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 .
Complex cobordism and Quillen's idempotent Brown–Peterson cohomology is a summand of MU(), which is complex cobordism MU localized at a prime . In fact MU() is a wedge product of suspensions of .
For each prime , Daniel Quillen showed there is a unique idempotent map of ring spectra ε from MUQ() to itself, with the property that is [CP] if is a power of , and otherwise. The spectrum is the image of this idempotent ε.
Structure of The coefficient ring is a polynomial algebra over on generators in degrees for .
is isomorphic to the polynomial ring over with generators in of degrees .
The cohomology of the Hopf algebroid is the initial term of the Adams–Novikov spectral sequence for calculating -local homotopy groups of spheres.
is the universal example of a complex oriented cohomology theory whose associated formal group law is -typical.
See also List of cohomology theories#Brown–Peterson cohomology
References
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