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 Ravenel (2003, Chapter 4). Its representing spectrum is denoted by BP.

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, 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.

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.

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

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