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.
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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, 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 vn 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 ti in BP2(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.