Brown–Peterson cohomology

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

• .
• .