of spheres describe how spheres of various dimensions can wrap around each other. They are examples of topological invariants, which reflect, in algebraic terms, the structure of spheres viewed as topological spaces, forgetting...

.

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